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UBC Theses and Dissertations

Random and deterministic (nonrandom) aspects of athletic behavior with special reference to National… MacDonald, Neil William 1990

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Random a n d D e t e r m i n i s t i c (Nonrandom) A s p e c t s o f A t h l e t i c B e h a v i o r w i t h S p e c i a l R e f e r e n c e t o N a t i o n a l L e a g u e H o c k e y by N e i l W i l l i a m M a c d o n a l d B . A . , M . A . T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 1958 , 1960 M . S . T h e U n i v e r s i t y o f O r e g o n 1967 P h . D . T h e U n i v e r s i t y o f M i n n e s o t a 1966 A T h e s i s S u b m i t t e d i n P a r t i a l F u l f i l l m e n t o f t h e R e q u i r e m e n t f o r t h e D e g r e e o f M a s t e r o f P h y s i c a l E d u c a t i o n i n T h e F a c u l t y o f G r a d u a t e S t u d i e s S c h o o l o f P h y s i c a l E d u c a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a D e c e m b e r 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T V a r i o u s p a r t s o f t h e q u e s t i o n c o n c e r n i n g how r a n d o m a n d d e t e r m i n i s t i c a t t r i b u t e s i n t e r t w i n e d u r i n g t h e c o u r s e o f a t h l e t i c c o n t e s t s h a v e b e e n e x p l o r e d by r e s e a r c h e r s . T h i s s t u d y a t t e m p t e d t o e x t e n d t h e r e s e a r c h d a t a b a s e a n d f o r m u l a t e t h e i n i t i a l p o s t u l a t e s f o r a m o d e l t o d e s c r i b e t h e r a n d o m / d e t e r m i n i s t i c i n t e r a c t i o n . T h e 1 9 8 8 - 8 9 N a t i o n a l H o c k e y L e a g u e s e a s o n was t h e p r i m a r y f o c u s o f a t t e n t i o n . S u p p l e m e n t a l e x a m i n a t i o n was made o f t h e 1 9 3 7 - 3 8 a n d 1 9 4 6 - 4 7 N H L , t h e l a s t 50% o f t h e 1 9 8 8 - 8 9 N a t i o n a l B a s k e t b a l l A s s o c i a t i o n s e a s o n a n d t h e 1 9 8 7 - 8 8 F o o t b a l l A s s o c i a t i o n E n g l i s h F i r s t D i v i s i o n s e a s o n s . T h e d a t a o v e r w h e l m i n g l y s u p p o r t e d e a r l i e r s t u d i e s w h i c h a r g u e d t h a t m a j o r o u t c o m e s ( w i n s / l o s s e s , g o a l s , s h o t s ) f o l l o w e d a r a n d o m s e q u e n c e . T h e a x i o m a t i c m o d e l d e v e l o p e d a r g u e d t h a t t h e r a n d o m p a t t e r n o f o u t c o m e s i s q u i t e p e r v a s i v e ( w i n s / l o s s e s , s h o t s a n d g o a l s f o r , a g a i n s t o r c o m b i n e d a r e d i s t r i b u t e d r a n d o m l y w h e t h e r home, away o r t o t a l games a r e e x a m i n e d ) . T h e p a t t e r n o f o u t c o m e s ( w i n / l o s s e s , g o a l s , s h o t s ) i s r e l a t i v e l y i n d e p e n d e n t o f t h e s i z e o f t h e u n i t o f m e a s u r e m e n t : r a n d o m p a t t e r n s h e l d w h e t h e r one p e r i o d , two p e r i o d , t h r e e p e r i o d games o r f o u r - g a m e s e t s were e x a m i n e d . C o n d i t i o n a l p r o b a b i l i t y t e s t s showed g a m e - t o - g a m e o u t c o m e s w e r e i n d e p e n d e n t (a w i n was no more l i k e l y t o be f o l l o w e d by a w i n t h a n by a l o s s ) . T h e p a t t e r n o f o u t c o m e s ( g o a l s ) i s d e p e n d e n t o n how t h e d a t a i s e x a m i n e d . I f a l l 21 t e a m ' s g o a l s a r e p l o t t e d t i m e - w i s e , g o a l s a r e d i s t r i b u t e d u n i f o r m l y m i n u t e b y m i n u t e ( e x c e p t f o r t h e l a s t m i n u t e o f p l a y ) . I f g o a l s o r s h o t s p e r game (or p e r i o d ) a r e t a l l i e d f o r home, away o r b o t h t e a m s , t h e r e s u l t a n t f r e q u e n c y d i s t r i b u t i o n w i l l a p p r o x i m a t e t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n . H o w e v e r , i f t h e t i m e - s p a c e s b e t w e e n g o a l s a r e t a l l i e d , a g e o m e t r i c d i s t r i b u t i o n w i l l e m e r g e . D e t e r m i n i s t i c e f f e c t s were d e m o n s t r a t e d when a r t i f i c i a l s e a s o n o u t c o m e s b a s e d o n f i r s t , s e c o n d o r t h i r d p e r i o d o n l y s e a s o n s were f o u n d t o c o r r e l a t e f a v o r a b l y w i t h r e a l s e a s o n o u t c o m e s ( w i n s , l o s s e s , p o i n t s , g o a l s f o r , g o a l s a g a i n s t ) . F i n a l l y , c o m p a r i s o n o f h o c k e y , b a s k e t b a l l a n d s o c c e r o u t c o m e s s u g g e s t e d t h a t u p s e t " r a t e s may v a r y f r o m one s p o r t t o a n o t h e r . i v T A B L E OF CONTENTS A b s t r a c t i i L i s t o f T a b l e s v L i s t o f A p p e n d i x T a b l e s v i L i s t o f F i g u r e s i x A c k n o w l e d g m e n t s x 1. I n t r o d u c t i o n 1 P u r p o s e 1 2 . L i t e r a t u r e R e v i e w 2 S t u d i e s S h o w i n g Randomness 2 U n i t o f M e a s u r e m e n t S t u d i e s 7 M o n t e C a r l o S t u d i e s 8 C h a n c e D i s t r i b u t i o n o f S c o r e s 9 G o o d n e s s - o f - f i t D i s t r i b u t i o n S t u d i e s 10 3 . P r o c e d u r e 11 D a t a S o u r c e s 11 A c c u r a c y o f M e a s u r e m e n t 13 B a s i c A s s u m p t i o n s 13 4 . P o s t u l a t e T e s t i n g 15 D i s t r i b u t i o n T e s t i n g 15 Runs T e s t i n g 37 C o n d i t i o n a l P r o b a b i l i t i e s 42 C o r r e l a t i o n a l D a t a 44 Game t o Game C o r r e l a t i o n s 48 Raw a n d B i n a r y C o r r e l a t i o n s 52 S t a t i o n a r i t y T e s t s 55 A r t i f i c i a l a n d R e a l S t a n d i n g s 58 U p s e t s 59 D e s c r i p t i v e D a t a 65 5 . D i s c u s s i o n 67 C o m p a r i s o n s w i t h O t h e r S t u d i e s 67 M o d e l B u i l d i n g 70 F i n a l Comments 73 A p p e n d i x 77 R e f e r e n c e s 184 V L I S T OF T A B L E S T a b l e Page 1 . Means a n d V a r i a n c e s f o r S h o t s 25 2 . C h i - s q u a r e C o m p a r i s o n o f S t r e a k F r e q u e n c i e s w i t h G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s . . . . 32 3 . Means a n d V a r i a n c e s f o r S t r e a k s 32 4 . E m p i r i c a l P r o b a b i l i t i e s o f R e p e a t i n g a W i n o r L o s s C o n d i t i o n e d o n t h e Outcome o f P r e v i o u s Game(s ) f o r a l l Teams C o m b i n e d 44 5 . Range o f Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r G o a l s a n d S h o t s ( P 1 G 1 . . P 2 G 1 . . P 3 . . G n ) . 53 6 . Range o f I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s a n d G o a l s F o r a n d A g a i n s t 54 7 . U p s e t R a t e i n V a r i o u s N a t i o n a l H o c k e y L e a g u e Y e a r s . . 61 8. C h i - s q u a r e C o m p a r i s o n o f U p s e t R a t e s i n H o c k e y a n d S o c c e r 63 9 . P r e - G a m e a n d S e a s o n U p s e t R a t e s f o r H o c k e y , S o c c e r a n d B a s k e t b a l l 64 1 0 . C h i - s q u a r e C o m p a r i s o n o f P r e - G a m e a n d S e a s o n U p s e t R a t e s b e t w e e n H o c k e y a n d S o c c e r 66 1 1 . C h i - s q u a r e C o m p a r i s o n o f P r e - G a m e a n d S e a s o n U p s e t R a t e s b e t w e e n H o c k e y a n d B a s k e t b a l l 66 VI L I S T OF T A B L E S IN APPENDIX T a b l e Page A l . C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s o f A l l Team G o a l s f o r A c t u a l P l a y i n g T i m e (One M i n u t e D i v i s i o n s ) . . 78 A 2 . C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s o f A l l Team G a o l s f o r A c t u a l P l a y i n g T i m e (Two M i n u t e D i v i s i o n s ) 80 A 3 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) B o t h Team G o a l s 81 A 4 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) G o a l s F o r . . 88 A 5 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) G o a l s A g a i n s t 95 A 6 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( T e n M i n u t e D i v i s i o n s : d f = 5) B o t h Team G o a l s 102 A 7 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( T e n M i n u t e D i v i s i o n s : d f = 5) G o a l s F o r . . . 109 A 8 . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( T e n M i n u t e D i v i s i o n s : d f = 5) G o a l s A g a i n s t . 116 A 9 . T o t a l Team C h i - s q u a r e C o m p a r i s o n o f S h o t D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s 123 A 1 0 . T o t a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l a n d S h o t D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s 125 A l l . I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F r o m T i m e o f F i r s t G o a l 126 VX1 T a b l e Page A 1 2 . Means a n d V a r i a n c e s f o r I n d i v i d u a l Teams Home, Away a n d T o t a l Games f o r F r e q u e n c y D i s t r i b u t i o n s F r o m T i m e o f F i r s t G o a l 139 A 1 3 . T o t a l Team C h i - s q u a r e C o m p a r i s o n o f W i n , L o s s a n d T o t a l S t r e a k F r e q u e n c i e s w i t h N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s 140 A 1 4 . C h i - s q u a r e C o m p a r i s o n o f B a c k - t o - B a c k a n d N o n - B a c k - t o -B a c k Game P o i n t R e c o r d s f o r 1 9 8 5 - 8 6 N a t i o n a l H o c k e y L e a g u e S e a s o n 143 A 1 5 . S i g n i f i c a n c e L e v e l s f o r Runs T e s t s on I n d i v i d u a l Teams 144 A 1 6 . I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r R u n s T e s t W i t h E a c h P e r i o d T r e a t e d as a Game 145 A 1 7 . S i g n i f i c a n c e L e v e l s f o r Runs T e s t On Number o f S h o t s By P e r i o d f o r I n d i v i d u a l Teams 147 A 1 8 . I n d i v i d u a l Team Runs T e s t R e s u l t s on T o t a l G o a l s F o r a n d A g a i n s t F o r Home, Away a n d T o t a l Games 148 A 1 9 . I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r R u n s T e s t s on A b o v e / B e l o w Mean V a r i a t i o n s ( P 1 G 1 . . . P 2 G 1 . . . P 3 G 4 0 ) . . . 149 A 2 0 . I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r R u n s T e s t s on A b o v e / B e l o w Mean V a r i a t i o n s ( P 1 G 1 . . . P 2 G 1 . . . P 3 G 8 0 ) . . . 150 A 2 1 . Runs T e s t L e v e l s o f S i g n i f i c a n c e f o r I n d i v i d u a l teams a n d S e t - t o - S e t C o r r e l a t i o n s on F o u r - G a m e S e t s T r e a t e d as Games ( T i e s = W i n s ) 151 A 2 2 . E m p i r i c a l P r o b a b i l i t y o f R e p e a t i n g a W i n , L o s s o r T i e C o n d i t i o n e d o n t h e Outcome o f P r e v i o u s Game(s ) . . . . 152 A 2 3 . I n d i v i d u a l Team I n t e r p e r i o d Outcome a n d Game Outcome C o r r e l a t i o n s 154 A 2 4 . P e r i o d t o P e r i o d C o r r e l a t i o n s o f G o a l s F o r , G o a l s A g a i n s t , S h o t s F o r a n d S h o t s A g a i n s t F o r I n d i v i d u a l Teams 155 A 2 5 . I n d i v i d u a l Team P e r i o d - t o - P e r i o d G o a l a n d S h o t C o r r e l a t i o n s 157 A 2 6 . I n d i v i d u a l Team C o r r e l a t i o n s o f G o a l D i f f e r e n c e s B e t w e e n P e r i o d s a n d w i t h T o t a l G o a l D i f f e r e n c e . . . . 158 v i i i T a b l e Page A 2 7 . I n d i v i d u a l Team C o r r e l a t i o n s o f S h o t D i f f e r e n c e B e t w e e n P e r i o d s a n d T o t a l S h o t D i f f e r e n c e 159 A 2 8 . C o r r e l a t i o n o f W i n s , G o a l s a n d S h o t s B e t w e e n P e r i o d s f o r A l l Teams C o m b i n e d 160 A 2 9 . I n d i v i d u a l Team L a g g e d C o r r e l a t i o n s o f W i n s , T i e s , L o s s e s f o r P e r i o d s 1, 2 a n d 3 B e t w e e n Games n a n d n + l . 161 A 3 0 . I n d i v i d u a l Team C o r r e l a t i o n s o f G o a l s F o r , G o a l s A g a i n s t a n d T o t a l G o a l s B e t w e e n P e r i o d s f o r Game l . . n v s . Game 2 . . n 162 A 3 1 . I n d i v i d u a l Team C o r r e l a t i o n s o f S h o t s F o r , S h o t s A g a i n s t a n d T o t a l S h o t s B e t w e e n P e r i o d s f o r Game l . . n v s . Game 2 . . n 163 A 3 2 . Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r G o a l s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) 164 o A 3 3 . Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) 166 A 3 4 . I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s a n d G o a l s F o r a n d A g a i n s t 168 A 3 5 . I n d i v i d u a l Team T e s t s o f S t a t i o n a r i t y C h i - s q u a r e A n a l y s i s o f Mean V a r i a t i o n s on F o u r - G a m e S e t s 170 A 3 6 . I n d i v i d u a l Team N o r m a l D i s t r i b u t i o n C h e c k s a n d S e t - t o -S e t C o r r e l a t i o n s f o r f o u r game s e t s 172 A 3 7 . A r t i f i c i a l a n d R e a l S e a s o n R e c o r d s 174 A 3 8 . C o r r e l a t i o n s f o r O u t c o m e s B a s e d on A r t i f i c i a l S e a s o n s as C o m p a r e d t o R e a l S e a s o n O u t c o m e s 177 A 3 9 . U p s e t C o m p a r i s o n s : 3 7 - 3 8 , 4 6 - 4 7 , 8 8 - 8 9 N a t i o n a l H o c k e y L e a g u e S e a s o n s 178 A 4 0 . C h i - s q u a r e C o m p a r i s o n s o f U p s e t s B a s e d o n P r e - G a m e v s . S e a s o n Rank P o s i t i o n 179 A 4 1 . C r o s s S p o r t C h i - S q u a r e C o m p a r i s o n s 180 A 4 2 . I n d i v i d u a l Team Means a n d S t a n d a r d D e v i a t i o n s F o r , A g a i n s t a n d T o t a l G o a l s a n d F o r , A g a i n s t a n d T o t a l S h o t s 182 i x L I S T OF F I G U R E S Page F i g u r e 1 . T o t a l L e a g u e G o a l s S c o r e d i n S e c o n d P e r i o d ( D a t a f i t U n i f o r m D i s t r i b u t i o n ) 19 F i g u r e 2 . T o t a l G o a l s D a t a f o r A l l Teams F i t t i n g N e g a t i v e B i n o m i a l a n d N o r m a l D i s t r i b u t i o n . . . . 26 F i g u r e 3 . G o a l s S c o r e d D a t a F o r B o s t o n B r u i n s F i t t i n g N e g a t i v e B i n o m i a l a n d G e o m e t r i c D i s t r i b u t i o n . . 29 X ACKNOWLEDGEMENTS I w o u l d l i k e t o t h a n k a l l t h o s e who h a v e h e l p e d me i n c o m p l e t i n g t h i s t h e s i s : my a d v i s o r , D r . R . W . S c h u t z , f o r b e i n g g e n e r o u s e n o u g h t o a c c e p t t h e t a s k o f s u p e r v i s i n g a n o n -t r a d i t i o n a l t y p e t h e s i s p r o j e c t (and d o i n g an e x c e l l e n t j o b as an a d v i s o r ) ; my c o m m i t t e e m e m b e r s , D r . I . M . F r a n k s a n d D r . M . D e l a m p a d y f o r s e r v i n g as h e l p f u l a n d a p p r e c i a t e d c r i t i c s ; D r . M u r r a y B e s l e r a n d t h e L a n g a r a R e s e a r c h F u n d C o m m i t t e e f o r t h e i r g e n e r o u s s u p p o r t a n d p a r t i c u l a r l y f o r D r . B e s l e r ' s a i d i n c o m p u t e r p r o g r a m m i n g ; work s t u d y s t u d e n t s who d i d much o f t h e d a t a c o d i n g a n d i n p u t , p a r t i c u l a r l y K e i t h B a i l e y , who t a u g h t me t h e a r t o f programme w r i t i n g . F i n a l l y I w i s h t o t h a n k a l l t h e f r i e n d s a n d f a m i l y members who p i t c h e d i n w i t h an i d e a h e r e , a comment t h e r e . T h a n k s . INTRODUCTION P u r p o s e A l t h o u g h t h e a v e r a g e s p o r t s f a n w i l l g r u d g i n g l y a t t r i b u t e t h e o c c a s i o n a l a t h l e t i c s u c c e s s o r f a i l u r e t o c h a n c e ( e . g . , t h e C a n u c k s o n l y won b e c a u s e o f t h a t f l u k e g o a l ) , mos t f a n s a p p e a r t o a t t r i b u t e t h e v a s t m a j o r i t y o f o u t c o m e s t o t h e r e l a t i v e s k i l l s a n d q u a l i t i e s o f t h e c o m p e t i t o r s i n v o l v e d . O u t c o m e s a r e d e f i n e d as p e r i o d a n d game s c o r e s , f i n a l s e a s o n s t a n d i n g s ; w i n n i n g a n d l o s i n g s t r e a k s , p e n a l t i e s , s h o t s on g o a l s , g o a l s . T h e r e i s , h o w e v e r , a d i s t i n c t p o s s i b i l i t y t h a t t h e p a t t e r n o f o u t c o m e s i s f a r more r a n d o m t h a n any f a n m i g h t e x p e c t . T h e l i t e r a t u r e on s p o r t a n d p r o b a b i l i t y i s e x t e n s i v e , d e t a i l e d a n d a c c u r a t e . T h e r e a r e s t u d i e s w i t h d a t a a n a l y s e s on v a r i o u s s p o r t s c o n c e r n i n g p r o b a b i l i t y , p r e d i c t i o n , s t r e a k s , e t c . , b u t t h e m a t e r i a l i s n e i t h e r s t r u c t u r e d n o r c o o r d i n a t e d ( p a r t l y b e c a u s e i t h a s b e e n p u b l i s h e d i n a w i d e r a n g e o f j o u r n a l s c o v e r i n g a d i v e r s i t y o f t o p i c s - - s t a t i s t i c s , e c o n o m i c s , p s y c h o l o g y , e t c . ) . No g e n e r a l r u l e s , l a w s o r " t h e o r i e s " h a v e b e e n d e v e l o p e d . T h e r e i s a n e e d f o r i n t e g r a t i o n . T h e p u r p o s e o f t h i s s t u d y i s : (1) t o s t a r t t o d e v e l o p a f r a m e w o r k by u s i n g t h e c o n c e p t s o f r a n d o m i z a t i o n a n d d e t e r m i n i s m as a b a s i s t o s t r u c t u r e a n d s y n t h e s i z e t h e p u b l i s h e d l i t e r a t u r e ; (2) t o t e s t t h e " t h e o r e t i c a l f r a m e w o r k " w i t h a d a t a s e t s u c h as t h e 1 9 8 8 / 8 9 N a t i o n a l H o c k e y L e a g u e s e a s o n . 2 L I T E R A T U R E REVIEW T h e p e r t i n e n t l i t e r a t u r e c a n be d i v i d e d i n t o f i v e g e n e r a l a r e a s : s t u d i e s s h o w i n g r a n d o m n e s s , u n i t o f m e a s u r e m e n t s t u d i e s , M o n t e C a r l o s t u d i e s , c h a n c e d i s t r i b u t i o n o f s c o r e s s t u d i e s a n d g o o d n e s s - o f - f i t d i s t r i b u t i o n s t u d i e s . I . S t u d i e s S h o w i n g Randomness T h e f i r s t g r o u p o f s t u d i e s , d e a l i n g d i r e c t l y w i t h t h e p r e v a l e n c e o f r a n d o m n e s s i n a t h l e t i c o u t c o m e s , a r e o f p a r t i c u l a r s i g n i f i c a n c e t o t h e p r o p o s e d i n v e s t i g a t i o n . T h e p r e m i e r e x a m p l e i s a s t u d y by G i l o v i c h , V a l l o n e a n d T v e r s k y who ( 1 9 8 5 , 1989) f o u n d b a s k e t b a l l s h o o t i n g s t r e a k s t o be r a n d o m . T h e i r s y s t e m o f a n a l y s i s i n c l u d e d t h e W a l d - W o l f o w i t z r u n s t e s t , a c h i - s q u a r e t e s t o f s t a t i o n a r i t y e x a m i n i n g n o n o v e r l a p p i n g s e t s o f f o u r a n d a s e r i a l c o r r e l a t i o n t e s t . L a r k e y , S m i t h a n d K a d a n e (1989) d i s a g r e e d w i t h G i l o v i c h e t a l . a r g u i n g t h a t t h e l a t t e r ' s s h o o t i n g d a t a a r e i n a v e r y d i f f e r e n t f o r m t h a n t h e d a t a u s u a l l y a v a i l a b l e t o a f a n . T h e G i l o v i c h d a t a c o n s i s t o f game by game i s o l a t e d i n d i v i d u a l p l a y e r s e q u e n c e s . L a r k e y e t a l . a r g u e d t h a t f a n s , p l a y e r s a n d c o a c h e s s e e i n d i v i d u a l p l a y e r ' s s h o o t i n g e f f o r t s i n a c o m p l e x game c o n t e x t . T h e y f o c u s s e d o n d e v e l o p i n g a m o d e l t h a t d e s c r i b e s a b y s t a n d e r ' s p e r c e p t i o n s o f s t r e a k a c t i v i t y r a t h e r t h a n a c t u a l s t r e a k a c t i v i t y . P l a y e r s A a n d V e a c h s c o r e 5 c o n s e c u t i v e f i e l d g o a l s i n a game i n w h i c h 220 f i e l d g o a l s a r e a t t e m p t e d . V s c o r e s s h o t s 7 0 , 7 1 , 72 , 73 a n d 74 i n t h e game w h i l e A s c o r e s s h o t s 1, 5 5 , 1 1 0 , 165 a n d 2 1 9 . L a r k e y e t a l . a r g u e d t h a t t h e p r o b a b i l i t y o f V s h i t t i n g 3 s h o t s 70 t h r o u g h 74 i s n o t e q u a l t o t h e p r o b a b i l i t y o f A h i t t i n g s h o t s 1, 5 5 , 1 1 0 , e t c . T h e y a s s e r t e d t h a t s t r e a k s o f s i m i l a r l e n g t h c a n be much more o r much l e s s p r o b a b l e t o o c c u r a n d t o be n o t i c e d . T h e i r s t u d y f o c u s s e d o n t h e f i e l d g o a l s h o o t i n g p a t t e r n s o f p l a y e r s w i t h r e p u t a t i o n s f o r s t r e a k i n e s s , a r g u i n g t h a t a s t r e a k s h o o t e r w o u l d s c o r e l o w - p r o b a b i l i t y , h i g h l y n o t i c e a b l e a n d m e m o r a b l e b a s k e t s w i t h g r e a t e r f r e q u e n c y t h a n p l a y e r s w i t h o u t r e p u t a t i o n s f o r s t r e a k s h o o t i n g . T h e y f o u n d o n l y one p l a y e r f i t t e d t h e i r d e f i n i t i o n o f a s t r e a k s h o o t e r . A l t h o u g h o n t h e s u r f a c e , L a r k e y e t a l . ' s r e s e a r c h a p p e a r s s i m i l a r t o t h a t o f G i l o v i c h e t a l . , t h e two s t u d i e s a c t u a l l y f o c u s o n v e r y d i f f e r e n t p h e n o m e n a . A d m i t t e d l y , G i l o v i c h e t a l . ' s r e s e a r c h i s a l s o b a s e d o n e x a m i n a t i o n o f p e r c e p t u a l e r r o r s . The o r i g i n a l e r r o r by t h e f a n t h a t W L L W L W i s more p r o b a b l e t h a n W W W L L L ( T v e r s k y & K a h n e m a n , 1973) i s a p e r c e p t u a l one b a s e d on a number o f p s y c h o l o g i c a l f u n c t i o n s o r more a p p r o p r i a t e l y , m a l f u n c t i o n s . B u t L a r k e y e t a l . ' s r e s e a r c h m a i n l y c o n c e r n s t h e a v e r a g e p e r s o n ' s p e r c e p t i o n s o f a p l a y e r ' s s h o o t i n g i n a t o t a l game c o n t e x t . G i l o v i c h e t a l . p r i m a r i l y d e a l w i t h i n d i v i d u a l s h o o t i n g b e h a v i o r i n a c o n t e x t i s o l a t e d f r o m t h e o n - g o i n g game. G o u l d (1989 , p . 11) p o i n t e d o u t t h a t mos t r e c o r d s a r e o n l y s l i g h t l y i n c r e m e n t a l l y s u p e r i o r t o t h e r u n n e r s - u p ( e . g . , M a r i s ' 61 t o R u t h ' s 60 home r u n s ) ; b u t t h a t J o e D i M a g g i o ' s 56-game h i t t i n g s t r e a k i s " r i d i c u l o u s l y " a b o v e K e e l e r a n d R o s e ' s 4 4 . G o u l d n o t e d t h a t P u r c e l l , a N o b e l l a u r e a t e i n p h y s i c s , d i d a 4 c o m p r e h e n s i v e s t u d y o f a l l b a s e b a l l s t r e a k s a n d c o n c l u d e d t h a t n o t h i n g e v e r h a p p e n e d i n b a s e b a l l a b o v e a n d b e y o n d t h e f r e q u e n c y p r e d i c t e d b y c o i n - t o s s i n g m e t h o d s . P u r c e l l n o t e d t h a t a h i t s t r e a k o f 50 games w o u l d be e x p e c t e d t o o c c u r b y c h a n c e o n c e s i n c e b a s e b a l l b e g a n . A n d t h a t i s a b o u t what h a s h a p p e n e d . B u t P u r c e l l a l s o c a l c u l a t e d t h a t t o make D i M a g g i o ' s s t r e a k l i k e l y w i t h a p r o b a b i l i t y g r e a t e r t h a n 50 p e r c e n t , b a s e b a l l w o u l d n e e d f o u r l i f e t i m e . 400 h i t t e r s o r 52 l i f e t i m e . 350 b a t t e r s o v e r c a r e e r s o f 1 , 0 0 0 g a m e s . I n a c t u a l i t y , t h e h i g h e s t l i f e t i m e a v e r a g e i s . 367 a n d o n l y t h r e e men h a v e l i f e t i m e a v e r a g e s o f . 3 5 0 . B a s e b a l l w r i t e r J o h n H o l w a y ( G o u l d , 1 9 8 9 , p . 15) a r g u e d t h a t D i M a g g i o ' s s t r e a k was a i d e d by two c h e a p h i t s , two " e r r o r s " s c o r e d as h i t s a n d a c o m p l e x n a r r o w e s c a p e ( H e n r i c h b u n t i n g s a f e l y t o a l l o w a 0 - f o r - 3 D i M a g g i o t o b a t ) . I n s h o r t , a s t r e a k may be a r t i f i c i a l l y p r o l o n g e d by c o n s c i o u s o r u n c o n s c i o u s d e s i r e s o f o f f i c i a l s t o s e e t h a t i t c o n t i n u e (a p o s s i b l e means o f c o n t a m i n a t i n g a n y p u r i t y o f p r o b a b i l i t y ) . S h o r t a n d W a s s e r m a n ( 1 9 8 9 , p . 13) a r g u e d t h a t t h e p r o b a b i l i t y o f D i M a g g i o ' s s t r e a k d e p e n d s on how one w i s h e s t o v i e w i t . P r o b a b i l i t y c a n be v a r i e d by c h a n g i n g t h e a s u m p t i o n s b e h i n d t h e a n a l y s i s . I f D i M a g g i o ' s . 3 2 5 l i f e t i m e b a t t i n g a v e r a g e i s u s e d , a n d 4 a t - b a t s p e r game i s a s s u m e d , D i M a g g i o h a d t h e p r o b a b i l i t y o f . 792 o f g e t t i n g a t l e a s t one h i t i n any s i n g l e game a n d t h e p r o b a b i l i t y o f . 0 0 0 0 0 2 1 3 o f h i t t i n g i n 56 s u c c e s s i v e g a m e s . I f a n a l y s i s i s r e s t r i c t e d t o t h e 32 p l a y e r s who b a t t e d . 3 2 5 o r b e t t e r o v e r t h e i r e n t i r e c a r e e r a n d who h a d 4 , 0 0 0 a t - b a t s o r m o r e , w i t h 5 a n a v e r a g e o f 3 . 8 1 a t - b a t s p e r game a n d 116 games p e r s e a s o n , t h e s e 32 p l a y e r s h a v e a c o m b i n e d a v e r a g e o f . 3 3 8 . T h e p r o b a b i l i t y o f a s t r e a k o f 56 f o r a .338 h i t t e r i n any s i n g l e s e a s o n i s . 0 0 0 0 2 7 7 . B u t o v e r t h e 571 p l a y e r - s e a s o n s r e p r e s e n t e d by t h e 32 p l a y e r s , t h e c h a n c e o f a t l e a s t one 56-game s t r e a k i s . 0 1 5 7 . A t f o u r a t - b a t s p e r game, t h e p r o b a b i l i t y becomes . 0 4 5 5 , w h i c h i s b a r e l y s i g n i f i c a n t . A n d a g a i n j u s t a b o u t what h a p p e n e d . C o o k (1964 , 1977) p i o n e e r e d an e x t e n s i v e m a t h e m a t i c a l s t u d y o f b a s e b a l l . S e v e r a l o f h i s o b s e r v a t i o n s a p p e a r r e l e v a n t t o t h e c u r r e n t i n v e s t i g a t i o n : 1) T h e d i s t r i b u t i o n o f games won i n W o r l d S e r i e s p l a y b e t w e e n 1903 a n d 1961 c l o s e l y f o l l o w e d t h e d i s t r i b u t i o n e x p e c t e d by c h a n c e . C o o k (1964 , p . 6) c o n c l u d e d t h a t " i n a W o r l d S e r i e s o f as few as s e v e n games b e t w e e n q u i t e e v e n l y m a t c h e d t e a m s , t h e c h a m p i o n s h i p i s l a r g e l y a m a t t e r o f c h a n c e . " S i m o n (1977) a g r e e d w i t h C o o k a n d a l s o f o u n d t h a t t h e r e h a s b e e n a t e n d e n c y f o r t h e W o r l d S e r i e s ( f r o m 1945 t o 1 9 7 6 , a t l e a s t ) t o l a s t s e v e n g a m e s . T h i s s u g g e s t s t h a t as t eams become t r u l y e q u a l , game o u t c o m e i s r a n d o m a n d s e r i e s a r e mos t l i k e l y t o e x t e n d t o t h e l i m i t . 2) T h e r e ( C o o k , 1 9 6 4 , p . 7) i s a n e x t e n s i v e i n f l u e n c e o f e x t r e m e l y s m a l l v a r i a t i o n s i n p e r f o r m a n c e on t h e s c o r i n g o f r u n s , w i n n i n g o f games a n d t h e f i n a l l e a g u e s t a n d i n g s ( e . g . , C i n c i n n a t i won t h e p e n n a n t by 4 games o v e r L o s A n g e l e s i n 1961 w i t h b a t t i n g , s l u g g i n g a n d f i e l d i n g p e r c e n t a g e s d i f f e r i n g o n l y i n t h e s e c o n d d e c i m a l p l a c e ) . 6 3) A p p l y i n g P a s c a l ' s b i n o m i a l e x p a n s i o n t o b a t t i n g s t r e a k s a n d s l u m p s , C o o k (1964 , p . 140) c o n c l u d e d t h a t " t h e r e i s no s u c h t h i n g as a b a t t i n g s l u m p f o r a n o r m a l , h e a l t h y . . . p l a y e r . " A .300 h i t t e r p a r t i c i p a t i n g i n 162 games may be e x p e c t e d t o h i t a b o u t . 131 i n 98 g a m e s ; . 400 f o r 45 games ; a n d .632 f o r 19 games , b u t ( C o o k , 1964 , p . 141) " t h e r e i s j u s t no way t o know i n w h i c h o f 98 games a . 3 0 0 -b a t s m a n w i l l i n e v i t a b l y p r o d u c e one o r l e s s b a s e h i t s p e r g a m e . " 4) A d d r e s s i n g t h e p r o b l e m o f how a b a t t e r ' s t r u e a v e r a g e may be d e t e r m i n e d , C o o k (1964 , p . 145) n o t e d t h a t two y e a r ' s p l a y (383 games) was n e e d e d t o d e t e r m i n e a p l a y e r ' s c a r e e r b a t t i n g a v e r a g e w i t h i n 10 p o i n t s a n d n e a r l y n i n e y e a r s (1412 games) was n e e d e d t o e s t i m a t e t h e i n d i v i d u a l ' s c a r e e r p e r f o r m a n c e c o r r e c t l y w i t h i n f i v e p o i n t s . 5) C o o k (1964 , p . 153) c o n c l u d e d t h a t w i n n i n g r e s u l t s f r o m " t h e mos t f a v o r a b l e a l i g n m e n t o f t h e g r e a t e s t p r o p o r t i o n o f a l a r g e number o f i n d i v i d u a l l y s m a l l v a r i a b l e s . " F o r 60 y e a r s , t h e w i n n i n g p e r c e n t a g e o f f i r s t p l a c e teams h a v e f o l l o w e d G a u s s i a n d i s t r i b u t i o n e q u a t i o n s b e c a u s e b a s e b a l l i s i n h e r e n t l y a game o f c h a n c e w i t h a l l i t s c o m p l e x i n t e r a c t i o n s b e i n g c o n f i d e n t l y i n t e r p r e t e d b y a p p l i c a t i o n o f t h e l a w s o f p r o b a b i l i t y ( C o o k , 1964 , p . 3 1 2 ) . T a r g e t i n g t h e c r u x o f h i s i n v e s t i g a t i o n s . C o o k ( 1 9 7 7 , p . 54) c o n c l u d e d , " W h i l e t h e p r o b a b i l i t y o f a b a s e h i t o r a s t r i k e - o u t may p r e c i s e l y be d e t e r m i n e d . . . no one w i l l e v e r know ' w h e n ' ! " T h e r e i s no r e a s o n t o b e l i e v e t h e same c o n c l u s i o n s s h o u l d n o t h o l d f o r h o c k e y o u t c o m e s . 7 S c h u t z a n d J a n s s e n (1987) u s e d m u l t i p l e l i n e a r r e g r e s s i o n a n a l y s i s t o c o m p a r e t h e r e c o r d s o f two h o c k e y teams b e f o r e a n d a f t e r a c h a n g e o f c o a c h e s a n d f o u n d no d e t e c t a b l e e f f e c t on team p e r f o r m a n c e . I n h e r e n t l y r a n d o m o u t c o m e s h a d b e e n m i s i n t e r p r e t e d as c o a c h i n g i n c o m p e t e n c e by h o c k e y owners a n d g e n e r a l m a n a g e r s . H a b e r m a n (1977) u s e d a t h r e e - w a y a n a l y s i s o f v a r i a n c e on t h e f i n a l s c o r e s o f a l l I v y L e a g u e f o o t b a l l games b e t w e e n 1960 a n d 1966 a n d f o u n d t h a t I v y L e a g u e games were v e r y u n p r e d i c t a b l e i n t e r m s o f s c o r e o r b o x - s c o r e s t a t i s t i c s , c o n c l u d i n g t h a t t h e d a t a s u g g e s t e d t h a t f o o t b a l l games a r e s u b j e c t t o much more r a n d o m f l u c t u a t i o n t h a n mos t o b s e r v e r s w o u l d s u s p e c t . I I . U n i t o f M e a s u r e m e n t S t u d i e s A s e c o n d g r o u p o f s t u d i e s a d d r e s s e d t h e u n i t o f m e a s u r e m e n t u s e d t o d e t e r m i n e a t h l e t i c p r o w e s s . Goodman (1969) f o u n d t h a t t h e o u t c o m e s o f d o u b l e h e a d e r s i n M a j o r L e a g u e b a s e b a l l were d i f f e r e n t f r o m t h e o u t c o m e s o f p a i r s o f s i n g l e games o n c o n s e c u t i v e d a y s . D o u b l e h e a d e r s were more l i k e l y t o be swept t h a n s p l i t , s u g g e s t i n g one game was n o t n e c e s s a r i l y a t r u e s i n g l e u n i t . Two games on one d a y m i g h t f o r m a s i n g l e u n i t as f a r as c o n s i s t e n c y o f team p l a y i s c o n c e r n e d . S p o r t s w r i t e r s F i s c h l e r a n d F i s c h l e r (1986 , p . 11) p r o v i d e d a summary o f b a c k - t o - b a c k v e r s u s r e g u l a r game d a t a f o r t h e 1 9 8 5 - 8 6 N a t i o n a l H o c k e y L e a g u e s e a s o n , w h i c h a l l o w e d t h e s i t u a t i o n c l o s e s t t o a d o u b l e h e a d e r i n h o c k e y t o be s t a t i s t i c a l l y a n a l y z e d . L i n d s e y (1961) f o u n d t h e d i s t r i b u t i o n o f r u n s i s n o t c o n s t a n t f r o m i n n i n g t o i n n i n g i n N a t i o n a l , A m e r i c a n a n d I n t e r n a t i o n a l 8 L e a g u e b a s e b a l l , t h e mean number o f r u n s b e i n g g r e a t e s t i n t h e f i r s t , t h i r d a n d s i x t h i n n i n g . G o o d m a n ' s a n d L i n d s e y ' s r e s e a r c h p i n p o i n t s t h e p r o b l e m o f d e t e r m i n i n g what a u n i t o f p e r f o r m a n c e i s (an i n n i n g , a game, a d o u b l e - h e a d e r ) a n d t h e p r o b l e m o f how r e s u l t s c h a n g e d e p e n d i n g o n what u n i t a r e s e a r c h e r s e l e c t s t o m e a s u r e . U n i t o f m e a s u r e m e n t i s a n e x c e e d i n g l y c o m p l e x t o p i c . I s o l a t e d i n t i m e , a game s t a r t s 0 - 0 , b u t v i e w e d i n t h e c o n t e x t o f p r e v i o u s g a m e s , game X w i l l h a v e a d i f f e r i n g i n i t i a l c o n d i t i o n f r o m game Y . T h e p r o g r e s s o f teams t h a t e n t e r game X w i t h 10 w i n s , 2 t i e s a n d 2 l o s s e s , s h o u l d be d i f f e r e n t f r o m teams t h a t e n t e r game X w i t h 2 w i n s , 0 t i e s a n d 12 l o s s e s . A n y c h a i n o f e v e n t s may h a v e p o i n t s o f c r i s i s w h e r e s m a l l c h a n g e s c a n l e a d t o s i g n i f i c a n t l y d i f f e r e n t r e s u l t s . T h e u n i t o f m e a s u r e m e n t c h o s e n may h i g h l i g h t o r o b s c u r e s u c h c h a n g e s . U n i t o f m e a s u r e m e n t i s a l s o r e l a t e d t o t h e q u e s t i o n : What d o e s a b a t t i n g a v e r a g e mean? A s X comes t o b a t , do we u s e . 320 ( t h i s s e a s o n ' s a v e r a g e ) , . 2 8 5 ( X ' s a v e r a g e a g a i n s t l e f t - h a n d e r s ) , . 267 ( X ' s l i f e t i m e a v e r a g e ) as a p r e d i c t o r ? I I I . M o n t e C a r l o S t u d i e s A t h i r d g r o u p o f s t u d i e s h a v e i n v o l v e d a t t e m p t s t o a p p r o x i m a t e r e a l - l i f e games by M o n t e C a r l o s i m u l a t i o n s u s i n g p r o b a b i l i t i e s e s t a b l i s h e d i n r e a l - l i f e g a m e s . F r e e z e (1977) u s e d a M o n t e C a r l o a n a l y s i s o f v a r i o u s h y p o t h e t i c a l b a t t i n g o r d e r s a n d c o n c l u d e d t h a t b a t t i n g o r d e r e x e r t e d a s m a l l i n f l u e n c e . T h e e f f e c t o f u s i n g t h e b e s t b a t t i n g o r d e r r a t h e r t h a n t h e w o r s t r e s u l t e d i n o n l y t h r e e e x t r a r u n s p e r 9 162-game s e a s o n . P e t e r s o n (1977) f o u n d t h a t t h e b e s t b a t t i n g o r d e r p r o d u c e d o n l y s e v e n more r u n s p e r s e a s o n p e r t e a m . S m i t h (1977) f o u n d t h a t a d j u s t m e n t s f o r s t r e n g t h o f o p p o n e n t l e d t o a l m o s t no c h a n g e i n p e r c e n t a g e s t a n d i n g s o f N a t i o n a l L e a g u e b a s e b a l l t e a m s . A l l t h i s i s s u g g e s t i v e o f e q u a l i t y i n t h e a b i l i t y o f p l a y e r s , a s o u r c e o f p o s s i b l e r a n d o m r e s u l t s , a n d t h e p o s s i b i l i t y t h a t S t e i n ' s p a r a d o x ( E f r o n & M o r r i s , 1977) may be a p p r o p r i a t e t o s u c h a n a l y s i s . A s a p r e d i c t o r , S t e i n ' s p a r a d o x " s h r i n k s " d i f f e r e n c e s b e t w e e n a v e r a g e s b y a b o u t 80% a n d was shown ( E f r o n & M o r r i s , 1977) t o be a b e t t e r p r e d i c t o r o f a g r o u p o f f u t u r e b a t t i n g a v e r a g e s t h a n were p r o j e c t i o n s f r o m r e a l - l i f e a v e r a g e s . A l l t h i s s u g g e s t s t h a t a p l a y e r ' s t r u e a b i l i t i e s a r e more t i g h t l y c l u s t e r e d t h a n any one s e a s o n ' s r a n g e o f a v e r a g e s w o u l d s u g g e s t . I V . C h a n c e D i s t r i b u t i o n o f S c o r e s A f o u r t h g r o u p o f s t u d i e s h a v e i n v e s t i g a t e d t h e n a t u r e o f s p o r t s s c o r e s t h e m s e l v e s . M o s t e l l e r (1970) e x a m i n e d w h i c h U . S . c o l l e g i a t e f o o t b a l l s c o r e s were more f r e q u e n t f o r t e a m s , games , w i n n e r s a n d l o s e r s a n d what t h e " o p e r a t i n g c h a r a c t e r i s t i c " f o r a s c o r e was ( e . g . , t h e c h a n c e any g i v e n s c o r e i s a w i n n i n g s c o r e , a l o s i n g s c o r e ) a n d f o u n d t h a t some s c o r e s a r e more l i k e l y t o w i n o r l o s e t h a n t h e i r n u m e r i c a l v a l u e a l o n e w o u l d s u g g e s t . He a l s o f o u n d t h a t r a n d o m l y a s s e m b l e d team s c o r e s were more c l o s e l y m a t c h e d t h a n a c t u a l t eam s c o r e s a n d a r g u e d t h a t t h i s s h o u l d be e x p e c t e d i n c o l l e g e f o o t b a l l w h e r e teams a r e o f t e n n o t e q u a l l y 10 s k i l l e d . C o l l e g e f o o t b a l l s h o u l d be one o f t h e l e a s t r a n d o m o f s p o r t s i n o u t c o m e s . V . G o o d n e s s - o f - f i t D i s t r i b u t i o n S t u d i e s A f i f t h g r o u p o f s t u d i e s a n a l y z e d s p e c i f i c k e y a t t r i b u t e s o f a s p o r t ( e . g . , g o a l s , r u n s ) m a t c h i n g t h e d i s t r i b u t i o n o f t h e a t t r i b u t e w i t h a known t h e o r e t i c a l d i s t r i b u t i o n . E l d e r t o n (1909) seems t o h a v e p i o n e e r e d d i s t r i b u t i o n i n v e s t i g a t i o n i n t h e s p o r t s a r e a when he s t u d i e d t h e f r e q u e n c y d i s t r i b u t i o n s o f f o u r i n d i v i d u a l c r i c k e t b a t s m e n ' s s c o r e s f o r t h e y e a r s 1 9 0 5 , 1906 a n d 1 9 0 7 . He (1909 , 1 9 1 5 , 1927) d e s c r i b e d t h e d i s t r i b u t i o n o f c r i c k e t s c o r e s as b e i n g a g e o m e t r i c c u r v e a n d c o n c l u d e d t h a t t h e r a t i o o f e a c h t e r m t o t h e p r e c e d i n g was . 5 4 . E x t e n d i n g E l d e r t o n ' s w o r k , w h i c h h a d l o o k e d a t o n l y f o u r b a t s m e n o v e r a t h r e e y e a r p e r i o d . Wood (1945) e x a m i n e d a l a r g e s a m p l e o f c r i c k e t b a t s m e n ' s p e r f o r m a n c e s a n d c o n c l u d e d t h a t a g e o m e t r i c d i s t r i b u t i o n e x i s t e d . Wood was c o n c e r n e d t h a t e x a c t f i t s w e r e n o t a t t a i n e d . A c c o r d i n g t o Wood, i f a s e r i e s i s i n s t r i c t g e o m e t r i c a l p r o g r e s s i o n : 1) t h e mean a n d s t a n d a r d d e v i a t i o n a r e e q u a l 2) t h e mean i s t o t h e m e d i a n as 1 . 4 4 3 t o 1 3) c e n t i l e s a n d q u a r t i l e s a r e i n u n c h a n g e d r e l a t i o n t o t h e m e d i a n , no m a t t e r what t h e m e d i a n i s 4) t h e c o m b i n a t i o n o f two g e o m e t r i c a l p r o g r e s s i o n s c a n n o t p r o d u c e a t h i r d S u c h d i s t i n c t i o n s c o u l d be o f u s e i n d i s t i n g u i s h i n g b e t w e e n d i s t r i b u t i o n s t h a t a r e d i f f i c u l t t o d e t e r m i n e b y c r u d e r m e t h o d s ( e . g . , a s e r i e s t h a t a p p e a r s f r o m c h i - s q u a r e a n a l y s i s t o f i t a n e g a t i v e b i n o m i a l a s w e l l a s a g e o m e t r i c d i s t r i b u t i o n ) . P o l l a r d (1973 , 1 9 8 5 , 1986) a n d R e e p , P o l l a r d a n d B e n j a m i n (1971) f o u n d t h a t t h e d i s t r i b u t i o n o f s c o r e s i n v a r i o u s s p o r t s ( s o c c e r , h o c k e y , b a s e b a l l , f o o t b a l l ) c l o s e l y f i t t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n . M u l l e t (1977) s t u d i e d N a t i o n a l H o c k e y L e a g u e s c o r i n g a n d c o n c l u d e d t h a t (1) mean g o a l s f o r a n d a g a i n s t ( b o t h home a n d away) were d i s t r i b u t e d a c c o r d i n g t o a member o f t h e P o i s s o n f a m i l y , a n d (2) t h a t g o a l s f o r a n d g o a l s a g a i n s t ( a t home a n d away) were i n d e p e n d e n t r a n d o m v a r i a b l e s . M u l l e t u s e d a t i m e s p a c e o f 60 m i n u t e s a n d d i d n o t c o n s i d e r s h o r t e r o r l o n g e r t i m e i n t e r v a l s . I n c o n t r a s t t o M u l l e t a n d P o l l a r d e t a l . , L a p i n (1984) n o t e d t h a t a r a n d o m d i s t r i b u t i o n c o u l d d e s c r i b e a u n i f o r m d i s t r i b u t i o n i f e v e n t p r o d u c t i o n ( e . g . , g o a l - s c o r i n g ) i s c o n s i d e r e d a k i n t o a r a n d o m - n u m b e r g e n e r a t o r . PROCEDURE I . D a t a S o u r c e s G a m e - b y - g a m e , p e r i o d - b y - p e r i o d d a t a o n t h e 1 9 8 8 - 8 9 N a t i o n a l H o c k e y L e a g u e was t a k e n f r o m 6 - p o i n t t y p e s i z e s u m m a r i e s i n The V a n c o u v e r S u n . T h e d a t a were i d e n t i c a l t o a s h o r t e r g a m e - b y - g a m e v e r s i o n o b t a i n e d f r o m NHL h e a d q u a r t e r s . I t t u r n e d o u t t h a t t h e NHL u s e d t h e same s o u r c e f o r i t s s t a t i s t i c a l r e c o r d s a s d i d t h e d a i l y p r e s s . When d i s c r e p a n c i e s ( e . g . , team X s u p p o s e d l y s c o r i n g 3 g o a l s w h i l e t a k i n g o n l y 2 s h o t s i n t h e f i r s t p e r i o d ) o r s e e m i n g l y o d d n u m b e r s ( e . g . , 18 g o a l s i n one p e r i o d ) were d i s c o v e r e d , t h e n a n e w s p a p e r i n t h e home c i t y o f e i t h e r o f t h e t eams i n v o l v e d was c o n s u l t e d ( e . g . , D e t r o i t F r e e P r e s s i f D e t r o i t , T o r o n t o G l o b e a n d M a i l i f T o r o n t o ) . U s u a l l y t h i s i n v o l v e d c a r e f u l r e a d i n g o f t h e game s t o r y s i n c e o f t e n t h e e r r o r was r e p e a t e d i n t h e s e c o n d s o u r c e . E v i d e n t l y a l l l e a g u e n e w s p a p e r s p i c k up 6 -p o i n t d a t a on games f r o m t h e same s o u r c e : T h e A s s o c i a t e d P r e s s o r i t s C a n a d i a n a f f i l i a t e , T h e C a n a d i a n P r e s s . T h e H o c k e y News was a l s o c o n s u l t e d . A l l d i s c o v e r e d e r r o r s were r e s o l v e d t o s a t i s f a c t i o n . M a t e r i a l f o r t h e 1937 -38 a n d 1 9 4 6 - 4 7 s e a s o n s was t a k e n f r o m T h e V a n c o u v e r S u n . M a t e r i a l f o r t h e s e c o n d h a l f o f t h e 1 9 8 8 - 8 9 N a t i o n a l B a s k e t b a l l A s s o c i a t i o n s e a s o n was t a k e n f r o m t h e 6 - p o i n t summary s e c t i o n o f t h e B e l l i n g h a m H e r a l d . D a t a f o r t h e F o o t b a l l A s s o c i a t i o n ' s E n g l i s h f i r s t d i v i s i o n ( R o l l i n , 1988) were t a k e n f r o m R o t h m a n ' s F o o t b a l l Y e a r b o o k 1 9 8 8 - 8 9 , t h e F A ' s o f f i c i a l p u b l i c a t i o n . D a t a were e n t e r e d i n t h e m a i n f r a m e c o m p u t e r a t V a n c o u v e r C o m m u n i t y C o l l e g e u n d e r a s e r i e s o f work s t u d y g r a n t s ( t h r e e p e r y e a r f o r f i v e y e a r s ) a n d two s p e c i a l r e s e a r c h g r a n t s f r o m t h e C o l l e g e ' s r e s e a r c h f u n d . S i n c e s t u d e n t s d i d t h e v a s t m a j o r i t y o f d a t a c o d i n g a n d c o m p u t e r i n p u t , t h e m a t e r i a l was d o u b l e c o d e d a n d d o u b l e i n p u t t o m a i n t a i n a c c u r a c y o f r e c o r d i n g . A l l c o m p u t e r p r o g r a m s w r i t t e n by t h e a u t h o r o r h i s c h i e f c o m p u t e r a s s i s t a n t , K e i t h B a i l e y , were c h e c k e d a g a i n s t known d a t a b a s e r e s u l t s b e f o r e b e i n g u s e d i n t h e s t u d y . A b o u t 25% o f t h e m a t e r i a l was h a n d l e d by t h e a u t h o r ' s p e r s o n a l c o m p u t e r . The b a s i c p r o g r a m u s e d was M i n i t a b . 13 I I . A c c u r a c y o f M e a s u r e m e n t T h e a s s u m p t i o n u n d e r l y i n g u s e o f t h e d a t a i n t h i s s t u d y i s t h a t a b s o l u t e p r e c i s e n e s s o f m e a s u r e m e n t i s n o t n e e d e d t o t e s t t h e v a r i o u s h y p o t h e s e s . Th e p r e c i s e s e c o n d o f s c o r i n g was n o t a v a i l a b l e . E s k e n a z i ( 1 9 6 9 , p p . 189 -190 ) n o t e d t h a t o c c a s i o n a l l y g o a l s a r e n o t s c o r e d b u t a r e c o u n t e d a n d o c c a s i o n a l l y g o a l s a r e s c o r e d b u t a r e n o t c o u n t e d : " . . . t h e ( g o a l ) j u d g e k e e p s h i s h a n d s away f r o m t h e ( r e d l i g h t ) s w i t c h . Once he i s c o n f i d e n t a g o a l h a s b e e n s c o r e d , he s i g n a l s . " B e i n g human, e r r o r s o c c u r . A c c u r a c y o f r e c o r d i n g i s n o t s p l i t - s e c o n d o r s p l i t - r e s p o n s e a c c u r a t e . E s k e n a z i ( 1 9 6 9 , p . 191) n o t e s t h a t " A l l c l u b s u s e a s t a t i s t i c i a n . . . H e k e e p s t r a c k o f h o m e - t e a m s h o t s o n n e t ( w h i c h u s u a l l y v a r y w i d e l y w i t h t h e o f f i c i a l s h o t s ) . " T h e p r o b l e m ( F i s c h l e r & F i s c h l e r , 1 9 8 6 , p . 8) may s t e m f r o m a l a c k o f a g r e e m e n t b e t w e e n s h o t s o n a n d s h o t s a t t h e n e t . T h e h i g h s p e e d a n d s m a l l s i z e o f a p u c k make t h e d i s t i n c t i o n o f w h e t h e r a s h o t w o u l d h a v e b e e n on g o a l o r n o t a h i g h l y s u b j e c t i v e j u d g m e n t . I I I . B a s i c A s s u m p t i o n s U n l e s s o t h e r w i s e s t a t e d , t h e 1 9 8 8 - 8 9 NHL s e a s o n was t h e d a t a b a s e u s e d i n t e s t i n g a s e r i e s o f p o s t u l a t e s w h i c h were b a s e d on t h e f o l l o w i n g a s s u m p t i o n s : 1) V a r i a t i o n s i n t h e l e n g t h o f . t h e u n i t o f m e a s u r e m e n t w i l l n o t a p p r e c i a b l y a f f e c t any o f t h e p o s t u l a t e s p r o p o s e d ( e . g . , i f o u t c o m e s a r e r a n d o m , t h e y w i l l be r a n d o m w h e t h e r p e r i o d s , p a r t s o f p e r i o d s , games o r s e t s o f games a r e u s e d as t h e u n i t o f m e a s u r e m e n t ) . 14 2) A l t h o u g h h o c k e y i s t h e m a i n s p o r t s t u d i e d , t h e p o s t u l a t e s a r e n o t s p o r t - s p e c i f i c . H o p e f u l l y , t h e y w o u l d e x t e n d t o any r e a s o n a b l y b a l a n c e d l e a g u e a n d n o t be l i m i t e d t o t i m e - b o u n d s p o r t s . A b e l s o n (1989) s u g g e s t s a means o f e s t i m a t i n g t h e number o f r u n s s c o r e d i n a h a l f - i n n i n g o f a b a s e b a l l game f r o m e l a p s e d r e a l t i m e . S i n c e b a s e b a l l i s n o t t i m e d , A b e l s o n ' s a n a l y s i s c o u l d be u s e d t o c o m p a r e ' n o n - t i m e d ' t o t i m e d s p o r t s . 3) A r e a s o n a b l y b a l a n c e d l e a g u e i s a s s u m e d . I t w o u l d be i n a p p r o p r i a t e t o a p p l y t h e p o s t u l a t e s t o t h e d e v e l o p m e n t s t a g e t h a t seems t o c h a r a c t e r i z e mos t e l i t e s p o r t s i n t h e i r e a r l y y e a r s . F o r e x a m p l e : T h o r n a n d P a l m e r (1989 , p . 9) n o t e t h a t t h e C i n c i n n a t i Red S t o c k i n g s , t h e f i r s t p r o f e s s i o n a l b a s e b a l l t e a m , won 60 s t r a i g h t i n 1 8 6 9 . N e m e c 1 s r e s e a r c h (1987) s u g g e s t s i t was 1 3 0 . E i t h e r way t h e t eam t o t a l l y d o m i n a t e d . S t r e a k s a r e m o s t l i k e l y t o o c c u r w i t h e x t r e m e l y g o o d o r e x t r e m e l y p o o r t e a m s . T h e 1 9 7 4 - 7 5 W a s h i n g t o n C a p i t a l s ( K l e i n & R e i f , 1 9 8 7 ) , who h a d t h e l o w e s t w i n n i n g p e r c e n t a g e e v e r ( . 1 3 1 ) , h a d t h e l o n g e s t r o a d w i n l e s s s t r e a k e v e r (37 g a m e s ) . 4) H o c k e y o u t c o m e s a r e d e t e r m i n e d by d e f i n i t e c a u s e s , b u t t h e i r o c c u r e n c e i s r a n d o m a c r o s s t i m e . P O S T U L A T E T E S T I N G A l a r g e number o f p o s t u l a t e s a r e p r o p o s e d i n t h i s s t u d y . The r a t i o n a l e b e h i n d t h e i r d e v e l o p m e n t , r e f i n e m e n t a n d u s e was as f o l l o w s : (1) e x p e c t a t i o n s f r o m t h e s c i e n t i f i c l i t e r a t u r e — a s i m i l a r p o s t u l a t e h a d o f t e n b e e n a d v a n c e d by o t h e r r e s e a r c h e r s who h a d e s t a b l i s h e d t h e p o s t u l a t e ' s v a l i d i t y i n a p a r t i c u l a r c o n t e x t ( e . g . , s i n c e G i l o v i c h e t a l . <1985, 1989> f o u n d i n d i v i d u a l b a s k e t b a l l p l a y e r ' s s h o o t i n g s t r e a k s d i d n o t e x c e e d c h a n c e e x p e c t a n c i e s b a s e d o n a p l a y e r ' s s h o o t i n g p e r c e n t a g e , t h i s i n v e s t i g a t i o n e x p a n d e d t h e s c o p e o f G i l o v i c h e t a l . ' s r e s e a r c h by l o o k i n g a t team s t r e a k s o n s h o t s , g o a l s , e t c . ) . (2) t h e i m p o r t a n c e o f t h e c o n c e p t o f u n i t o f m e a s u r e m e n t — t h i s l e d t o a n e x t e n s i o n o f many p o s t u l a t e s f r o m c o v e r i n g one u n i t ( e . g . , a p e r i o d ) t o c o v e r i n g o t h e r u n i t s ( e . g . , two p e r i o d s , a O o game, f o u r g a m e s ) . (3) p e r s o n a l e x p e r i e n c e — t h e a u t h o r ' s b a c k g r o u n d i n p s y c h o l o g y a n d c o m m u n i c a t i o n s p r o b a b l y p r e d i s p o s e d t h e d e v e l o p m e n t o f s e v e r a l p o s t u l a t e s ( e . g . , a p p l y i n g t h e c o n c e p t s i n v o l v e d i n i n t e l l i g e n c e t e s t c o n s t r u c t i o n t o an e x a m i n a t i o n o f how e a c h p e r i o d c o r r e l a t e d w i t h o t h e r p e r i o d s a n d t o t a l game o u t c o m e ) . I . D i s t r i b u t i o n T e s t i n g T h e f i r s t s e r i e s o f p o s t u l a t e s e x a m i n e d t h e d i s t r i b u t i o n o f b e h a v i o r a l o u t c o m e s ( e . g . , g o a l s ) a c r o s s t i m e . S i n c e a n e x t r e m e l y l a r g e number o f v a r i a b l e s mus t come t o g e t h e r f o r a n i n t e n d e d a c t ( e . g . , a g o a l ) t o o c c u r , i t i s i m p o s s i b l e t o p r e d i c t p r e c i s e l y 16 when t h i s " r i g h t " c o m b i n a t i o n w i l l o c c u r . T h i s g e n e r a l h y p o t h e s i s was e x a m i n e d by t e s t i n g t h e f o l l o w i n g s p e c i f i c p o s t u l a t e s : l a . G o a l s i n NHL h o c k e y a r e s c o r e d e v e n l y a c r o s s t i m e . ( i n c l u d e s : g o a l s f o r , g o a l s a g a i n s t , t o t a l g o a l s f o r a n d a g a i n s t f o r e a c h team) l b . T h e p a t t e r n o f o u t c o m e s i n any p e r i o d i s t h e same as t h e p a t t e r n o f o u t c o m e s f o r any o t h e r p e r i o d a n d an e n t i r e game. ( i n c l u d e s : g o a l s f o r , g o a l s a g a i n s t , t o t a l g o a l s f o r a n d a g a i n s t f o r e a c h team) a n d t h e s p e c i f i c p o s t u l a t e : l c . T h e r e i s a n i n c r e a s e i n t h e r a t e o f g o a l s c o r i n g d u r i n g t h e f i n a l two m i n u t e s o f game t i m e . T h i s i s n o t c o n s i d e r e d a n e g a t i o n o f p o s t u l a t e s l a a n d l b , b e c a u s e t h e p u l l i n g o f a g o a l t e n d e r i s a s i g n i f i c a n t e n o u g h c h a n g e as t o c o n s t i t u t e an e n t i r e l y new a n d d i f f e r e n t e v e n t s e r i e s . Games were d i v i d e d i n t o v a r i o u s t i m e s e g m e n t s . C o m p a r i s o n o f o b s e r v e d a n d e x p e c t e d f r e q u e n c i e s o f g o a l s f o r n o r m a l , u n i f o r m , P o i s s o n a n d n e g a t i v e b i n o m i a l d i s t r i b u t i o n s were made . The p a t t e r n o f g o a l s a c r o s s t i m e c o u l d be r a n d o m l y d i s t r i b u t e d a c c o r d i n g t o a P o i s s o n ( M u l l e t , 1 9 7 7 ) , n e g a t i v e b i n o m i a l ( P o l l a r d , 1 9 7 3 , 1 9 8 5 , 1986; R e e p , P o l l a r d & B e n j a m i n , 1971) o r u n i f o r m d i s t r i b u t i o n ( L a p i n , 1 9 8 4 ) . T h e r a t i o n a l e f o r t e s t i n g e a c h d i s t r i b u t i o n was as f o l l o w s : 1) t h e P o i s s o n d i s t r i b u t i o n was t e s t e d f o r a f i t b e c a u s e i t t e l l s w h e t h e r a d i s t r i b u t i o n i s r a n d o m l y d i s t r i b u t e d o v e r t i m e a t an a v e r a g e r a t e , 2) t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n was t e s t e d f o r a f i t b e c a u s e i t i s t h e Compound P o i s s o n ( i t a s s u m e s a P o i s s o n 17 d i s t r i b u t i o n f o r i n d i v i d u a l g a m e s ) , 3) t h e g e o m e t r i c d i s t r i b u t i o n was l a t e r t e s t e d f o r a f i t b e c a u s e i t i s r e l a t e d t o t h e n e g a t i v e b i n o m i a l , 4) t h e u n i f o r m d i s t r i b u t i o n was t e s t e d f o r a f i t b e c a u s e i t s a p p e a r a n c e ( L a p i n , 1984) may be i n d i c a t i v e o f a r a n d o m p r o c e s s , a n d 4) t h e n o r m a l d i s t r i b u t i o n was e m p l o y e d f o r a p o s s i b l e f i t s i n c e i t i s t h e " p a r e n t " o f d i s t r i b u t i o n s — i t h a s an o v e r w h e l m i n g f r e q u e n c y o f a p p e a r e n c e i n a w i d e r a n g e o f s i t u a t i o n s ( e . g . , t h e n o r m a l d i s t r i b u t i o n m i g h t be e x p e c t e d t o d e s c r i b e g o a l s - p e r - g a m e s c o r i n g ) . T h e a s s u m p t i o n a t t h e s t a r t o f t h e s t u d y was t h a t a g o a l o r w i n h a d no a f t e r e f f e c t i n t h e s e n s e t h a t t h e p a s t h i s t o r y p e r m i t s no c o n c l u s i o n s as t o t h e f u t u r e . A t e a c h a r r i v a l ( g o a l ) , t h e p r o c e s s s t a r t s f r o m s c r a t c h as a p r o b a b i l i s t i c r e p l i c a o f t h e w h o l e p r o c e s s . T h e s u c c e s s i v e w a i t i n g t i m e s b e t w e e n a r r i v a l s ( F e l l e r , 1 9 5 0 , 1966) must be i n d e p e n d e n t a n d must h a v e t h e same d i s t r i b u t i o n . I n t h e t e r m i n o l o g y o f s t o c h a s t i c p r o c e s s e s , t h e s e q u e n c e c o n s t i t u t e s a r e n e w a l p r o c e s s , where w l , w2 ( C i n l a r , 1975) a r e i d e n t i c a l l y d i s t r i b u t e d n o n - n e g a t i v e r a n d o m v a r i a b l e s . T h e p r o c e s s may be a M a r k o v c h a i n o r a P o i s s o n p r o c e s s . I t becomes a M a r k o v c h a i n i f t h e s o j o u r n t i m e s a r e a l l e q u a l t o one a n d i t becomes a r e n e w a l p r o c e s s i f t h e r e i s o n l y one s t a t e ( t h e i d e a i s o f a p a r t i c l e m o v i n g f r o m one s t a t e t o a n o t h e r w i t h r a n d o m s o j o u r n t i m e s i n b e t w e e n ) . 18 T h r e e m e t h o d s o f m e a s u r i n g t h e d i s t r i b u t i o n o f g o a l s were e m p l o y e d a n d d a t a f o r a l l t h r e e s h o u l d d e m o n s t r a t e r a n d o m n e s s . (A) r e a l - t i m e o u t c o m e c o u n t s : c o u n t i n g t h e number o f g o a l s s c o r e d i n e a c h o f a s e r i e s o f t i m e - s p a c e s ( e . g . , how many g o a l s s c o r e d e a c h m i n u t e a c r o s s r e a l t i m e ) , (B) f r e q u e n c y c o u n t s : c o u n t i n g t h e number o f g o a l s p e r game. (C) b e t w e e n e v e n t t i m e c o u n t s : c o u n t i n g t h e number o f g o a l s s c o r e d 1, 2 , 3 , e t c . m i n u t e s a p a r t . A . R e s u l t s R e a l - T i m e Outcome C o u n t s T h e f i r s t s e r i e s o f r e a l - t i m e o u t c o m e c o u n t s i n v o l v e d e x a m i n a t i o n o f t o t a l g o a l s f o r a l l 21 t e a m s . T h e f o c u s was on t h e number o f g o a l s s c o r e d d u r i n g e a c h m i n u t e o f p l a y i n g t i m e . A u n i f o r m d i s t r i b u t i o n was t h e m o s t l i k e l y d i s t r i b u t i o n e x p e c t e d s i n c e t eam g o a l s a r e p r e d i c t e d t o o c c u r a t r a n d o m a c r o s s t i m e , a c t i n g l i k e a r a n d o m - n u m b e r g e n e r a t o r ( L a p i n , 1 9 8 4 , p . 1 0 ) . G i v e n e n o u g h g o a l s , a u n i f o r m d i s t r i b u t i o n s h o u l d a r i s e . C h i - s q u a r e a n a l y s i s o f m i n u t e - b y - m i n u t e r e a l - t i m e s c o r i n g s u g g e s t e d t h a t t h e d a t a f i t t e d a u n i f o r m d i s t r i b u t i o n f o r t h e f i r s t a n d s e c o n d p e r i o d , b u t n o t f o r t h e t h i r d . When t h e l a s t m i n u t e o f p l a y was r e m o v e d , h o w e v e r , t h e f i r s t 19 m i n u t e s o f t h e t h i r d p e r i o d c o n f o r m e d t o a u n i f o r m d i s t r i b u t i o n . T h e l a s t m i n u t e o f a h o c k e y game ( p r o b a b l y d u e t o t h e e f f e c t o n g o a l s c o r i n g o f p u l l i n g t h e g o a l t e n d e r ) d o e s n o t a p p e a r t o h a v e t h e same p r o b a b i l i t y o f g o a l s c o r i n g as t h e r e s t o f t h e game. T h e d a t a a r e s u m m a r i z e d i n T a b l e A l ( p a g e s 7 9 ) . F i g u r e 1 (page 19) i l l u s t r a t e s g o a l s i n t h e s e c o n d p e r i o d . A s t r a i g h t , r e l a t i v e l y F i g u r e 1 20 h o r i z o n t a l l i n e c o u l d be drawn through the sawtooth c o n f i g u r a t i o n to demonstrate the a c t u a l uniform nature of the data. The p a t t e r n p r e v a i l e d when two-minute i n t e r v a l s were examined. A f i t with the uniform d i s t r i b u t i o n was found f o r the f i r s t and second p e r i o d , but not the t h i r d . But when the l a s t two minutes of the t h i r d p e r i o d were dropped from the a n a l y s i s , the 18-minute t h i r d p e r i o d f i t t e d a uniform d i s t r i b u t i o n . The data are summarized i n Table A2 (page 80). The data i n v o l v e p o o l i n g of a l l teams s c o r i n g . The l a s t one-minute i n c r e a s e i n s c o r i n g d i d not appear i n p e r i o d 3 when i n d i v i d u a l team data were examined. I t appears goals are scored u n i f o r m l y across time (except f o r the l a s t one or two minutes of the game). T h i s u n i f o r m i t y occurs d e s p i t e the f a c t c e r t a i n l i n e s and p l a y e r s score much more f r e q u e n t l y than o t h e r s . A h i g h - s c o r i n g l i n e or p l a y e r i s e v i d e n t l y not on i c e d u r i n g the same time frame from game to game. The i n t e r v a l between g o a l s f o r home and away games was examined under two c o n d i t i o n s : (a) s t a r t i n g measurement at the s t a r t of each game, thereby c o u n t i n g the time from s t a r t of game to f i r s t g o a l and (b) s t a r t i n g measurement at the time of f i r s t g o a l . Minute-by-minute goal frequency d i s t r i b u t i o n s were c o n s t r u c t e d f o r home and away games. The c o r r e l a t i o n between minute-to-minute g o a l frequency d i s t r i b u t i o n s f o r home games and minute-to-minute g o a l frequency d i s t r i b u t i o n s f o r away games was .965 when measurement began at the s t a r t of each game ( c o n d i t i o n a) and .973 when measurement began at the time of the f i r s t g o a l ( c o n d i t i o n 21 b ) . T h i s s u g g e s t s t h a t f r e q u e n c y o f s c o r i n g i s r e l a t i v e l y c o n s t a n t w h e t h e r a t e a m i s p l a y i n g away o r a t home . D a t a on F i v e - M i n u t e D i v i s i o n s . D a t a f o r i n d i v i d u a l t e ams was e x a m i n e d u s i n g f i v e a n d 10 m i n u t e d i v i s i o n s . The P o i s s o n , u n i f o r m a n d n o r m a l d i s t r i b u t i o n s w e r e t e s t e d f o r g o o d n e s s - o f - f i t . A r a n d o m s a m p l e o f 35 i n d i v i d u a l t e am d i s t r i b u t i o n s o n 1 0 - m i n u t e d i v i s i o n s was s e l e c t e d a n d t e s t e d f o r a f i t w i t h t h e n e g a t i v e b i n o m i a l . When o n l y 1 o f 35 ( 2 . 9 % ) y i e l d e d s i g n i f i c a n t r e s u l t s a t t h e . 0 5 l e v e l o r l e s s , e x a m i n a t i o n o f t h e n e g a t i v e b i n o m i a l was a b a n d o n e d f o r b o t h f i v e - a n d t e n - d i v i s i o n s . The d a t a on f i v e - m i n u t e t i m e d i v i s i o n s s u p p o r t e d t h e a r g u m e n t t h a t g o a l s o f a l l k i n d s a r e d i s t r i b u t e d e v e n l y a c r o s s t i m e . G o o d n e s s - o f - f i t t e s t s r e g a r d i n g t o t a l g o a l s f o r , t o t a l g o a l s a g a i n s t a n d t o t a l g o a l s f o r a n d a g a i n s t c o m b i n e d s h o w e d a c l o s e f i t b e t w e e n o b s e r v e d a n d e x p e c t e d f r e q u e n c i e s b a s e d on t h e u n i f o r m d i s t r i b u t i o n f o r e v e r y t e a m ' s home , away a n d t o t a l g a m e s , 189 o f 189 t e s t s s h o w i n g no s i g n i f i c a n t d i f f e r e n c e b e t w e e n o b s e r v e d a n d e x p e c t e d f r e q u e n c i e s . T o t a l s o n o t h e r p h a s e s o f t h i s s e c t i o n do n o t a l w a y s a d d t o t h e e x p e c t e d number b e c a u s e n was n o t a l w a y s l a r g e e n o u g h t o a c c o m m o d a t e t h e e x p e c t a n c y c o n d i t i o n s f o r c h i - s q u a r e a n a l y s i s . The t o t a l o f 189 i s a t t a i n e d b y m u l t i p l y i n g 3 game c o n d i t i o n s (home, a w a y , t o t a l ) t i m e s 3 g o a l c o n d i t i o n s ( power p l a y , n o n p o w e r p l a y , t o t a l ) t i m e s 21 t e a m s . The d i s t r i b u t i o n o f n o n - p o w e r p l a y g o a l s a l s o a p p e a r s t o be r e l a t i v e l y u n i f o r m f o r home , away a n d t o t a l g a m e s . T o t a l n o n -22 power p l a y goals f o r , a g a i n s t and combined a l l f i t t e d a uniform d i s t r i b u t i o n , 63 of 63 t e s t s showing no s i g n i f i c a n t d i f f e r e n c e between observed and expected f r e q u e n c i e s . Where n allowed a 2 X 12 chi - s q u a r e a n a l y s i s to be made without any c e l l s having f r e q u e n c i e s of l e s s than 5, f i t s f o r non-power p l a y goals f o r and non-power p l a y goals a g a i n s t supported a uniform d i s t r i b u t i o n , 51 of 51 t e s t s showing no s i g n i f i c a n t d i f f e r e n c e between observed and expected f r e q u e n c i e s . Examination of t o t a l power p l a y goals (combining f o r and against) l e d to 21 of 21 t e s t s showing no s i g n i f i c a n t d i f f e r e n c e between observed and expected f r e q u e n c i e s . The n f o r power p l a y goals f o r was o f t e n inadequate f o r each team to conduct a proper Goodness-of-Fit t e s t . Those comparisons t h a t were p o s s i b l e showed th a t 9 of 9 combined f o r and a g a i n s t home power p l a y g o a l s , 6 of 6 f o r and a g a i n s t away, 11 of 11 goals a g a i n s t home and away combined and 4 of 4 power p l a y goals f o r a l l games showed a f i t w i t h the uniform d i s t r i b u t i o n . In t o t a l , 354 of 354 f i t s were made f o r the uniform d i s t r i b u t i o n . Comparison with the Poisson d i s t r i b u t i o n r e v e a l e d t h a t only 12 of 311 comparisons (3.9%) showed no s i g n i f i c a n t d i f f e r e n c e between observed and expected f r e q u e n c i e s , whereas 254 of 311 comparisons (81.7%) showed a chi - s q u a r e d i f f e r e n c e at the .001 l e v e l of s i g n i f i c a n c e . The comparison with the normal d i s t r i b u t i o n was l e s s c l e a r c u t with 172 of 349 comparisons (49.3%) showing no s i g n i f i c a n t d i f f e r e n c e s , while 42, 29, 55 and 51 comparisons were at the .05, .02, .01 and .001 l e v e l s r e s p e c t i v e l y . The c o n c l u s i o n that goals are not normally d i s t r i b u t e d across time i s strengthened by examination of t o t a l game data ( r a t h e r than j u s t home or away). T o t a l game data had 109 of 140 comparisons (77.9%) showing s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s . The data are comprehensively l i s t e d i n Tables A3, A4 and A5 (pages 81-101). Data on Ten-minute D i v i s i o n s . The data on 10-minute d i v i s i o n s p a r a l l e l e d the data on 5-minute d i v i s i o n s : 481 of 483 (99.6%) comparisons between observed and expected f r e q u e n c i e s r e v e a l e d no s i g n i f i c a n t d i f f e r e n c e s f o r the uniform d i s t r i b u t i o n ; 283 of 483 (58.6%) showed no s i g n i f i c a n t d i f f e r e n c e s f o r the Poisson d i s t r i b u t i o n ; and 207 of 483 (42.9%) l e d to no s i g n i f i c a n c e d i f f e r e n c e f o r the Normal d i s t r i b u t i o n . The data are l i s t e d i n Tables A6, A7 and A8 (pages 101-122). R e s u l t s : Event (Goal, Shot) D i s t r i b u t i o n . The d i s t r i b u t i o n s were analyzed i n both a c o n v e n t i o n a l and unconventional manner. The c o n v e n t i o n a l manner i n v o l v e d adding any remaining numbers i n a 'strung out' sequence to the l a s t number and then making a comparison with expected n's f o r the v a r i o u s d i s t r i b u t i o n s . Since e x c e s s i v e p o o l i n g can d e s t r o y the t a i l s of a d i s t r i b u t i o n and make a l l f u n c t i o n s appear uniform. M u l l e t (1977) argued that i t was best to pool j u s t enough to e l i m i n a t e time-spaces with e x p e c t a n c i e s of l e s s than 5. T h i s i s the c o n v e n t i o n a l procedure. The unconventional method, which leaves the t a i l numbers out, may not s a t i s f y the mathematic p u r i s t , but i t has the h e u r i s t i c value 24 of s e e i n g what a d i s t r i b u t i o n might be i f numbers were s u f f i c i e n t l y l a r g e to keep the extreme ends of the t a i l from v i o l a t i n g c h i - s q u a r e ' s need expectancies of 5 or more i n any one c e l l . The d i f f e r e n c e s i n r e s u l t s were not g r e a t whichever method was used. Zar (1984, p. 409) noted t h a t the g o o d n e s s - o f - f i t f o r a Poisson can be t e s t e d by c h i - s q u a r e : the f r e q u e n c i e s i n the t a i l s of the d i s t r i b u t i o n should be pooled to a r r i v e at a t a b u l a t i o n having no expected f r e q u e n c i e s l e s s than 1.0. Zar a l s o argued t h a t with the P o i s s o n i t i s d e s i r a b l e to use a sample d i s t r i b u t i o n with a f a i r l y s m a l l mean, c e r t a i n l y below 10, p r e f e r a b l y below 5, i d e a l l y near 1. T h i s was attempted throughout the a n a l y s i s . B. Frequency D i s t r i b u t i o n The second s e r i e s of r e a l - t i m e outcome counts i n v o l v e d examination of t o t a l g o a l s f o r a l l 21 teams and focussed on the frequency d i s t r i b u t i o n of the number of outcomes (e.g., g o a l s ) . The frequency d i s t r i b u t i o n s of shots and goals were examined f o r t h e i r f i t with the normal, uniform, Poisson, geometric and n e g a t i v e b i n o m i a l d i s t r i b u t i o n s . The data r e v e a l e d t h a t the d i s t r i b u t i o n of t o t a l shots (X2 = 5.087), shots f o r (X2 = 6.330, shots a g a i n s t (X2 = 7.374), t o t a l g o als (X2 = 2.807) and goals f o r (X2 = 2.668) a l l f i t t e d the n e g a t i v e b i n o m i a l d i s t r i b u t i o n . The d i s t r i b u t i o n of t o t a l shots (X* = 3.702) and t o t a l g o als (X2 = 5.932) a l s o f i t t e d a normal d i s t r i b u t i o n and the d i s t r i b u t i o n of g o a l s f o r (X2 = 3.088) a l s o f i t t e d a Poisson d i s t r i b u t i o n . Except f o r t o t a l shots, where the lowest chi-square was f o r the normal 25 d i s t r i b u t i o n , a l l other comparisons y i e l d e d lowest c h i - s q u a r e s (the b e s t f i t ) f o r the n e g a t i v e b i n o m i a l d i s t r i b u t i o n . T h i s agreed with P o l l a r d (1973, 1985, 1987) and Reep et a l . (1971) on the d i s t r i b u t i o n of goals and shots and c o u l d be expected s i n c e the r a t e of outcome probably v a r i e s from game to game due to the d i f f e r e n t l e v e l s of c o m p e t i t i o n f a c e d . F i g u r e 2 (page 26) i l l u s t r a t e s t o t a l g o als per game i n a frequency d i s t r i b u t i o n . The data i n t h i s example f i t both a normal and a n e g a t i v e b i n o m i a l d i s t r i b u t i o n . A l l the data are summarized i n Tables A9 and A10 (pages 123-125). The data may a l s o be c o n s i d e r e d from the p o i n t - o f - v i e w of the mean and the v a r i a n c e s i n c e the v a r i a n c e i s expected to be l e s s than the mean with the uniform d i s t r i b u t i o n , equal to the mean with the Poisson and g r e a t e r than the mean with the negative b i n o m i a l . Means and v a r i a n c e s are l i s t e d i n Table 1. Table 1 Means and V a r i a n c e s f o r Shots Mean Va r i a n c e T o t a l Shots on s c a l e 3.54 0.90 on raw data 60.95 89.83 Shots For on s c a l e 4.28 2.08 on raw data 30.38 52.12 Shots A g a i n s t on s c a l e 4.27 2.10 on raw data 7.49 8.02 26 F i g u r e 2 T o t a l Goals Data f o r A l l Teams F i t t i n g Negative Binomial and Normal D i s t r i b u t i o n 250 -NUMBER OF GAMES 225 -200 175 150 -125 -100 -75 -50 -25 -0 -0*2 3*4 5*6 7*8 9*10 11*12 13*14 15*16 Number of Goals TIME BETWEEN GOALS IN MINUTES 27 The mean-variance data supported a uniform d i s t r i b u t i o n when i t was based on the converted u n i t s c a l e by which d i s t r i b u t i o n s are t e s t e d (1, 2, 3, e t c . ) ; but supported a negative binomial d i s t r i b u t i o n when based on the raw data. The l a t t e r agreement i s probably the more r e a l i s t i c s ince the raw data r e f l e c t the a c t u a l mean/variance whereas the former i s a f u n c t i o n of the numbers used i n s c a l i n g . C. Between Event Time Counts The t h i r d s e r i e s of outcome counts examined the number of outcomes o c c u r r i n g various time spaces apart (e.g., 1 minute apart, 2 minutes a p a r t ) . E i t h e r a Poisson or a negative binomial d i s t r i b u t i o n was expected w i t h the l a t t e r being p a r t i c u l a r l y l i k e l y s ince i t had shown a c l o s e f i t using event time counts. The geometric was not expected to f i t , the chance of a c c i d e n t l y approximating the case where the negative binomial goes over to the geometric (r = 1) not being a n t i c i p a t e d . Team by team examination of the d i s t r i b u t i o n of goals f o r home, away and t o t a l games measured from time of f i r s t goal showed that 60 of 63 (95.2%) comparisons followed a geometric d i s t r i b u t i o n and 63 of 63 (100%) f i t t e d a negative binomial d i s t r i b u t i o n . Two of the three discrepant chi-square r e s u l t s f o r the geometric d i s t r i b u t i o n i n v o l v e d home games. The other was f o r t o t a l games. But the s i z e of chi-square was smaller f o r the geometric f i t than the negative binomial i n 49 of 63 (77.8%) cases: 14 of 21 (66.7%) f o r home games, 20 of 21 (95.2%) f o r away games and 15 of 21 (71.4%) f o r t o t a l games. A f i t f o r the Poisson d i s t r i b u t i o n was 28 found i n 3 of 63 (4.7%) ch i - s q u a r e comparisons. No f i t was found f o r the normal or uniform d i s t r i b u t i o n s . The mean exceeded the v a r i a n c e i n a l l 63 comparisons (home, away and t o t a l games) f o r i n d i v i d u a l teams. The data suggest the geometric d i s t r i b u t i o n i s the most l i k e l y f i t and i s summarized i n Tables A l l and A12 (pages 126-139). F i g u r e 3 (see page 29) i l l u s t r a t e s the r e v e r s e J c o n f i g u r a t i o n of the geometric d i s t r i b u t i o n . The data i n F i g u r e 3 are f o r t o t a l Boston B r u i n goals (as l i s t e d i n Table A l l , p. 117) scored 5, 10, 15, 20 and 25-plus minutes a p a r t . Summary; D i s t r i b u t i o n Data. Since the data f i t the negative b i n o m i a l d i s t r i b u t i o n i n the m a j o r i t y of cases both f o r shots and goa l s and y i e l d e d the lowest c h i - s q u a r e s (and best f i t s ) on a l l but t o t a l shots, the c o n c l u s i o n that these outcomes occur randomly, but at a v a r y i n g r a t e from game to game seems the most f e a s i b l e e x p l a n a t i o n . P o l l a r d et a l . (1977) presented a two argument e x p l a n a t i o n to e x p l a i n the f i t between d i s t r i b u t i o n of goa l s i n hockey and the neg a t i v e b i n o m i a l d i s t r i b u t i o n . In " C o n d i t i o n One", each team has a d i f f e r e n t r a t e of s c o r i n g goals which v a r i e s from game to game, but which remains constant w i t h i n a game." In " C o n d i t i o n Two", each team has the same r a t e of s c o r i n g at the s t a r t of each game, but the r a t e changes, depending on the occurrence of p r e v i o u s g o a l s . A c c o r d i n g to F e l l e r (1943), " C o n d i t i o n One" i s "apparent contagion" and " C o n d i t i o n Two" i s "t r u e contagion." "True contagion" i n v o l v e s the e f f e c t of p r i o r p l a y — each goal i n f l u e n c e s and changes the r a t e of f u r t h e r s c o r i n g as a game pr o g r e s s e s . "Apparent c o n t a g i o n " i s where the TIME BETWEEN GOALS IN MINUTES rate of scoring during a game appears to change, but does not. Both conditions w i l l r e s u l t i n a negative binomial d i s t r i b u t i o n . P o l l a r d et a l . argue that both conditions are operant i n the r e a l -l i f e hockey s i t u a t i o n . In short, each team w i l l have a rate of scoring goals (or taking shots) which varies from game to game and which may vary within each game depending on the pattern of scoring (or shooting) that develops during the game. The present data support t h i s argument f a i r l y well. Explaining why the geometric rather than the negative binomial d i s t r i b u t i o n provides the better f i t i s d i f f i c u l t . Theoretically, the negative binomial was expected to f i t the data (as i t did), but why did the geometric also f i t ? The answer may stem from the fact that the negative binomial d i s t r i b u t i o n i s related to the geometric. Rothschild and Logothetis (1986) note that i f X/, X^ . . .X_3 are independent and i d e n t i c a l l y d i s t r i b u t e d random variables, each having a geometric d i s t r i b u t i o n with parameter p, r £xi w i l l have a negative binomial d i s t r i b u t i o n with parameters p L~l and r. The present analysis may have approximated th i s condition. What i s precisely being measured may also help explain the data. Measuring from goal to goal i s exactly that: the procedure measures the number of goals scored i n a certain time period. The analysis i s based on five-minute i n t e r v a l s . Goals (by either team) were scored on an average of .624 every 5 minutes of real time. The goal per game average of 7.5 goals a game translated to a goal every 8 minutes. But t h i s p a r t i c u l a r measurement procedure began at the time of the f i r s t goal and ended at the time of the 31 l a s t g o a l . The t i m e - s p a c e l o s t c r a m s an a v e r a g e 7 . 5 g o a l s i n t o a b o u t 52 m i n u t e s . T h e e f f e c t may move t h e r e m a i n i n g t i m e d i s t r i b u t i o n t o w a r d t h e g e o m e t r i c . P o l l a r d ' s e x p l a n a t i o n i s i n o p p o s i t i o n t o t h e a s s u m p t i o n t h a t a g o a l o r w i n h a s no a f t e r e f f e c t i n t h e s e n s e t h a t t h e p a s t h i s t o r y p e r m i t s no c o n c l u s i o n s a s t o t h e f u t u r e : a t e a c h a r r i v a l ( g o a l ) t h e p r o c e s s s t a r t s f r o m s c r a t c h as a p r o b a b i l i s t i c r e p l i c a o f t h e w h o l e p r o c e s s . T h e s u c c e s s i v e w a i t i n g t i m e s b e t w e e n a r r i v a l s ( F e l l e r , 1966) must be i n d e p e n d e n t and must h a v e t h e same d i s t r i b u t i o n . I n t h e t e r m i n o l o g y o f s t o c h a s t i c p r o c e s s e s , t h e s e q u e n c e c o n s t i t u t e s a r e n e w a l p r o c e s s . F e l l e r (1950) n o t e s t h a t a s u c c e s s o c c u r s i n a s e q u e n c e o f B e r n o u l l i t r i a l s a n d t h a t t h e w a i t i n g t i m e up t o t h e f i r s t s u c c e s s h a s a g e o m e t r i c d i s t r i b u t i o n ; when t h e f i r s t s u c c e s s o c c u r s , t h e number o f t r i a l s b e t w e e n t h e r t h and t h e ( r + l ) s t s u c c e s s h a s t h e same g e o m e t r i c d i s t r i b u t i o n . S t r e a k D i s t r i b u t i o n S t r e a k s may a l s o be e x a m i n e d f r o m a f r e q u e n c y d i s t r i b u t i o n p o i n t - o f - v i e w ( e . g . , B o s t o n h a d 5 ' s t r e a k s ' o f 1 w i n i n a r o w , 10 s t r e a k s o f 2 , 3 o f 3 , 1 o f 8 and 1 o f 9 w i n s i n a r o w ) . Due t o t h e s m a l l n f o r e a c h t e a m , c u m u l a t i v e t o t a l s f o r w i n n i n g s t r e a k s , l o s i n g s t r e a k s and w i n n i n g / l o s i n g s t r e a k s t a k e n t o g e t h e r were e x a m i n e d f o r t h e i r a g r e e m e n t / d i s a g r e e m e n t w i t h t h e n o r m a l , u n i f o r m , P o i s s o n , g e o m e t r i c a n d n e g a t i v e b i n o m i a l d i s t r i b u t i o n . T h e d i s t r i b u t i o n s were a g a i n a n a l y z e d i n b o t h a c o n v e n t i o n a l a n d u n c o n v e n t i o n a l m a n n e r . 32 R e s u l t s . T h e d a t a f a v o r e d t h e g e o m e t r i c d i s t r i b u t i o n f o r l o s i n g a n d t o t a l s t r e a k s a n d t h e n e g a t i v e b i n o m i a l o r d i s t r i b u t i o n f o r w i n n i n g s t r e a k s . I t i s s u m m a r i z e d i n T a b l e 2 . T a b l e 2 C h i - s q u a r e C o m p a r i s o n o f S t r e a k F r e q u e n c i e s w i t h G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s C o n v e n t i o n a l U n c o n v e n t i o n a l 2 2 W i n n i n g S t r e a k s X d f l s X d f I s G e o m e t r i c 1 1 . 5 1 9 6 n s 2 . 4 1 3 5 n s N e g a t i v e B i n o m i a l 7 . 2 9 6 6 n s 3 . 7 5 6 5 ns L o s i n g S t r e a k s G e o m e t r i c 2 . 6 4 3 4 n s 2 . 2 3 6 4 n s N e g a t i v e B i n o m i a l 8 . 2 1 1 4 n s 7 . 1 8 2 4 n s T o t a l S t r e a k s G e o m e t r i c 7 . 1 5 4 6 n s 1 . 1 1 6 6 n s N e g a t i v e B i n o m i a l 7 . 7 1 1 6 ns 3 . 6 8 6 6 ns C o m p l e t e r e s u l t s may be f o u n d i n T a b l e A13 ( p a g e s 1 4 0 - 1 4 2 ) . Means a n d v a r i a n c e s a r e l i s t e d i n T a b l e 3 . T a b l e 3 Means a n d V a r i a n c e s f o r S t r e a k s C o n v e n t i o n a l U n c o n v e n t i o n a l S t r e a k s Mean V a r i a n c e Mean V a r i a n c e w i n n i n g 2 . 2 1 2 . 5 9 2 . 2 7 3 . 3 1 l o s i n g 1 .84 1 .29 1 .87 1 .58 t o t a l 2 . 0 4 2 . 0 5 2 . 0 7 2 . 5 4 33 T h e m e a n / v a r i a n c e d a t a d i d n o t c l e a r l y s u p p o r t t h e p r e v a l e n c e o f a n y p a r t i c u l a r d i s t r i b u t i o n . A n o t h e r c l u e t o t h e g e o m e t r i c c h a r a c t e r o f t h e d a t a i s f o u n d by l o o k i n g a t ' T h e G a m b l e r ' s F a l l a c y . ' I f a team w i n s s i x i n a r o w , what i s t h e p r o b a b i l i t y o f t h e team w i n n i n g t h e n e x t game, ( a s s u m i n g t h e team p l a y s . 500 h o c k e y ) . T h e c h a n c e o f w i n n i n g s i x s t r a i g h t was . 0 1 5 6 , t h e p r o b a b i l i t y o f s e v e n i n a row was . 0 0 7 8 , b u t o n c e t h e s i x t h s t r a i g h t game was w o n , t h e p r o b a b i l i t y t h e team w i l l w i n t h e 7 t h game i s . 5 0 . A l l teams d o n ' t p l a y a t a .500 a v e r g e , b u t t h e d i s t r i b u t i o n w o u l d s t i l l be g e o m e t r i c i n g e n e r a l d e s i g n s i n c e t h e c h a n c e o f w i n n i n g game X + l w o u l d t h e o r e t i c a l l y be a b o u t t h e same a s i t s w i n n i n g X . A c h e c k w i t h s e v e r a l teams b a s e d o n t h e i r s e a s o n w i n - l o s s - t i e p e r c e n t a g e d i d n o t a p p r e c i a b l y o r s i g n i f i c a n t l y c h a n g e t h e g e o m e t r i c c h a r a c t e r o f t h e d i s t r i b u t i o n o r c h i - s q u a r e . B a c k - t o - B a c k Game D a t a S p o r t s w r i t e r s F i s c h l e r and F i s c h l e r (1986) p r o v i d e d a c o n v e n i e n t s e t o f raw d a t a i n v o l v i n g b a c k - t o - b a c k g a m e s . T h i s i s t h e c l o s e s t t h i n g t o a ' d o u b l e h e a d e r ' i n h o c k e y a n d a l l o w e d some c o m p a r i s o n w i t h G o o d m a n ' s (1969) f i n d i n g s on b a s e b a l l . A l t h o u g h F i s c h l e r a n d F i s c h l e r a d v a n c e d no s c i e n t i f i c p o s t u l a t e s , t h e i r d a t a were c o n v e n i e n t l y a r r a n g e d f o r a n a l y s i s . C h i - s q u a r e c o m p a r i s o n s f o r t h e 21 teams ( 1 9 8 5 - 8 6 NHL s e a s o n ) r e v e a l e d t h a t : a) o n l y 1 o f 21 (4.8%) t e a m s h a d a s i g n i f i c a n t l y d i f f e r e n t ( . 05 l e v e l ) p o i n t s r e c o r d f o r t h e f i r s t n i g h t a s c o m p a r e d t o t h e s e c o n d n i g h t . b) o n l y 1 o f 21 (4.8%) teams h a d a s i g n i f i c a n t l y d i f f e r e n t ( .02 l e v e l ) p o i n t s r e c o r d f o r t h e f i r s t n i g h t as o p p o s e d t o t h e i r o t h e r games p o i n t r e c o r d ( o t h e r game b e i n g a n y game n o t a p a r t o f a b a c k - t o - b a c k s c h e d u l i n g ) . c) 2 o f 21 (9.6%) teams h a d a s i g n i f i c a n t l y d i f f e r e n t ( . 05 and .02 l e v e l ) p o i n t s r e c o r d f o r t h e s e c o n d n i g h t a s o p p o s e d t o t h e i r o t h e r games p o i n t r e c o r d . B o t h teams d i d b e t t e r d u r i n g t h e i r o t h e r ( n o n - b a c k - t o - b a c k ) g a m e s . d) 3 o f 21 (14.3%) teams h a d s i g n i f i c a n t l y d i f f e r e n t ( . 05 l e v e l ) p o i n t s when a l l t h r e e c o n d i t i o n s were c o m p a r e d . T h e d a t a s u g g e s t e d t h a t o u t c o m e s i n b a c k - t o - b a c k games a r e n o t c o m p a r a b l e t o d o u b l e h e a d e r s i n b a s e b a l l . The d a t a a r e s u m m a r i z e d i n T a b l e A14 (page 1 4 3 ) . Summary: A l l D i s t r i b u t i o n T e s t i n g D i s t r i b u t i o n s o f o u t c o m e s ( e . g . , g o a l s , s h o t s ) were c o m p a r e d w i t h s e v e r a l d i f f e r e n t t h e o r e t i c a l d i s t r i b u t i o n s p r e v i o u s l y f o u n d a p p r o p r i a t e — g e o m e t r i c ( E l d e r t o n , 1 9 1 5 , 1 9 2 7 ; E l d e r t o n a n d E l d e r t o n , 1 9 0 9 ) , n e g a t i v e b i n o m i a l ( P o l l a r d , 1 9 7 3 , 1 9 8 5 , 1 9 8 6 ; R e e p e t a l . , 1 9 7 1 ) ; P o i s s o n ( M u l l e t , 1 9 7 7 ) , u n i f o r m ( L a p i n , 1984) — o r s p e c u l a t e d a s p o s s i b l y s i g n i f i c a n t ( t h e n o r m a l d i s t r i b u t i o n was o f t e n a d d e d b e c a u s e o f t h e n o r m a l d i s t r i b u t i o n ' s g e n e r a l p r e v a l e n c e i n a w i d e v a r i e t y o f s i t u a t i o n s ) . 35 T h e d a t a s t r o n g l y s u p p o r t e d p o s t u l a t e l b . T h e p a t t e r n o f o u t c o m e s i n any p e r i o d was t h e same t h e p a t t e r n o f o u t c o m e s f o r a n y o t h e r p e r i o d a n d an e n t i r e game ( e . g . , g o a l s were s c o r e d e v e n l y a c r o s s t h e f i r s t a n d s e c o n d p e r i o d , and a c r o s s t h e f i r s t 18 m i n u t e s o f t h e t h i r d p e r i o d ) . T h e d a t a p a r t i a l l y s u p p o r t e d p o s t u l a t e l c . A l t h o u g h t h e r e was n o t an i n c r e a s e i n t h e r a t e o f g o a l s c o r i n g d u r i n g t h e f i n a l two m i n u t e s o f game t i m e , t h e r e was an i n c r e a s e i n t h e f i n a l m i n u t e . P o s t u l a t e l c may t h e r e f o r e be m o d i f i e d a c c o r d i n g l y . T h e d a t a a p p e a r e d l e s s c l e a r - c u t c o n c e r n i n g p o s t u l a t e l a . When m i n u t e - b y - m i n u t e s e a s o n g o a l s c o r i n g was c o n s i d e r e d f o r a l l 21 t e a m s p o o l e d t o g e t h e r o r f o r i n d i v i d u a l t e a m s , g o a l s were u n i f o r m l y d i s t r i b u t e d a c r o s s t h e f i r s t 59 m i n u t e s o f p l a y i n g t i m e f o r p o o l e d teams a n d a c r o s s t h e e n t i r e 60 m i n u t e s f o r i n d i v i d u a l t e a m s . T h e d i f f e r e n c e was due t o t h e f a c t t h a t an i n d i v i d u a l team d o e s n o t s c o r e e n o u g h g o a l s t o a l l o w d e t e c t i o n o f a l a s t - m i n u t e i n c r e a s e i n s c o r i n g . G r o u p i n g team s c o r e s t o g e t h e r r e v e a l s t h e c h a n g e i n s c o r i n g p a t t e r n . T h e u n i f o r m d i s t r i b u t i o n was e x p e c t e d s i n c e g o a l s were p r e d i c t e d t o o c c u r e v e n l y a c r o s s t i m e (a g o a l was j u s t a s l i k e l y t o o c c u r i n t h e X m i n u t e a s t h e X + 1 m i n u t e ) . I f e n o u g h n u m b e r s ( L a p i n , 1984) a r e g e n e r a t e d , e a c h u n i t s p a c e w i l l be e v e n t u a l l y a b o u t e q u a l l y f i l l e d . When e v e n t d i s t r i b u t i o n ( e . g . , s h o t s o r g o a l s p e r game) was e x a m i n e d t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n p r e v a i l e d , s t r o n g l y s u p p o r t i n g a random d i s p e r s a l o f g o a l s o r s h o t s a c r o s s g a m e s . E x a m i n a t i o n o f t h e d i s t r i b u t i o n o f t h e number o f g o a l s one m i n u t e a p a r t , two m i n u t e s a p a r t , e t c . , y i e l d e d f i t s w i t h b o t h t h e n e g a t i v e b i n o m i a l and t h e g e o m e t r i c d i s t r i b u t i o n . T h i s a l s o a f f i r m e d a random d i s t r i b u t i o n o f e v e n t s , t h e n e g a t i v e b i n o m i a l s u g g e s t i n g a v a r y i n g e v e n t r a t e f r o m game t o game (and p e r h a p s w i t h i n g a m e s ) ; t h e g e o m e t r i c d i s t r i b u t i o n f i t s u g g e s t i n g t h e a r b i t r a r y c h o i c e o f f i v e m i n u t e s and t h e p r o c e d u r e o f c o u n t i n g t i m e b e t w e e n g o a l s a p p r o x i m a t e d t h e c a s e w h e r e r = 1 when t h e n e g a t i v e b i n o m i a l b e c o m e s a g e o m e t r i c d i s t r i b u t i o n . T h i s was p r o b a b l y a l s o a f u n c t i o n o f t r e a t i n g c o n t i n u o u s d a t a a s i f i t were d i s c r e t e d a t a . T h e g e o m e t r i c and n e g a t i v e b i n o m i a l d i s t r i b u t i o n s a l s o were f o u n d t o f i t w i n n i n g , l o s i n g a n d t o t a l s t r e a k s . A g a i n , i t a p p e a r s t h e r a t e o f w i n n i n g o r l o s i n g f r o m g a m e - t o - g a m e was n o t c o n s t a n t . T h i s was p o o l e d d a t a w h i c h may h a v e h a d t h e e f f e c t o f e s t a b l i s h i n g t h e g e o m e t r i c f i t . T h e p o o l o f t eams w o u l d a v e r a g e o u t t o h a v i n g r o u g h l y a 5 0 - 5 0 c h a n c e o f w i n n i n g o r l o s i n g ( t h e t o p t e a m ' s 6 0 - 4 0 b e i n g b a l a n c e d b y t h e b o t t o m t e a m ' s 4 0 - 6 0 ) . S u c h an e f f e c t w o u l d l e a d t o a r o u g h g e o m e t r i c d i s t r i b u t i o n ( e . g . , c h a n c e o f w i n n i n g game 1 = . 5 0 ; c h a n c e o f w i n n i n g game 1 + game 2 = .50 x . 50 = . 2 5 ; c h a n c e o f w i n n i n g games 1 + 2 + 3 = .50 x .50 x . 5 0 = . 1 2 5 ) . The r e l a t i o n s h i p b e t w e e n t h e n e g a t i v e b i n o m i a l a n d g e o m e t r i c s u g g e s t s a s e q u e n t i a l s t r e a k s i t u a t i o n c l o s e l y a p p r o x i m a t i n g t h e s i t u a t i o n w h e r e r = 1 . F i n a l l y , t h e c o m p a r i s o n o f t h e F i s c h l e r d a t a on h o c k e y games s t r o n g l y s u p p o r t s t h e n o t i o n t h a t b a c k - t o - b a c k games a r e n o t 37 s i m i l a r i n o u t c o m e t o b a s e b a l l d o u b l e h e a d e r s , a f i n d i n g t h a t s u g g e s t s t h e u n i t o f m e a s u r e m e n t i n a t h l e t i c c o n t e s t s i s n o t more t h a n 24 h o u r s l o n g . Goodman (1969) f o u n d teams p l a y i n g two games i n one d a y t e n d e d t o w i n o r l o s s b o t h games r a t h e r t h a n w i n o n e , l o s e o n e ; b u t b a s e b a l l t eams were no more l i k e l y t o w i n o r l o s e two i n a row t h a n s p l i t two games when games w e r e p l a y e d one d a y a f t e r a n o t h e r . I I . Runs T e s t i n g T h e s e c o n d s e r i e s o f p o s t u l a t e s was n o t d i r e c t l y t i m e o r i e n t e d . E a c h p r o b e d t h e d a t a s e t f o r t h e e x i s t e n c e o f " r a n d o m n e s s " o f r e s p o n s e . Randomness i n t h e f o l l o w i n g p o s t u l a t e s r e f e r s t o t h e c h a n c e d e p a r t u r e f r o m t h e s t r e a k e x p e c t a n c i e s t h a t may be e x p e c t e d f r o m a team w i n n i n g X % o f i t s games ( e . g . , t h e p a t t e r n o f w i n -l o s s e s f o r a team t h a t won X % o f i t s games w i l l n o t d e v i a t e s i g n i f i c a n t l y i n s e q u e n c e s o f w i n s o r l o s s e s t h a n m i g h t be e x p e c t e d by c h a n c e f o r a team w i n n i n g X o f e v e r y 100 g a m e s ) . 2 a . T h e p a t t e r n o f w i n s a n d l o s s e s f o r e a c h team f o r t h e s e a s o n w i l l be " r a n d o m " . 2 b . I s o l a t i n g e a c h p e r i o d a s i f i t were a s e p a r a t e e n t i t y , t h e p a t t e r n o f f i r s t , s e c o n d and t h i r d p e r i o d o u t c o m e s f o r e a c h team w i l l be " r a n d o m " . ( e . g . , c o m p a r e p e r i o d 1 , game 1; p e r i o d 1 , game 2 . . . p e r i o d 1, game n ) . 2 c . I f any two p e r i o d s a r e c o m b i n e d ( P e r i o d 1 + P e r i o d 2 , P e r i o d 1 + P e r i o d 3 , P e r i o d 2 + P e r i o d 3) a s i f t h e y were a s e p a r a t e game, t h e p a t t e r n o f o u t c o m e s w i l l be " r a n d o m " . 2 d . A s s i g n i n g a " w i n " f o r m o s t , a " l o s s " f o r l e a s t , s h o t s on g o a l p e r game a n d p e r p e r i o d ( e . g . , p e r i o d 1, game 1; p e r i o d 1 , game 2 . . . p e r i o d 1 , game n) f o r e a c h team w i l l be " r a n d o m " . 38 T h e s e p o s t u l a t e s were e x a m i n e d ( f o r home, away a n d t o t a l games) u s i n g t h e W a l d - W o l f o w i t z r u n s t e s t . T h e d a t a were a n a l y z e d f r o m two v i e w p o i n t s : (a) u n d e f e a t e d s t r e a k s - t i e s and w i n s c o n s i d e r e d as w i n (b) " p s y c h o l o g i c a l " s t r e a k s - a t i e c o u n t e d a s a l o s s when a team was a t home; a t i e c o u n t e d a s a w i n when a team was on t h e r o a d . R e s u l t s . T h e d a t a s u p p o r t e d p o s t u l a t e 2 a : w i n / l o s s s t r e a k s w e r e a s e x p e c t e d f o r 101 o f 105 r u n s ( o n l y 4 o f 105 (3.8%) o f r u n s t e s t r e s u l t s were s i g n i f i c a n t a t t h e . 0 5 o r l e s s l e v e l ) . T h e d a t a a r e s u m m a r i z e d i n T a b l e A15 (page 1 4 4 ) . A s G i l o v i c h e t a l . ( 1 9 8 5 , 1989) a r g u e d , s t r e a k s a r e p e r c e p t u a l l y d e c e p t i v e : C a l g a r y h a d w i n s t r e a k s o f 12 a n d 1 3 ; B o s t o n , s t r e a k s o f 8 and 9 ; W a s h i n g t o n , 7 and 8; P i t t s b u r g h , 7 a n d 8 a n d M o n t r e a l , a s t r e a k o f 9 , 3 s t r e a k s o f 6 a n d 3 o f 5 . Y e t n o t one o f t h e s e t e a m ' s p a t t e r n o f w i n s - l o s s e s d e v i a t e d f r o m c h a n c e . C a l g a r y a n d M o n t r e a l , f o r e x a m p l e , s i m p l y won t o o many games o v e r a l l f o r t h e i r s t r e a k s t o be a n y t h i n g o t h e r t h a n c h a n c e o c c u r e n c e s . T h e f o u r r u n s t e s t s t h a t were s t a t i s t i c a l l y s i g n i f i c a n t were f o r t h e more i s o l a t e d c a s e s o f away games ( H a r t f o r d and L o s A n g e l e s ) a n d home games ( C a l g a r y and P h i l a d e l p h i a ) . F i n d i n g 4 r u n s t e s t s s i g n i f i c a n t i n 105 t r i e s i s , o f i t s e l f , n o t b e y o n d t h e r e a l m o f c h a n c e . W i t h 21 teams p l a y i n g a 80 -game s c h e d u l e , t h e o c c a s i o n a l s t r e a k o f 15 s t r a i g h t w o u l d n o t be b e y o n d t h e r e a l m o f c h a n c e f o r a team w i t h many w i n s o r l o s s e s . T h e d a t a a l s o s u p p o r t e d p o s t u l a t e s 2b a n d 2 c . T h e p a t t e r n 39 o f r a n d o m n e s s o f w i n s / l o s s e s p r e v a i l e d w h e t h e r p e r i o d s were t r e a t e d a s s e p a r a t e games o r p a i r e d t o g e t h e r ( p e r i o d s 1 a n d 2; p e r i o d s 1 a n d 3 ; p e r i o d s 2 a n d 3 ) . O n l y 20 o f 378 (5.3%) r u n s t e s t r e s u l t s were s i g n i f i c a n t a t t h e . 0 5 l e v e l : 5 were a t t h e . 0 5 l e v e l , 3 a t t h e . 0 4 , 2 a t t h e . 0 3 , 4 a t t h e . 0 2 , 5 a t t h e .01 and 1 a t t h e . 0 0 0 3 . T h e 20 s i g n i f i c a n t r e s u l t s were s p r e a d a c r o s s t h e s i x a r t i f i c i a l game c o n d i t i o n s ( s e e T a b l e A 1 6 , p a g e s 1 4 5 - 1 4 6 ) : P e r i o d 1 g a m e s : 6 o f 63 (9.5%) P e r i o d 2 g a m e s : 1 o f 63 (1.6%) P e r i o d 3 g a m e s : 4 o f 63 (6.3%) P e r i o d 1 and 2 g a m e s : 2 o f 63 (3.2%) P e r i o d 1 and 3 g a m e s : 2 o f 63 (3.2%) P e r i o d 2 and 3 g a m e s : 5 o f 63 (7.9%) a n d a c r o s s 12 o f t h e 21 t e a m s : 3 o f t h e 18 (16.7%) f o r 2 teams 2 o f t h e 18 (11.1%) f o r 4 teams 1 o f t h e 18 (5.5%) f o r 6 teams T h e d a t a s u p p o r t t h e h y p o t h e s i s t h a t r a n d o m p a t t e r n s p r e v a i l . T h e few r u n s t e s t r e s u l t s t h a t were s i g n i f i c a n t c o u l d be c o n s i d e r e d a s m o s t l i k e l y b e i n g c h a n c e o c c u r r e n c e s , b u t a few m i g h t a l s o be t r u e d e v i a t i o n s f r o m c h a n c e . T h e l a t t e r i n t e r p r e t a t i o n g a i n s some s u p p o r t f r o m t h e f a c t t h a t one t e a m , C a l g a r y , t h e team w i t h t h e b e s t o v e r a l l w i n - l o s s r e c o r d , h a d 2 o f 5 s i g n i f i c a n t r e s u l t s o n o v e r a l l w i n s - l o s s e s a n d 3 o f 18 s i g n i f i c a n t r e s u l t s on t h e a r t i f i c i a l game r u n s t e s t s . T h e p r o b l e m may be s o l v e d by f u t u r e r e s e a r c h . O v e r a number o f s e a s o n s , d o e s t h e b e s t ( o r w o r s t ) team c o n s i s t e n t l y d e v i a t e f r o m t h e p a t t e r n s shown b y t h e o t h e r t e a m s ? 40 T h e d a t a a l s o s u p p o r t e d p o s t u l a t e 2 d : t h e p a t t e r n s o f s h o t d o m i n a n c e were r a n d o m . O n l y 4 o f 84 (4.8%) r u n s t e s t r e s u l t s were s i g n i f i c a n t i n t h e n o n - r a n d o m d i r e c t i o n . T h e d a t a a r e s u m m a r i z e d i n T a b l e A17 (page 1 4 7 ) . S i n c e p o s t u l a t e 2d e x a m i n e d t h e p a t t e r n o f s h o t s o n a " g r o s s " b a s i s ( s e e i n g i f a n y team c o u l d d o m i n a t e i n s h o t s f o r a s k e i n o f p e r i o d s g r e a t e r t h a n w o u l d be e x p e c t e d by c h a n c e ) , a more " m i c r o " a p p r o a c h seemed w a r r a n t e d . T h i s l e d t o p o s t u l a t e 2 e . 2 e . T h e p a t t e r n o f g o a l s f o r and g o a l s a g a i n s t t r a c k e d f r o m p e r i o d t o p e r i o d , game t o game a s a c o n t i n u o u s s t r e a m o f 240 p e r i o d s f o r t o t a l g a m e s , 120 p e r i o d s f o r home o r away g a m e s , w i l l be " r a n d o m " ( w i l l n o t e x c e e d e x p e c t a t i o n s b a s e d on e a c h t e a m ' s w i n - l o s s p e r c e n t a g e ) . e . g . , C a l g a r y Team Run T e s t P e r i o d S c o r i n g T i m e Code Number 1 C a l g a r y 1 5 . 0 5 GF 1 2 D e t r o i t 4 . 2 1 GA 0 3 C a l g a r y 1 0 . 5 6 GF 1 4 D e t r o i t 1 4 . 2 7 GA 0 T h e d a t a s u p p o r t e d p o s t u l a t e 2 e . O n l y 4 o f 63 (6.3%) r u n s t e s t r e s u l t s were s i g n i f i c a n t i n t h e n o n - r a n d o m d i r e c t i o n . T h e r e s u l t s f o r H a r t f o r d ' s home a n d away game s e q u e n c e s were b o t h s i g n i f i c a n t , b u t H a r t f o r d ' s t o t a l game s e q u e n c e was n o t s i g n i f i c a n t . T h e r e i s a r e m o t e p o s s i b i l i t y t h a t t h i s may, l i k e C a l g a r y ' s m o d e s t p r e p o n d e r a n c e o f s i g n i f i c a n c e on r u n s t e s t d a t a , be i m p o r t a n t . F u t u r e r e s e a r c h s h o u l d e a s i l y s o l v e t h e p r o b l e m . O v e r a number o f s e a s o n s , t eams s h o u l d c o n t i n u e t o r e v e a l r andom s e q u e n c e s and t h e number o f teams w i t h home, away o r t o t a l game s e q u e n c e s m a t c h i n g ( l i k e H a r t f o r d ' s ) i n t h e s i g n i f i c a n t d i r e c t i o n 41 s h o u l d be no more t h a n t h e number e x p e c t e d by c h a n c e . The d a t a f o r t h i s s t u d y a r e s u m m a r i z e d i n T a b l e 18 (page 1 4 8 ) . G o a l s a n d s h o t s f o r a n d a g a i n s t f r o m p e r i o d t o p e r i o d , game t o game w e r e a l s o e x a m i n e d o n an a b o v e / b e l o w t h e mean b a s i s . T h i s p r o c e d u r e may a l s o be c a r r i e d o u t by v i e w i n g r u n s b e l o w a n d a b o v e t h e m e d i a n ( p r o b a b l y t h e more f r e q u e n t l y u s e d p r o c e d u r e ) . A r a n d o m c h e c k o f team s h o t a n d g o a l m e d i a n s a n d means r e v e a l e d t h a t t h e y d i d n o t p r o f o u n d l y d i f f e r s o t h e s i m p l e r mean s p l i t p r o c e d u r e was u s e d . T h e a r g u m e n t u n d e r l y i n g t h i s t e s t i n g p r o c e d u r e i s t h a t a l t h o u g h a b o u t as many m e a s u r e m e n t s s h o u l d be a b o v e t h e mean a s b e l o w t h e m e a n , t h e s e q u e n c e n e e d n o t be r a n d o m . The i d e a i s t h a t a h i d d e n s t r e a k o f a b o v e a v e r a g e o r b e l o w a v e r a g e b e h a v i o r may l u r k b e h i n d an o v e r a l l a v e r a g e . A team a v e r a g i n g 30 s h o t s p e r game f o r 80 games c o u l d t h e o r e t i c a l l y h a v e a g a m e - b y - g a m e p a t t e r n w h e r e t h e team was b e l o w t h e i r s e a s o n a v e r a g e f o r t h e i r f i r s t 40 games a n d a b o v e t h e a v e r a g e f o r t h e i r f i n a l 4 0 . I f t h i s were s o , a team w o u l d t h e o r e t i c a l l y p l a y b e l o w a v e r a g e f o r a s i g n i f i c a n t t i m e p e r i o d a n d a b o v e a v e r a g e f o r a s i g n i f i c a n t t i m e p e r i o d . T h i s l e d t o p o s t u l a t e 2 f . 2 f . O u t c o m e s ( e . g . , s h o t s , g o a l s ) f o r a n d a g a i n s t w i l l be d i s t r i b u t e d i n r a n d o m s e q u e n c e a b o v e a n d b e l o w t h e m e a n . T h e d a t a s u p p o r t e d a r a n d o m i n t e r p r e t a t i o n o f t h e p a t t e r n s i n v o l v i n g g o a l s and s h o t s , f o r a n d a g a i n s t , f o r home, away and t o t a l g a m e s . O n l y 16 o f 273 (5.9%) r u n s t e s t r e s u l t s were s i g n i f i c a n t i n t h e n o n - r a n d o m d i r e c t i o n . T h e d a t a a r e i n T a b l e s A19 (page 149) and A20 (page 1 5 0 ) . P o s t u l a t e 2g e x a m i n e d t h e e f f e c t o f i n c r e a s i n g s c a l e . The f o u r - g a m e s e t s e m p l o y e d i n t h e t e s t o f s t a t i o n a r i t y s e c t i o n o f t h e s t u d y were t r e a t e d a s s i n g l e games (4 p o i n t s o r more e q u a l l i n g a ' w i n ' ; 3 p o i n t s o r l e s s , a ' l o s s ' ) . 2 g . F o u r - g a m e s e t s t r e a t e d a s s i n g l e games w i l l show a r a n d o m p a t t e r n o f ' w i n s ' and ' l o s s e s ' . None o f t h e 21 r u n s t e s t s were s i g n i f i c a n t . L a g g e d c o r r e l a t i o n s o f s e t 1 t o s e t 2 , s e t 2 t o s e t 3 , e t c . , a l s o s u g g e s t e d a random r e l a t i o n s h i p b e t w e e n s e t s . C o r r e l a t i o n s b e t w e e n s e t s r a n g e d f r o m - . 3 5 7 t o . 3 2 1 . T h e d a t a a r e i n T a b l e A21 (page 1 5 1 ) . Summary: A l l Runs T e s t i n g T h e r u n s t e s t a n a l y s i s o v e r w h e l m i n g l y s u p p o r t e d a l l f o u r p o s t u l a t e s : s t r e a k s o f w i n s o r l o s s e s do n o t e x c e e d c h a n c e e x p e c t a t i o n s ( b a s e d on a t e a m ' s o v e r - a l l r e c o r d ) . T h i s h e l d f o r p e r i o d s , c o m b i n a t i o n s o f p e r i o d s , games a n d c o m b i n a t i o n s o f g a m e s , f o r g o a l s a n d s h o t s , w h e t h e r p e r i o d s were i s o l a t e d a s d i s c r e t e e n t i t i e s o r o u t c o m e s f o r a n d a g a i n s t were f o l l o w e d s t r a i g h t t h r o u g h g a m e - a f t e r - g a m e , w h e t h e r o n l y home, away o r t o t a l games w e r e e x a m i n e d . No m a t t e r how t h e d a t a were a n a l y z e d ( e . g . , l e n g t h o f r u n s a b o v e o r b e l o w mean o r l e n g t h o f c o n s e c u t i v e w i n s o r l o s s e s ) r a n d o m n e s s a c r o s s t i m e p r e v a i l e d . I I I . C o n d i t i o n a l P r o b a b i l i t i e s C l o s e l y a l l i e d w i t h r u n s t e s t i n g i s t h e e x a m i n i n g o f c o n d i t i o n a l p r o b a b i l i t i e s . T h e l a t t e r p r o c e d u r e was a l s o a d o p t e d d i r e c t l y f r o m G i l o v i c h , V a l l o n e and T v e r s k y (1985) who s u g g e s t e d 43 t h a t d a t a o n t h e p r o b a b i l i t y o f a w i n a f t e r a w i n , a w i n a f t e r two w i n s , e t c . w o u l d be u s e f u l i n e x a m i n i n g s t r e a k b e h a v i o r . 3 a . T h e p r o b a b i l i t y o f a w i n a f t e r 1 w i n , 2 w i n s o r 3 w i n s w i l l be t h e same a s a t e a m ' s w i n / l o s s p e r c e n t a g e . 3 b . T h e p r o b a b i l i t y o f a l o s s a f t e r 1 l o s s , 2 l o s s e s o r 3 l o s s e s w i l l be t h e same a s a t e a m ' s w i n / l o s s p e r c e n t a g e . 3 c . T h e p r o b a b i l i t y o f a t i e a f t e r a t i e w i l l be t h e t h e same a s a t e a m ' s t i e p e r c e n t a g e . R e s u l t s . T h e d a t a o n c o n d i t i o n a l p r o b a b i l i t i e s f o r t h e 1 9 8 8 - 8 9 NHL s e a s o n s u p p o r t e d t h e c o n c e p t o f r a n d o m n e s s i n g e n e r a l — o u t c o m e s o f games a p p e a r t o be i n d e p e n d e n t f r o m one game t o t h e n e x t . T h e p r o b a b i l i t y o f s e q u e n t i a l w i n s e x c e e d e d a t e a m ' s w i n n i n g p e r c e n t a g e on 16 o f 62 (25.8%) c o m p a r i s o n s ( t h e r e was one t i e ) . T h e i n d i v i d u a l p e r c e n t a g e s were 2 o f 21 (9.5%) f o r a w i n f o l l o w i n g a w i n , 3 o f 20 (15.0%) f o r a w i n f o l l o w i n g 2 w i n s , a n d 11 o f 21 (52.4%) f o r a w i n f o l l o w i n g 3 w i n s . T h e l a t t e r w o u l d be o f i n t e r e s t i f i t o c c u r r e d o v e r a number o f s e a s o n s . F i n a l l y , t h e a v e r a g e s e q u e n t i a l w i n s a n d l o s s e s f o r a l l t eams h o v e r e d a r o u n d a 5 0 - 5 0 s p l i t . T h e p e r c e n t a g e s a r e d e t a i l e d i n T a b l e 4 . C o m p a r i n g a t e a m ' s s e a s o n p e r c e n t a g e o f t i e s w i t h i t s s e q u e n t i a l t i e p e r c e n t a g e s h o w e d t h a t t h e s e a s o n p e r c e n t a g e was h i g h e r t h a n t h e s e q u e n t i a l p e r c e n t a g e i n 11 o f 20 (55%) o f c o m p a r i s o n s . O v e r a l l , t h e r e were 97 t i e s i n 840 games ( 1 1 . 5 % ) . T h i s c o m p a r e d c l o s e l y w i t h a c h a n c e o f a t i e f o l l o w i n g a t i e w h i c h was 1 1 . 9 % . T h i s s u p p o r t e d p o s t u l a t e 3 c . T h e d a t a a r e s u m m a r i z e d i n T a b l e A22 ( p a g e s 1 5 2 - 1 5 3 ) . 44 T a b l e 4 E m p i r i c a l P r o b a b i l i t i e s o f R e p e a t i n g a Win o r L o s s C o n d i t i o n e d on t h e Outcome o f P r e v i o u s Game(s) F o r a l l Teams C o m b i n e d P e t . W i n s n 1 w i n a f t e r a w i n . 439 322 733 2 w i n s a f t e r a w i n . 465 147 316 3 w i n s a f t e r a w i n . 556 80 144 T o t a l . 460 549 1193 P e t . L o s s e s n 1 l o s s a f t e r a l o s s .452 331 733 2 l o s s e s a f t e r a l o s s .491 160 326 3 l o s s e s a f t e r a l o s s . 4 6 5 73 157 T o t a l .464 564 1216 Summary: C o n d i t i o n a l P r o b a b i l i t i e s T h e a n a l y s i s o f c o n d i t i o n a l p r o b a b i l i t i e s s u p p o r t e d t h e c o n c e p t o f i n d e p e n d e n t g a m e s : t h e o u t c o m e o f game X i s n o t a s s o c i a t e d w i t h t h e o u t c o m e o f game X + 1 . F o r t h e w h o l e l e a g u e , a l o s s i s a s l i k e l y t o f o l l o w a w i n a s a w i n . A w i n i s a s l i k e l y t o f o l l o w a l o s s a s a l o s s . F o r i n d i v i d u a l t e a m s , t h e l i k e l i h o o d o f a w i n f o l l o w i n g a w i n , a l o s s f o l l o w i n g a l o s s i s a f u n c t i o n o f e a c h t e a m ' s w i n - l o s s p e r c e n t a g e . • I V . C o r r e l a t i o n a l D a t a T h e f o u r t h s e r i e s o f p o s t u l a t e s p r o b e d t h e c o r r e l a t i o n a l r e l a t i o n s h i p b e t w e e n o u t c o m e s f r o m one p e r i o d t o t h e n e x t . 4 . T h e c o r r e l a t i o n b e t w e e n o u t c o m e s i n a n y one p e r i o d w i l l n o t be c o r r e l a t e d w i t h o u t c o m e s i n a n y o t h e r p e r i o d o f t h e same game. T h i s i n c l u d e s : 45 p e r i o d 1 , game 1 . . 8 0 v s . p e r i o d 2 , game 1 . . 8 0 p e r i o d 1 , game 1 . . 8 0 v s . p e r i o d 3 , game 1 . . 8 0 p e r i o d 2 , game 1 . . 8 0 v s . p e r i o d 3 , game 1 . . 8 0 a n d i n v o l v e s p o s t u l a t e s c o n c e r n i n g t h e o u t c o m e s : 4 a . w i n s - t i e s / l o s s e s 4 b . g o a l s f o r 4 c . g o a l s a g a i n s t 4 d . s h o t s f o r 4 e . s h o t s a g a i n s t 4 f . t o t a l g o a l s 4 g . t o t a l s h o t s 4 h . g o a l d i f f e r e n c e 4 i . s h o t d i f f e r e n c e A f i n a l s e r i e s o f w i t h i n game c o r r e l a t i o n s i n v o l v e d t h e c o r r e l a t i o n b e t w e e n e a c h p e r i o d o u t c o m e a n d game o u t c o m e . No h y p o t h e s i s was a d v a n c e d . R e s u l t s . T h e d a t a s u p p o r t e d p o s t u l a t e s 4a t h r o u g h 4 i : 4 a . F o r t h e 21 t e a m s , i n d i v i d u a l c o r r e l a t i o n s b e t w e e n p e r i o d s f o r w i n s - t i e s / l o s s e s r a n g e d f r o m .150 t o - . 2 3 2 f o r p e r i o d 1 v s . p e r i o d 2 . 3 9 3 t o - . 1 1 1 f o r p e r i o d 1 v s . p e r i o d 3 . 323 t o - . 1 4 9 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s f o r t h e s e t h r e e c o n d i t i o n s were - . 0 3 4 , . 0 8 4 a n d . 0 6 0 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A23 (page 154) . 4 b . F o r t h e 21 t e a m s , i n d i v i d u a l c o r r e l a t i o n s b e t w e e n p e r i o d s o n t h e b a s i s o f g o a l s f o r r a n g e d f r o m .222 t o - . 1 7 2 f o r p e r i o d 1 v s . p e r i o d 2 .257 t o - . 3 4 3 f o r p e r i o d 1 v s . p e r i o d 3 . 204 t o - . 2 2 6 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were . 0 0 9 , - . 0 0 2 and . 0 7 5 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A24 ( p a g e s 1 5 5 - 1 5 6 ) . 4 c . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s on t h e b a s i s o f g o a l s a g a i n s t r a n g e d f r o m .204 t o - . 1 9 9 f o r p e r i o d 1 v s . p e r i o d 2 . 2 3 9 t o - . 1 3 4 f o r p e r i o d 1 v s . p e r i o d 3 . 2 7 5 t o - . 1 4 6 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were . 0 1 4 , . 0 3 5 and . 0 7 3 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A24 ( p a g e s 1 5 5 - 1 5 6 ) . 4 d . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s o n t h e b a s i s o f s h o t s f o r r a n g e d f r o m .324 t o - . 0 6 6 f o r p e r i o d 1 v s . p e r i o d 2 . 176 t o - . 1 4 3 f o r p e r i o d 1 v s . p e r i o d 3 . 2 8 3 t o - . 1 2 3 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were . 1 1 8 , .034 a n d . 1 2 4 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A24 ( p a g e s 1 5 5 - 1 5 6 ) . 4 e . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s o n t h e b a s i s o f s h o t s a g a i n s t r a n g e d f r o m .286 t o - . 2 6 2 f o r p e r i o d 1 v s . p e r i o d 2 . 260 t o - . 1 1 7 f o r p e r i o d 1 v s . p e r i o d 3 . 246 t o - . 1 3 5 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s w e r e . 0 7 7 , .054 a n d . 1 0 6 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A24 ( p a g e s 1 5 5 - 1 5 6 ) . 4 f . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s o n t h e b a s i s o f t o t a l g o a l s r a n g e d f r o m .268 t o - . 1 7 7 f o r p e r i o d 1 v s . p e r i o d 2 .261 t o - . 1 2 0 f o r p e r i o d 1 v s . p e r i o d 3 . 1 8 9 t o - . 1 9 3 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were . 0 5 7 , .052 a n d . 0 1 3 . The d a t a a r e s u m m a r i z e d i n T a b l e A25 (page 1 5 7 ) . 4 g . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s o n t h e b a s i s o f t o t a l s h o t s r a n g e d f r o m . 269 t o - . 1 0 7 f o r p e r i o d 1 v s . p e r i o d 2 . 278 t o - . 1 2 7 f o r p e r i o d 1 v s . p e r i o d 3 .297 t o . 019 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were . 0 9 8 , .087 a n d . 1 4 1 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A25 (page 1 5 7 ) . 4 h . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s on t h e b a s i s o f g o a l d i f f e r e n c e r a n g e d f r o m . 1 6 9 t o - . 2 6 7 f o r p e r i o d 1 v s . p e r i o d 2 . 337 t o - . 1 8 3 f o r p e r i o d 1 v s . p e r i o d 3 . 302 t o - . 1 5 6 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s w e r e - . 0 3 6 , .061 and . 0 5 1 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A26 (page 1 5 8 ) . 4 i . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s on t h e b a s i s o f s h o t d i f f e r e n c e r a n g e d f r o m .372 t o - . 1 5 6 f o r p e r i o d 1 v s . p e r i o d 2 . 167 t o - . 2 0 6 f o r p e r i o d 1 v s . p e r i o d 3 . 3 5 9 t o - . 1 8 2 f o r p e r i o d 2 v s . p e r i o d 3 A v e r a g e c o r r e l a t i o n s were .101 , ' . 0 1 5 and . 1 0 0 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A27 (page 1 5 9 ) . Summary: C o r r e l a t i o n a l D a t a T h e 567 p e r i o d - t o - p e r i o d w i t h i n game c o r r e l a t i o n s f o r i n d i v i d u a l teams r a n g e d f r o m . 3 9 3 t o - . 3 4 3 . T h i s s t r o n g l y s u g g e s t e d t h a t p e r i o d o u t c o m e s a r e r e l a t i v e l y i n d e p e n d e n t o f one a n o t h e r . T h e o v e r a l l c o r r e l a t i o n s , b a s e d o n t o t a l s f o r 21 t e a m s , r a n g e d f r o m .141 t o - . 0 3 4 . T h e d a t a a r e s u m m a r i z e d i n T a b l e A28 (page 1 6 0 ) . T h e c o r r e l a t i o n a l d a t a c o m p a r i n g i n d i v i d u a l p e r i o d s w i t h game t o t a l s on w i n - t i e / l o s s e s , g o a l d i f f e r e n c e and s h o t d i f f e r e n c e r a n g e d f r o m . 2 1 5 t o .811 f o r i n d i v i d u a l t e a m s . T h e o v e r a l l 48 c o r r e l a t i o n s were a s f o l l o w s : P e r i o d 1 P e r i o d 2 P e r i o d 3 T o t a l g o a l d i f f e r e n c e . 5 3 5 .593 .646 T o t a l s h o t d i f f e r e n c e .584 .652 .616 T h e g e n e r a l t r e n d s u g g e s t e d e a c h p e r i o d ' s o u t c o m e was m o d e r a t e l y c o r r e l a t e d w i t h game o u t c o m e . T h i s s h o u l d be e x p e c t e d s i n c e e a c h p e r i o d r e p r e s e n t e d 33% o f t h e t o t a l a g a i n s t w h i c h i t was b e i n g c o r r e l a t e d . T h e d a t a a r e s u m m a r i z e d i n T a b l e s A 2 3 , A26 a n d A27 ( p a g e s 1 5 4 , 158 a n d 1 5 9 ) . Game t o Game C o r r e l a t i o n s L a g g e d c o r r e l a t i o n a l c o m p a r i s o n s were a l s o made b e t w e e n p e r i o d o u t c o m e s f r o m one game t o t h e t h e n e x t ( e . g . , p e r i o d 1 , game 1 t o p e r i o d 1 , game 2 ) . S e v e n p o s t u l a t e s , i n v o l v i n g o u t c o m e s (5a) w i n -t i e / l o s s , (5b) g o a l s f o r , (5c) g o a l s a g a i n s t , (5d) t o t a l g o a l s , (5e) s h o t s f o r , ( 5 f ) s h o t s a g a i n s t a n d (5g) t o t a l s h o t s a r g u e d t h a t : T h e c o r r e l a t i o n b e t w e e n an o u t c o m e i n p e r i o d 1 , 2 , 3 o r an e n t i r e game w i l l n o t be c o r r e l a t e d w i t h i t s c o u n t e r p a r t o u t c o m e i n t h e f o l l o w i n g game. R e s u l t s . T h e g a m e - t o - g a m e c o m p a r i s o n s w e r e e x c e p t i o n a l l y s i m i l a r t o t h e w i t h i n - g a m e c o m p a r i s o n s : 5 a . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f " w i n - t i e / l o s s " r a n g e d f r o m . 1 4 2 t o - . 3 3 1 f o r p e r i o d 1, game i v s . p e r i o d 1, game i+1 . 1 8 5 t o - . 2 4 8 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 . 248 t o - . 2 9 3 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e c o r r e l a t i o n s f o r game t o game w i n - t i e / l o s s o u t c o m e s f o r i n d i v i d u a l t e a m ' s t o t a l games r a n g e d f r o m .167 t o - . 2 0 0 . 49 T h e d a t a a r e s u m m a r i z e d i n T a b l e A29 (page 1 6 1 ) . 5 b . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f g o a l s f o r r a n g e d f r o m .046 t o - . 1 8 9 f o r p e r i o d 1 , game i v s . p e r i o d 1, game i+1 . 326 t o - . 2 7 8 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 .217 t o - . 2 3 6 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A30 (page 1 6 2 ) . 5 c . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f g o a l s a g a i n s t r a n g e d f r o m .298 t o - . 1 8 7 f o r p e r i o d 1, game i v s . p e r i o d 1, game i+1 . 1 1 5 t o - . 3 0 7 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 . 190 t o - . 2 0 0 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A30 (page 1 6 2 ) . 5 d . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f t o t a l g o a l s r a n g e d f r o m .176 t o - . 2 4 5 f o r p e r i o d 1, game i v s . p e r i o d 1 , game i+1 . 206 t o - . 2 5 4 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 .178 t o - . 3 2 5 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A30 (page 1 6 2 ) . 5 e . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game on t h e b a s i s . o f s h o t s f o r r a n g e d f r o m .187 t o - . 2 2 7 f o r p e r i o d 1 , game i v s . p e r i o d 1 , game i+1 .198 t o - . 1 2 7 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 . 156 t o - . 1 3 5 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A31 (page 1 6 3 ) . 5 f . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f s h o t s a g a i n s t r a n g e d f r o m . 2 3 5 t o - . 1 5 3 f o r p e r i o d 1 , game i v s . p e r i o d 1 , game i+1 .157 t o - . 1 7 7 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 .154 t o - . 1 5 5 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A31 (page 1 6 3 ) . 50 5 g . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d s f r o m game t o game o n t h e b a s i s o f t o t a l s h o t s r a n g e d f r o m . 1 6 3 t o - . 1 8 6 f o r p e r i o d 1, game i v s . p e r i o d 1, game i+1 .202 t o - . 2 3 5 f o r p e r i o d 2 , game i v s . p e r i o d 2 , game i+1 . 180 t o - . 1 7 6 f o r p e r i o d 3 , game i v s . p e r i o d 3 , game i+1 T h e d a t a a r e i n T a b l e A31 (page 1 6 3 ) . Summary: G a m e - t o - G a m e C o r r e l a t i o n s . O u t c o m e - t o - o u t c o m e c o r r e l a t i o n s f r o m game t o game r a n g e d f r o m .326 t o - . 3 3 1 , a g a i n s h o w i n g o n l y a m o d e s t r e l a t i o n s h i p b e t w e e n m e a s u r e m e n t s . The c o r r e l a t i o n s c o m p a r i n g an o u t c o m e f r o m one game t o a n o t h e r were w i t h i n t h e same r a n g e a s c o r r e l a t i o n s b e t w e e n an o u t c o m e i n one p e r i o d a n d an o u t c o m e i n a n o t h e r p e r i o d i n t h e same game. I n o t h e r w o r d s , a c t i v i t y i n p e r i o d X , game 1 seems no more o r l e s s r e l a t e d t o a c t i v i t y i n p e r i o d X + l , game 1 t h a n i t i s t o a c t i v i t y i n p e r i o d X , game 2 . T h e maximum r a n g e f o r a l l 1029 c o r r e l a t i o n s was o n l y . 3 9 3 t o - . 3 4 3 , a s p r e a d t h a t c o u l d p o s s i b l y s u p p o r t t h e c o n c e p t o f i n d e p e n d e n c e o f p e r i o d t o p e r i o d a n d game t o game o u t c o m e s . C o r r e l a t i o n : P e r i o d Outcome t o Game O u t c o m e . A n o t h e r s e t o f c o r r e l a t i o n s e x a m i n e d t h e r e l a t i o n s h i p b e t w e e n p e r i o d o u t c o m e and game o u t c o m e . S i n c e a p e r i o d i s o n e - t h i r d o f a game, t h e r e s u l t s o f p l a y i n a n y g i v e n p e r i o d h a s t h e p o t e n t i a l o f c o n t r i b u t i n g c o n s i d e r a b l y t o game o u t c o m e . T h e b a s i c p o s t u l a t e was t h e r e f o r e : 5 h . P e r i o d o u t c o m e s h o u l d be p o s i t i v e l y c o r r e l a t e d w i t h game o u t c o m e . R e s u l t s . T h e d a t a , i n g e n e r a l , s u p p o r t e d p o s t u l a t e 5 h . 5 h . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d o u t c o m e and game o u t c o m e on t h e b a s i s o f w i n s - t i e s / l o s s e s r a n g e d f r o m 51 . 2 1 5 t o .624 f o r p e r i o d 1 a n d game o u t c o m e .306 t o .668 f o r p e r i o d 2 and game o u t c o m e .328 t o .672 f o r p e r i o d 3 and game o u t c o m e T h e d a t a a r e s u m m a r i z e d i n T a b l e A23 (page 1 5 4 ) . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d o u t c o m e and game o u t c o m e on t h e b a s i s o f g o a l d i f f e r e n c e s r a n g e d f r o m .408 t o . 662 f o r p e r i o d 1 and game o u t c o m e . 4 2 3 t o .718 f o r p e r i o d 2 and game o u t c o m e .496 t o . 7 7 5 f o r p e r i o d 3 and game o u t c o m e T h e o v e r a l l c o r r e l a t i o n s f o r a l l teams w e r e . 5 3 5 , . 5 9 3 a n d . 6 4 6 . T h e d a t a a r e i n T a b l e A26 (page 1 5 8 ) . F o r t h e 21 t e a m s , c o r r e l a t i o n s b e t w e e n p e r i o d o u t c o m e and game o u t c o m e on t h e b a s i s o f s h o t d i f f e r e n c e r a n g e d f r o m .429 t o . 723 f o r p e r i o d 1 and game o u t c o m e .518 t o .811 f o r p e r i o d 2 and game o u t c o m e .426 t o . 709 f o r p e r i o d 3 a n d game o u t c o m e T h e o v e r a l l c o r r e l a t i o n s f o r a l l t eams were . 5 8 4 , .642 and . 6 1 6 . T h e d a t a a r e i n T a b l e A27 (page 1 5 9 ) . Summary: P e r i o d t o Game O u t c o m e . A n a l y s i s o f t h e c o r r e l a t i o n s b e t w e e n p e r i o d a n d game o u t c o m e f o r w i n s - t i e s / l o s s e s , g o a l d i f f e r e n c e s and s h o t d i f f e r e n c e s r e v e a l e d a d e f i n i t e c o r r e l a t i o n b e t w e e n p e r i o d and game o u t c o m e . T h i s s u g g e s t s a h o c k e y game c l o s e l y r e s e m b l e s t h e i d e a l s e l e c t i o n t e s t . I t e m s ( i n h o c k e y ' s c a s e , p e r i o d s ) a r e n o t c o r r e l a t e d w i t h e a c h o t h e r (a b a s i c r e q u i r e m e n t f o r i t e m c h o i c e ) , b u t i t e m s ( p e r i o d o u t c o m e s ) a r e c o r r e l a t e d f a i r l y w e l l w i t h f i n a l s c o r e (game o u t c o m e ) . T h i s ( A n a s t a s i , 1954) i s t h e " p e r f e c t " t e s t made o f i n d e p e n d e n t i t e m s , e a c h c o r r e l a t e d w i t h f i n a l s c o r e , b u t n o t w i t h e a c h o t h e r . V I . Raw and B i n a r y C o r r e l a t i o n s E a r l i e r o u t c o m e r e s e a r c h was b a s e d on c o m p a r i n g t h e t o t a l o v e r a l l d i f f e r e n c e b e t w e e n t h e o u t c o m e f o r one team and a n o t h e r d u r i n g one p e r i o d o f p l a y : i f team X o u t s h o o t s team Y , t h e n team X i s g i v e n a ' w i n . ' T h e c o r r e l a t i o n i s a s i m p l e o n e - s t e p l a g c o m p a r i s o n w i t h ' o u t c o m e ' i n game Z b e i n g c o m p a r e d w i t h ' o u t c o m e ' i n game Z + l . Runs t e s t d a t a were b a s e d on t h i s p r o c e d u r e ( s e e p a g e 3 7 ) . Two p r o c e d u r e s were e m p l o y e d t o d e t e r m i n e p e r i o d - t o - p e r i o d , g a m e - t o - g a m e c o r r e l a t i o n s . T h e f i r s t u s e d t h e o r i g i n a l raw m a t e r i a l i n a s i m p l e o n e - s t e p l a g c o m p a r i s o n . T h i s meant t h e 2 g o a l s o r 9 s h o t s t a k e n i n t i m e - s p a c e W by team X w o u l d be c o m p a r e d w i t h t h e 1 g o a l o r 11 s h o t s t a k e n by team X i n t i m e - s p a c e W+l . T h e s e c o n d p r o c e d u r e i n v o l v e d c o n v e r t i n g d a t a t o a b i n a r y s y s t e m : i f t h e e v e n t a p p e a r e d i n t h e t i m e - s p a c e , a ' 1 ' was a w a r d e d ; i f n o t , a ' 0 * was a w a r d e d ( e . g . , no g o a l i n p e r i o d = 0 , 1 o r more g o a l s i n p e r i o d = 1 ) . L i k e t h e f i r s t p r o c e d u r e , t h i s d o e s n o t i n v o l v e a c o m p a r i s o n w i t h t h e o p p o s i n g t e a m . T h e p r o c e d u r e t r a n s f o r m i n g s h o t s i n t o a b i n a r y c o d e i n v o l v e d a s p l i t a b o v e o r b e l o w t h e mean o f t h a t t e a m ' s s h o t s . T h e two a n a l y s e s c o m p a r e d p e r i o d 1 , game 1 w i t h p e r i o d 2 , game 1 d a t a . The e x p e c t a n c y i s t h a t p e r i o d - t o - p e r i o d , g a m e - t o - g a m e r e s u l t s , w h e t h e r o n raw o r b i n a r y d a t a , w i l l n o t be c o r r e l a t e d . T h e p o s t u l a t e s e x a m i n e d w e r e : T h e p e r i o d - t o - p e r i o d , g a m e - t o - g a m e c o r r e l a t i o n s o f raw and b i n a r y c o n v e r t e d s c o r e s w i l l be a p p r o x i m a t e l y z e r o . T h i s w i l l h o l d f o r home, away a n d t o t a l games f o r : 53 6 a . g o a l s f o r 6 b . g o a l s a g a i n s t 6 c . s h o t s f o r 6 d . s h o t s a g a i n s t R e s u l t s . T h e d a t a s u p p o r t e d t h e f o u r p o s t u l a t e s : t h e c o r r e l a t i o n s were v e r y l o w a n d t h e r a n g e c e n t e r e d a r o u n d z e r o . A summary o f t h e r a n g e o f c o r r e l a t i o n s o s d e t a i l e d i n T a b l e 5 ( s e e b e l o w ) . T h e d a t a c o n c e r n i n g g o a l s a r e s u m m a r i z e d i n T a b l e A32 ( p a g e s 1 6 4 - 1 6 5 ) a n d c o n c e r n i n g s h o t s i n T a b l e A33 ( p a g e s 1 6 6 - 1 6 7 ) . F o r g e n e r a l i n f o r m a t i o n , c o r r e l a t i o n s b e t w e e n v a r i o u s a t t r i b u t e s ( e . g . , g o a l s f o r a n d s h o t s f o r ) were c a l c u l a t e d u s i n g raw s c o r e s . T h e r a n g e o f i n d i v i d u a l team c o r r e l a t i o n s i s s u m m a r i z e d i n T a b l e 6 (page 5 4 ) . T a b l e 5 Range o f Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r G o a l s and S h o t s ( P 1 G 1 . . . P 2 G 1 . . . P 3 G n ) G o a l s F o r Home Away T o t a l Low H i g h Low H i g h Low H i g h Raw - . 1 9 8 . 1 1 3 - . 1 2 2 . 183 - . 1 3 4 .142 B i n - . 1 3 7 .214 - . 0 9 2 . 219 - . 1 2 1 .161 G o a l s A g a i n s t Raw - . 1 9 1 . 2 1 3 - . 1 9 6 .070 - . 0 9 2 .090 B i n - . 2 0 2 . 1 9 9 - . 1 6 0 . 093 - . 1 2 0 .151 S h o t s F o r Raw - . 0 6 5 . 2 1 5 - . 1 6 2 .197 - . 1 0 7 .176 B i n .147 .157 - . 1 2 9 .211 - . 1 6 3 .277 S h o t s A g a i n s t Raw -.144 . 180 - . 0 0 6 . 1 7 5 - . 1 0 4 .197 B i n - . 1 4 4 .142 - . 1 3 1 . 159 - . 0 4 7 .127 54 T h e d a t a show t h a t g o a l s f o r a n d s h o t s f o r , a s w e l l as g o a l s a g a i n s t and s h o t s a g a i n s t , were m o d e s t l y c o r r e l a t e d a s m i g h t be T a b l e 6 Range o f I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s a n d G o a l s F o r a n d A g a i n s t Home Away T o t a l GF v SF GA v SA GF v GA SF v SA GA v SF Low .157 .112 .151 .318 . 2 6 5 H i g h . 4 2 3 .482 .168 .072 . 088 Low .240 .112 .198 .362 .130 H i g h . 476 .498 .140 .064 .194 Low .247 .152 . 1 2 9 . 300 .129 H i g h .418 .410 .116 .124 .087 e x p e c t e d . G o a l s f o r and g o a l s a g a i n s t a p p e a r e d n o t t o be c o r r e l a t e d , a g a i n a s m i g h t be e x p e c t e d . S h o t s f o r a n d s h o t s a g a i n s t were m o d e s t l y n e g a t i v e l y c o r r e l a t e d , w h i c h c o u p l e d w i t h t h e f a c t g o a l s f o r / g o a l s a g a i n s t a p p e a r e d u n r e l a t e d , s u g g e s t s a s u p e r i o r team may be more l i k e l y t o o u t s h o o t t h a n o u t s c o r e an i n f e r i o r t e a m . T h i s m i g h t c o n t r i b u t e t o t h e r a n d o m n e s s o f r e s u l t . T h i s i s a p o s t u l a t e w o r t h c o n s i d e r i n g i n f u t u r e r e s e a r c h . The d a t a a r e s u m m a r i z e d i n T a b l e A34 ( p a g e s 1 6 8 - 1 6 9 ) . Summary: Raw a n d B i n a r y C o r r e l a t i o n s T h e c o r r e l a t i o n a l d a t a s u p p o r t e d t h e c o n c e p t t h a t p e r i o d - t o -p e r i o d , g a m e - t o - g a m e r e s u l t s a r e u n c o r r e l a t e d a n d p o s s i b l y i n d e p e n d e n t e n t i t i e s . I f v a l i d , t h e c o n c e p t o f i n d e p e n d e n c e a g r e e s w i t h t h e f i n d i n g t h a t s t a n d i n g s b a s e d one p e r i o d games c a n a p p r o x i m a t e a r e a l s e a s o n ' s f i n a l s t a n d i n g s . 55 T h e g e n e r a l c o r r e l a t i o n s were a s e x p e c t e d : g o a l s f o r and s h o t s f o r , g o a l s a g a i n s t a n d s h o t s a g a i n s t were m o d e s t l y c o r r e l a t e d . C r o s s - c o r r e l a t i o n s ( e . g . , s h o t s f o r v s . s h o t s a g a i n s t ) were n e g a t i v e l y c o r r e l a t e d . A f t e r a l l , g o a l s a r e a r e s u l t o f s h o t s and f o r team X t o w i n , i t h a s t o o u t s c o r e (and m o s t t i m e s o u t s h o o t ) t e a m Y . V I I . S t a t i o n a r i t y T e s t s T h e s e v e n t h s e r i e s o f p o s t u l a t e s i n v o l v e d an e x a m i n a t i o n o f two c o n c e p t s : (a) t h e u n i t o f m e a s u r e m e n t p r o b l e m - r e l a t i o n s h i p s s h o u l d h o l d i f a u n i t o f m e a s u r e m e n t i s e x t e n d e d i n s i z e ( j u s t a s t h e y h o l d i f t h e u n i t i s s h o r t e n e d ) . T h i s t i e s t h e p r e s e n t r e s e a r c h w i t h G o o d m a n ' s (1969) d a t a on swept d o u b l e h e a d e r s a n d L i n d s e y ' s (1961) o n d i f f e r e n t i a l s c o r i n g i n b a s e b a l l i n n i n g s . (b) t h e s t r e a k p r o b l e m - t h e p o s s i b i l i t y t h a t t e a m s ( G i l o v i c h , V a l l o n e & T v e r s k y , 1985) move f r o m one p l a t e a u t o a n o t h e r i n p l a y i n g tempo (a ' w i n n i n g ' s t r e a k i s s e e n a s w i n n i n g 3 o f 4 , 4 o f 4 a s o p p o s e d t o , s a y , 1 o f 4 , 0 o f 4 , 2 o f 4) . I f games (home, away , t o t a l ) f o r e a c h team a r e g r o u p e d i n t o n o n - o v e r l a p p i n g s e t s o f f o u r f o r c o m p a r i s o n a g a i n s t c h a n c e e x p e c t a n c i e s , t h e r e w i l l be no s i g n i f i c a n t d i f f e r e n c e s f o r 7 a . game o u t c o m e ( w i n s , t i e s , l o s s e s ) 7 b . g o a l s ( f o r o r a g a i n s t ) 7 c . s h o t s ( f o r o r a g a i n s t ) R e s u l t s . T h e p r o c e d u r e u s e d by G i l o v i c h , V a l l o n e & T v e r s k y (1985) a s a t e s t o f s t a t i o n a r i t y was e m p l o y e d . T h e c o n c e p t o f s t a t i o n a r i t y ( N e l s o n , 1 9 7 3 , p . 22) h a s t h e g e n e r a l i m p l i c a t i o n t h a t a g o a l d o e s n o t d e p e n d on t i m e b u t i s a c o n s t a n t v a r y i n g a r o u n d a mean ( e . g . , o n t h e a v e r a g e a team w i l l s c o r e a g o a l e v e r y X m i n u t e s ) . T h e p r o c e s s o f g o a l s c o r i n g may be l o c a t e d w i t h i n a r e g i o n o f t i m e f r o m w h i c h i t w i l l o n l y r a r e l y d e p a r t . A t t h i s s t a g e o f i n v e s t i g a t i o n i t i s n o t p a r a m o u n t w h e t h e r g o a l s f o l l o w a s t a t i o n a r y m o d e l w h i c h (Box & J e n k i n s , 1 9 7 6 , p . 7) a s s u m e s t h e p r o c e s s r e m a i n s i n e q u i l i b r i u m a b o u t a c o n s t a n t mean l e v e l o r a n o n s t a t i o n a r y m o d e l h a v i n g no n a t u r a l mean w h i c h r e q u i r e s u s i n g e x p o n e n t i a l l y w e i g h t e d m o v i n g a v e r a g e s . A s w i t h p o s t u l a t e 2g (page 4 2 ) , e a c h t e a m ' s 80 -game s e a s o n was d i v i d e d i n t o 20 n o n - o v e r l a p p i n g s e t s o f f o u r games ( i n a s e n s e , t h i s c r e a t e d 20 ' a r t i f i c i a l ' g a m e s ) . T o a s c e r t a i n t h e ' e x p e c t e d ' f r e q u e n c i e s f o r t h e f o u r - g a m e s e t s , t h e d a t a w e r e f i r s t c h e c k e d t o a s c e r t a i n i f t h e f r e q u e n c y d i s t r i b u t i o n o f e a c h t e a m ' s f o u r - g a m e s e t s d e v i a t e d f r o m n o r m a l d i s t r i b u t i o n e x p e c t a n c i e s . N o r m a l p r o b a b i l i t y p l o t s ( C r u z e & W e l d o n , 1 9 8 9 , p . 4 -8 ) b a s e d on t h e S h a p i r o - W i l k t e s t ( S h a p i r o & W i l k , 1965) u s i n g F i l l i b e n ' s p r o b a b i l i t y p l o t c o r r e l a t i o n c o e f f i c i e n t t e s t f o r n o r m a l i t y ( F i l l i b e n , 1975) were r e c o r d e d . A l t h o u g h n i s e x c e e d i n g l y s m a l l , C r u z e a n d W e l d o n ( 1 9 8 9 , p . 4 -8 ) p r o v i d e a t a b l e f o r t e s t i n g s m a l l n ' s a n d a r g u e t h a t t h e p r o b a b i l i t y p l o t i s o f v a l u e . T h e d a t a d i d n o t e x c e e d t h e l i m i t s t h e i r t a b l e s p e c i f i e d f o r a n o r m a l d i s t r i b u t i o n . T h i s met t h e a s s u m p t i o n s f o r d e t e r m i n i n g t h e w i n -57 t i e - l o s s p o i n t p r o b a b i l i t i e s by u s i n g l i n e a r i n t e r p o l a t i o n s f r o m t h e b i n o m i a l d i s t r i b u t i o n were m e t . C h i - s q u a r e a n a l y s i s and t h e W a l d - W o l f o w i t z r u n s t e s t were u s e d t o e x a m i n e t h e d a t a . T h e s t a t i o n a r i t y t e s t d a t a were n o t a s s a t i s f a c t o r y a s o t h e r t e s t s i n d e m o n s t r a t i n g t h e r a n d o m n a t u r e o f t h e d a t a . S e v e n o f t h e 21 (33.3%) t e a m ' s 20 -game s e t s p o i n t s d i s t r i b u t i o n v a r i e d s i g n i f i c a n t l y f r o m c h a n c e : 4 a t t h e . 0 5 , 1 a t t h e . 0 2 , 2 a t t h e . 0 0 1 . A n a l y s i s f o r g o a l s a n d s h o t s r e q u i r e d two c o m p a r i s o n s p e r team s i n c e t h e m i d d l e number o f t h r e e c o u l d a r b i t a r i l y be p l a c e d i n t h e u p p e r o r l o w e r row o f c e l l s . None o f t h e 168 c h i - s q u a r e c o m p a r i s o n s c o n c e r n i n g g o a l s f o r , g o a l s a g a i n s t , s h o t s f o r a n d s h o t s a g a i n s t were s i g n i f i c a n t . T h i s was i n a g r e e m e n t w i t h 7b a n d 7 c . T h e d a t a c o n c e r n i n g g o a l s a n d s h o t s were a l s o e x a m i n e d f o r s e t -b y - s e t c o r r e l a t i o n s . T h e n o r m a l d i s t r i b u t i o n p l o t s showed t h e d a t a n o t t o d e v i a t e f r o m a p o s s i b l e n o r m a l d i s t r i b u t i o n . T h e s e t -t o - s e t team c o r r e l a t i o n s c o n c e r n i n g g o a l s r a n g e d : f r o m - . 6 4 5 t o . 3 3 5 a s r e g a r d s g o a l s f o r f r o m - . 3 2 3 t o . 3 2 9 a s r e g a r d s g o a l s a g a i n s t f r o m - . 5 3 0 t o . 256 a s r e g a r d s t o t a l g o a l s T h e s e t - t o - s e t team c o r r e l a t i o n s s t r o n g l y s u g g e s t t h e r e i s no c o r r e l a t i o n b e t w e e n one s e t a n d t h e n e x t c o n c e r n i n g g o a l s f o r , g o a l s a g a i n s t , o r t o t a l g o a l s . A c t u a l l y , 39 o f 63 (61.9%) f a v o r e d t h e n e g a t i v e s i d e , d e f i n i t e l y s u g g e s t i n g a p o s i t i v e c o r r e l a t i o n i s n o t o p e r a t i n g . T h e s e t - t o - s e t team c o r r e l a t i o n s c o n c e r n i n g s h o t s r a n g e d : f r o m - . 2 4 1 t o . 456 a s r e g a r d s s h o t s f o r f r o m - . 6 3 0 t o . 554 a s r e g a r d s s h o t s a g a i n s t f r o m - . 3 3 9 t o . 4 8 9 a s r e g a r d s t o t a l s h o t s T h e d a t a a r e s u m m a r i z e d i n T a b l e s A35 ( p a g e s 170-171} and A36 ( p a g e s 1 7 2 - 1 7 3 ) . F u t u r e a n a l y s i s s h o u l d p r o b a b l y e m p l o y t i m e s e r i e s a n a l y s i s . T i m e and c o s t l i m i t e d t h e p r e s e n t s t u d y t o p r o c e d u r e s u s e d by p r e v i o u s r e s e a r c h e r s , e s p e c i a l l y G i l o v i c h , V a l l o n e a n d T v e r s k y ( 1 9 8 5 ) . E x a m i n a t i o n o f u n i t o f m e a s u r e m e n t a s s u m p t i o n s ( e . g . , s i z e o f u n i t was a s s u m e d i m m a t e r i a l t o any p o s t u l a t e d e f f e c t ) r e q u i r e d a l a r g e r t h a n one game s e t be e x a m i n e d . V I I I . A r t i f i c i a l a n d R e a l S t a n d i n g s A n o t h e r s e t o f p o s t u l a t e s i n v o l v e d a c o m p l e x o f u n d e r l y i n g i s s u e s : r a n d o m n e s s , d e t e r m i n i s m , u n i t o f m e a s u r e m e n t a n d t h e c o n c e p t t h a t t h e l o n g e r t h e t e s t , t h e more v a l i d t h e o u t c o m e ( a s s u m i n g t h e i n d i v i d u a l " i t e m s " w i t h i n t h e t e s t a r e n o t b i a s e d ) . T h e b a s i c p o s t u l a t e s w e r e : 8 a . O u t c o m e s ( e . g . , f i n a l s t a n d i n g s ) b a s e d o n a 1 - p e r i o d game o f 1 s t , 2nd o r 3 r d p e r i o d o n l y w i l l be p o s i t i v e l y c o r r e l a t e d w i t h r e a l f i n a l s t a n d i n g s ; 8 b . b u t w i l l be l e s s c l o s e l y c o r r e l a t e d t h a n w i l l s t a n d i n g s b a s e d o n o u t c o m e s o f 2 - p e r i o d games o f 1 s t a n d 2nd p e r i o d , 1 s t and 3 r d p e r i o d , 2nd and 3 r d p e r i o d . R e s u l t s . T h e s i n g l e p e r i o d d a t a were a n a l y z e d by c o n s t r u c t i n g an a r t i f i c i a l s e a s o n o f w i n s - t i e s - l o s s e s a n d c o m p a r i n g team s t a n d i n g s i n t h e a r t i f i c i a l s e a s o n w i t h a c t u a l team s t a n d i n g s d u r i n g t h e r e a l 1 9 8 8 - 8 9 s e a s o n . T h e c o r r e l a t i o n s b e t w e e n a r t i f i c i a l and r e a l s e a s o n were .569 b a s e d on f i r s t p e r i o d o u t c o m e s , . 693 b a s e d o n s e c o n d p e r i o d o u t c o m e s a n d . 8 4 9 b a s e d on t h i r d p e r i o d o u t c o m e s . T h e d a t a s u g g e s t t h a t t h e t h i r d p e r i o d i s a more a c c u r a t e i n d i c a t o r o f team d i f f e r e n c e s i n a b i l i t y t h a n t h e f i r s t o r s e c o n d p e r i o d . T h e c o r r e l a t i o n s b e t w e e n a r t i f i c i a l a n d r e a l s e a s o n s t a n d i n g s b a s e d o n a c o m b i n a t i o n o f p e r i o d s were c o n s i d e r a b l y h i g h e r : .921 f o r p e r i o d s 1 and 2 , . 887 f o r p e r i o d s 1 a n d 3 , a n d .917 f o r p e r i o d s 2 a n d 3 . T h i s s u p p o r t e d p o s t u l a t e s 8a and 8 b . T h e l a t t e r , o f c o u r s e , s h o u l d be e x p e c t e d s i n c e t h e c o m b i n a t i o n o f p e r i o d s a l l o w s more " i n p u t " (66% o f t h e t o t a l i s b e i n g s a m p l e d ) . P o i n t s a p p e a r e d t o be t h e b e s t i n d i c a t o r , y i e l d i n g t h e h i g h e s t c o r r e l a t i o n s on a l l c a t e g o r i e s b u t p e r i o d 1 . T h e raw d a t a f o r e a c h team i s s u m m a r i z e d i n T a b l e A37 ( p a g e s 1 7 4 -176) , w i t h t h e o v e r - a l l c o r r e l a t i o n s s u m m a r i z e d i n T a b l e A38 (page 177) . T h e o u t c o m e o f an a r t i f i c i a l s e a s o n b a s e d o n f o u r - g a m e s e t s c o m p a r e d t o r e a l s e a s o n o u t c o m e l e d t o a P e a r s o n p r o d u c t - m o m e n t c o r r e l a t i o n ( u s i n g w i n - t i e - l o s s p o i n t s y s t e m ) o f . 976 and a S p e a r m a n r a n k c o r r e l a t i o n o f . 956 u s i n g a s y s t e m o f r a n k s . I X . U p s e t s I f g o a l s and s h o t s a r e r a n d o m a c r o s s t i m e , u p s e t s become o f i n t e r e s t . I n g e n e r a l , i t w o u l d a p p e a r t h a t t h e more h o m o g e n e o u s t e a m s a r e i n a b i l i t y , t h e more random s h o t s a n d g o a l s a c r o s s t i m e s h o u l d become and t h e more l i k e l y u p s e t s s h o u l d b e c o m e . An u p s e t i s d e f i n e d a s a l o w e r r a n k i n g team d e f e a t i n g an o p p o n e n t w i t h a . h i g h e r r a n k i n g . Rank p o s i t i o n was d e f i n e d i n two w a y s : (a) a t e a m ' s r a n k p o s i t i o n p r i o r t o g a m e - t i m e a n d (b) a t e a m ' s r a n k p o s i t i o n a t t h e e n d o f t h e s e a s o n . I t c o u l d be a r g u e d t h a t t h e 60 o l d s i x a n d e i g h t - t e a m NHL were more e l i t i s t t h a n t h e p r e s e n t 2 1 -team NHL and t h a t an a r t i f i c i a l l e a g u e f o r m e d by o n l y t a l l y i n g t h e games b e t w e e n t h e t o p s i x o r e i g h t teams i n t h e 1 9 8 8 - 8 9 s e a s o n s h o u l d a l s o be more e l i t i s t t h a n t h e 2 1 - t e a m l e a g u e . I f s o , t h e f o l l o w i n g p o s t u l a t e s s h o u l d h o l d : T h e r e w i l l be more u p s e t s i n t h e : 9 a . 8 - t e a m NHL o f 1 9 3 7 - 3 8 t h a n t h e 2 1 - t e a m NHL o f 1 9 8 8 - 8 9 9 b . 6 - t e a m NHL o f 1 9 4 7 - 4 8 t h a n t h e 2 1 - t e a m NHL o f 1 9 8 8 - 8 9 9 c . 8 - t e a m NHL o f 1 9 3 7 - 3 8 t h a n games b e t w e e n t h e t o p 8 teams o f t h e 2 1 - t e a m NHL o f 1 9 8 8 - 8 9 9 d . 6 - t e a m NHL o f 1 9 4 7 - 4 8 t h a n games b e t w e e n t h e t o p 6 teams o f t h e 2 1 - t e a m NHL o f 1 9 8 8 - 8 9 S i n c e s c o r i n g c h a n c e s i n b a s k e t b a l l e x c e e d t h o s e i n h o c k e y a n d s c o r i n g c h a n c e s i n h o c k e y e x c e e d t h o s e i n s o c c e r , a c o m p a r i s o n o f u p s e t s i n t h e t h r e e s p o r t s s h o u l d show more u p s e t s i n : 9 e . E n g l i s h F i r s t D i v i s i o n s o c c e r t h a n N a t i o n a l L e a g u e h o c k e y p l a y 9 f . E n g l i s h F i r s t D i v i s i o n s o c c e r t h a n N a t i o n a l B a s k e t b a l l A s s o c i a t i o n p l a y 9 g . N a t i o n a l L e a g u e h o c k e y t h a n N a t i o n a l B a s k e t b a l l A s s o c i a t i o n p l a y N o t e : T h e 1 9 8 7 - 8 8 F o o t b a l l A s s o c i a t i o n E n g l i s h F i r s t D i v i s i o n was t h e l a t e s t d a t a r e a d i l y a v a i l a b l e a t t h e t i m e o f t h e s t u d y . Due t o t i m e a n d f u n d r e s t r i c t i o n s , o n l y t h e l a s t 50% o f t h e 1 9 8 8 - 8 9 NBA s e a s o n c o u l d be c o d e d , i n p u t a n d u s e d . R e s u l t s . The d a t a d i d n o t s u p p o r t p o s t u l a t e s 9a o r 9 b . U p s e t r a t e s i n t h e NHL a r e o u t l i n e d i n T a b l e 7 . 61 U p s e t R a t e i n V a r i o u s 100% 8 8 - 8 9 NHL 4 6 - 4 7 NHL 3 7 - 3 8 NHL T a b l e 7 N a t i o n a l H o c k e y L e a g u e S e a s o n s Rank P r e g a m e S e a s o n 43 .3% 37 .3% 40.4% 35 .9% 35.6% • 31 .1% T h e o l d s i x and e i g h t - t e a m NHLs h a d f e w e r u p s e t s t h a n t h e p r e s e n t 2 1 - t e a m 1 9 8 8 - 8 9 N H L . T h e c h i - s q u a r e d i f f e r e n c e s were n o t s i g n i f i c a n t , s u g g e s t i n g t h e r e w a s , a t l e a s t , no d i f f e r e n c e i n u p s e t s . T h e r e was no s i g n i f i c a n t d i f f e r e n c e b e t w e e n t h e s i x - t e a m a n d t h e e i g h t - t e a m l e a g u e s e i t h e r . The s i m i l a r i t y o f u p s e t r e m a i n e d when t h e ' 8 8 - 8 9 s e a s o n was s h o r t e n e d t o i t s l a s t 75% a n d 50% o f g a m e s . I t a p p e a r s t h a t t h e r a t e o f u p s e t s may h a v e b e e n r e l a t i v e l y c o n s t a n t f r o m d e c a d e t o d e c a d e d e s p i t e many c h a n g e s i n t h e game o v e r t h e y e a r s . I t c o u l d be a r g u e d t h a t c h a n g e s o f v a r i o u s s o r t s a l t e r e d t h e game b e t w e e n 1 9 3 7 - 3 8 a n d 1 9 4 6 - 4 7 and b e t w e e n 1 9 4 6 - 4 7 and 1 9 8 8 - 8 9 . I t may a l s o be a r g u e d t h a t t h e teams e s s e n t i a l l y p l a y e d on an ' e q u a l ' c h a n c e o f w i n n i n g b a s i s f r o m e r a t o e r a . T h e e s s e n t i a l c h a n g e s b e t w e e n 1 9 3 7 - 3 8 a n d 1 9 4 6 - 4 7 ( M c F a r l a n e , 1 9 6 7 ; D i a m o n d & S t u b b s , 1987) w e r e : o T h e b l u e l i n e s were w i d e n e d t o 12 i n c h e s , o T h e c e n t e r l i n e was i n t r o d u c e d . o T h e s c h e d u l e was i n c r e a s e d f r o m 48 t o 60 g a m e s . o P l a y e r s t a k i n g p e n a l t y s h o t s were a l l o w e d t o go t o t h e mouth o f t h e g o a l i f t h e y w i s h e d . 62 o T h e number o f t eams d e c l i n e d f r o m e i g h t t o s i x . o One r e f e r e e p l u s two l i n e s m e n were i n t r o d u c e d . o T h e f l o o d i n g o f t h e i c e b e t w e e n p e r i o d s became o b l i g a t o r y . o G o a l l i g h t s were s y c h r o n i z e d . T h e e s s e n t i a l c h a n g e s b e t w e e n 1 9 4 6 - 4 7 a n d 1 9 8 8 - 8 9 ( M c F a r l a n e , 1 9 6 7 ; D i a m o n d & S t u b b s , 1987) w e r e : o R o s t e r s i z e was i n c r e a s e d . o T h e s c h e d u l e was i n c r e a s e d t o 80 g a m e s . o T h e number o f t eams i n c r e a s e d f r o m 6 t o 2 1 . o T h e i c e s u r f a c e was p a i n t e d w h i t e . o T h e d r a f t was i n t r o d u c e d . L i f e - l o n g p r o t e c t i o n o f j u n i o r a n d m i n o r l e a g u e p l a y e r s was d i s c o n t i n u e d . T h e p l a y e r p o o l was g r o s s l y e n l a r g e d as 100s o f i n d o o r r i n k s were b u i l t , y o u t h l e a g u e s were b e t t e r o r g a n i z e d , p l a y e r s were r e c r u i t e d f r o m E u r o p e . o H i g h t e c h a n a l y s i s a n d t r a i n i n g was i n t r o d u c e d . A n i n t e r n a l e x a m i n a t i o n o f t h e ' 8 8 - 8 9 NHL showed no d i f f e r e n c e i n u p s e t s e x c e p t f o r p r e - g a m e c h i - s q u a r e c o m p a r i s o n o f t h e e n t i r e s e a s o n a n d t h e l a s t h a l f o f t h e s e a s o n . T h e l a s t 50% p r e - g a m e u p s e t r a t e was 39% a s c o m p a r e d t o a 43 .3% f u l l s e a s o n p r e - g a m e u p s e t r a t e . T h e d i f f e r e n c e was s i g n i f i c a n t a t t h e .01 l e v e l . I n t e r n a l e x a m i n a t i o n o f t h e ' 3 7 - 3 8 and ' 4 6 - 4 7 f u l l NHL s e a s o n s v e r s u s t h e l a s t h a l v e s o f e a c h d i d n o t r e v e a l any s i g n i f i c a n t d i f f e r e n c e s i n u p s e t r a t e . A s s e m b l i n g an a r t i f i c i a l T o p S i x and T o p E i g h t ' 8 8 - 8 9 NHL a l l o w e d u p s e t r a t e s t o be d e t e r m i n e d f o r i n t e r p l a y b e t w e e n t h e t o p s i x and t o p e i g h t t e a m s i n 1 9 8 8 - 8 9 . T h e d a t a d i d n o t s u p p o r t p o s t u l a t e s 8 a , 8 b , 8c o r 8 d . T h e t o p e i g h t showed an u p s e t r a t e 63 o f 45 .5% (as c o m p a r e d t o a 43 .3% r a t e f o r t h e f u l l - t e a m ' 8 8 - 8 9 NHL) a n d t h e t o p s i x showed an u p s e t r a t e o f 4 4 . 0 % . T h e r e was no s t a t i s t i c a l d i f f e r e n c e b e t w e e n t h e s e r a t e s a n d t h e f u l l - t e a m r a t e . I t was e x p e c t e d t h a t t h e u p p e r s i x o r e i g h t teams w o u l d h a v e a h i g h e r r a t e b e c a u s e o f t h e i r h o m o g e n i e t y . T h e i r u p s e t r a t e s were h i g h e r , b u t n o t s i g n i f i c a n t l y s o . C o m p a r i s o n w i t h t h e ' 3 7 - 3 8 a n d ' 4 6 - 4 7 NHL y i e l d e d o n l y one c h i -s q u a r e r e s u l t t h a t was s i g n i f i c a n t . T h i s was a c o m p a r i s o n o f t h e t o p e i g h t ' 8 8 - 8 9 NHL and t h e 50% ' 3 7 - 3 8 NHL u p s e t s on a p r e - g a m e r a n k c o m p a r i s o n . T h e c h i - s q u a r e d i f f e r e n c e was s i g n i f i c a n t a t t h e . 0 5 l e v e l . T h e d a t a f o r h o c k e y u p s e t c o m p a r i s o n s a r e s u m m a r i z e d i n T a b l e A39 (page 1 7 8 ) . P r e - g a m e and s e a s o n r a n k u p s e t r a t e s were c o m p a r e d : 4 o f 15 (26.7%) y i e l d i n g s i g n i f i c a n t c h i - s q u a r e d i f f e r e n c e s . A l l 4 s i g n i f i c a n t d i f f e r e n c e s s h o w e d t h e s e a s o n u p s e t r a t e t o be s i g n i f i c a n t l y l o w e r t h a n t h e p r e - g a m e u p s e t r a t e . T h e d a t a a r e s u m m a r i z e d i n T a b l e 8 . T a b l e 8 C h i - s q u a r e C o m p a r i s o n o f U p s e t R a t e s i n H o c k e y and S o c c e r 2 P r e - g a m e S e a s o n 43 .3% 37 .3% 3 7 . 8 2 9 . 0 3 8 . 9 2 7 . 1 3 5 . 1 2 2 . 8 X l s 100% 8 8 - 8 9 NHL 5 . 3 4 2 4 . 8 0 6 7 . 4 0 4 4 . 9 3 5 . 0 5 . 0 5 .01 . 0 5 100% 3 P t 8 7 - 8 8 S o c c e r 100% 2 P t 8 7 - 8 8 S o c c e r 50% 2 P t 8 7 - 8 8 S o c c e r 64 I f d i f f e r e n c e s were t o o c c u r , s e a s o n s t a n d i n g w o u l d be e x p e c t e d t o be more s t a b l e t h a n p r e - g a m e s t a n d i n g s i n c e t h e l a t t e r i n c l u d e s t h e e a r l y s e a s o n s c r a m b l i n g o f teams b e f o r e t h e y s e t t l e i n t o a r e l a t i v e l y s t a b l e r a n k p o s i t i o n . A two a n d t h r e e p o i n t s o c c e r l e a g u e s t a n d i n g was u s e d b e c a u s e t h e E n g l i s h F i r s t D i v i s i o n p r e s e n t l y u s e s a d i f f e r e n t s y s t e m o f s c o r i n g p o i n t s t h a n t h e NHL o r NBA. T h e E n g l i s h l e a g u e g i v e s 3 p o i n t s f o r a w i n a n d 1 p o i n t f o r a t i e . A 2 p o i n t w i n , 1 p o i n t t i e s y s t e m was a p p l i e d t o t h e E n g l i s h l e a g u e t o c r e a t e a c o m p a r a b l y s c o r e d l e a g u e a s t h e NHL and N B A . B o t h 2 and 3 p o i n t r a n k i n g s were u s e d i n c o m p a r i s o n s . T h e p r e - g a m e a n d s e a s o n c o m p a r i s o n s a r e s u m m a r i z e d i n T a b l e A40 (page 1 7 9 ) . I n t e r - s p o r t c o m p a r i s o n s d i d n o t s u p p o r t 9e o r 9 f . U p s e t s were n o t more f r e q u e n t i n s o c c e r t h a n h o c k e y . T h e r e v e r s e h e l d . U p s e t s were more f r e q u e n t i n h o c k e y t h a n s o c c e r . U p s e t s were e q u a l i n s o c c e r a n d b a s k e t b a l l . T h e d a t a d i d s u p p o r t p o s t u l a t e 9 g . U p s e t s were more f r e q u e n t i n h o c k e y t h a n b a s k e t b a l l . P r e - g a m e a n d s e a s o n u p s e t r a t e s f o r a l l t h r e e s p o r t s a r e l i s t e d i n T a b l e 9 . T a b l e 9 P r e - G a m e and S e a s o n U p s e t R a t e s f o r H o c k e y , S o c c e r a n d B a s k e t b a l l U p s e t R a t e P r e - g a m e S e a s o n 3 7 . 3 3 3 . 1 3 1 . 1 2 8 . 1 2 9 . 0 3 1 . 3 8 8 - 8 9 NHL 4 6 - 4 7 NHL 3 7 - 3 8 NHL 4 3 . 3 4 0 . 4 3 5 . 6 3 8 . 9 37 . 8 3 3 . 5 2 p t S o c c e r 3 p t S o c c e r 50% NBA 65 C h i - s q u a r e c o m p a r i s o n s f o r u p s e t r a t e s a r e l i s t e d i n T a b l e 10 ( s e e p a g e 6 6 ) . None o f t h e p r e - g a m e c h i - s q u a r e c o m p a r i s o n s were s i g n i f i c a n t . Two o f t h e s i x s e a s o n u p s e t c o m p a r i s o n s were s i g n i f i c a n t : t h e ' 8 8 -89 NHL a n d s o c c e r 2 p o i n t a n d 3 p o i n t s y s t e m s . None o f t h e c o m p a r i s o n s b e t w e e n t h e E n g l i s h f i r s t d i v i s i o n s o c c e r and NBA b a s k e t b a l l were s i g n i f i c a n t . NHL and NBA c o m p a r i s o n s a r e l i s t e d i n T a b l e 11 ( s e e p a g e 6 6 ) . U p s e t c o m p a r i o n s b e t w e e n t h e ' 4 6 - 4 7 and ' 3 7 - 3 8 NHL a n d t h e l a s t 50% NBA w e r e n o t s i g n i f i c a n t . C h i - s q u a r e c o m p a r i s o n s b e t w e e n t h e ' 8 8 - 8 9 NHL a n d t h e l a s t 50% NBA w e r e : t h e 43 .3% p r e - g a m e u p s e t r a t e f o r t h e NHL a s o p p o s e d t o t h e N B A ' s 33 .5% was s i g n i f i c a n t a t t h e . 001 l e v e l a n d t h e N H L ' s 37 .3% s e a s o n u p s e t r a t e a s o p p o s e d t o t h e N B A ' s 31 .3% was s i g n i f i c a n t a t t h e . 0 5 l e v e l . Summary: U p s e t s I t seems h o c k e y u p s e t s a r e more f r e q u e n t t h a n b a s k e t b a l l u p s e t s a n d a r e m o s t l i k e l y more f r e q u e n t t h a n s o c c e r u p s e t s . S o c c e r and b a s k e t b a l l a p p e a r a b o u t e q u a l i n u p s e t s . T h e d a t a a r e s u m m a r i z e d i n T a b l e A41 ( p a g e s 1 8 0 - 1 8 1 ) . X . D e s c r i p t i v e D a t a I t s e e m e d s e n s i b l e t o d e v e l o p a g e n e r a l d e s c r i p t i v e p i c t u r e o f t h e 1 9 8 8 - 8 9 NHL s e a s o n t o c o m p l e t e t h e s t a t i s t i c a l a n a l y s i s . Mean g o a l s f o r p e r game r a n g e d f r o m 3 . 1 1 2 t o 4 . 7 0 0 . T h e d i f f e r e n c e b e t w e e n t h e h i g h e s t s c o r i n g and l o w e s t s c o r i n g teams was a n a v e r a g e o f 1 .588 g o a l s . T h e s t a n d a r d d e v i a t i o n f o r g o a l s f o r e v e r y team e x c e e d e d 1 . 5 8 8 . 66 T a b l e 10 C h i - s q u a r e C o m p a r i s o n o f P r e - G a m e and S e a s o n U p s e t R a t e s f o r H o c k e y and S o c c e r P r e - g a m e U p s e t R a t e C o m p a r i s o n s 2 U p s e t R a t e X d f I s 88 - 8 9 NHL v s . 2 P t S o c c e r 4 3 . 3 V 3 8 . 9 1 .232 1 ns v s . 3 p t S o c c e r 4 3 . 3 V 37 . 8 2 . 426 1 n s 46 - 4 7 NHL v s . 2 P t S o c c e r 4 0 . 4 V 3 8 . 9 0 . 0 7 9 1 ns v s . 3 P t S o c c e r 4 0 . 4 V 3 7 . 8 0 . 273 1 n s 37 - 3 8 NHL v s . 2 P t S o c c e r 3 5 . 6 V 3 8 . 9 0 . 410 1 n s v s . 3 P t S o c c e r 3 5 . 6 V 3 7 . 8 0 . 2 0 9 1 ns S e a s o n U p s e t R a t e C o m p a r i s o n s 2 U p s e t R a t e X d f I s 8 8 - 8 9 NHL v s . 2 p t S o c c e r 3 7 . 3 v 2 8 . 1 7 . 3 2 8 1 .01 v s . 3 p t S o c c e r 3 7 . 3 v 2 9 . 0 6 . 1 1 2 1 .02 4 6 - 4 7 NHL v s . 2 p t S o c c e r 3 3 . 1 v 2 8 . 1 1 .154 1 ns v s . 3 p t S o c c e r 3 3 . 1 v 2 9 . 0 0 . 7 8 1 1 n s 2 U p s e t R a t e X d f I s 3 7 - 3 8 NHL v s . 2 p t S o c c e r 3 1 . 1 v 2 8 . 1 0 . 4 1 5 1 n s v s . 3 p t S o c c e r 3 1 . 1 v 2 9 . 0 0 . 2 0 5 1 ns T a b l e 11 C h i - s q u a r e C o m p a r i s o n o f P r e - G a m e a n d S e a s o n R a t e s B e t w e e n H o c k e y and B a s k e t b a l l P r e - g a m e U p s e t R a t e C o m p a r i s o n s P r e - g a m e Rank 2 U p s e t R a t e X d f I s 50% NBA v s . 100% 8 8 - 8 9 NHL 3 3 . 5 v 4 3 . 3 1 2 . 164 1 .001 4 6 - 4 7 NHL 3 3 . 5 v 3 3 . 5 2 . 438 1 ns 3 7 - 3 8 NHL 3 3 . 5 v 3 5 . 6 S e a s o n Rank 0 . 217 1 ns 50% NBA v s . 100% 8 8 - 8 9 NHL 3 1 . 3 v 3 7 . 3 5 . 004 1 . 0 5 4 6 - 4 7 NHL 3 1 . 3 v 3 3 . 1 0 . 192 1 ns 3 7 - 3 8 NHL 3 1 . 3 v 3 2 . 0 0 . 312 1 ns 67 Mean s h o t s f o r p e r game r a n g e d f r o m a l o w o f 2 7 . 0 8 9 t o a h i g h o f 3 4 . 6 3 8 . T h e d i f f e r e n c e b e t w e e n t h e m o s t and l e a s t f r e q u e n t s h o o t i n g t e a m ' s a v e r a g e s was 7 . 5 4 9 . T h e c l o s e n e s s o f t e a m s i n o u t p u t i s s u g g e s t e d r a t h e r g r a p h i c a l l y when t h e d a t a a r e r e d u c e d t o s h o t s p e r p e r i o d — t h e d i f f e r e n c e f r o m h i g h t o l o w b e i n g o n l y 2 . 5 s h o t s a p e r i o d o r . 5 g o a l s . T h e p a t t e r n , a s e x p e c t e d , was s i m i l a r f o r g o a l s a g a i n s t . Mean g o a l s a g a i n s t p e r game r a n g e d f r o m a low o f 2 . 7 1 3 t o a h i g h o f 4 . 4 3 8 . T h e d i f f e r e n c e b e t w e e n t h e l e a s t a n d t h e m o s t on team a v e r a g e s was 1 . 7 4 5 g o a l s . T h e s t a n d a r d d e v i a t i o n f o r g o a l s a g a i n s t f o r 16 o f 21 (76.2%) teams e x c e e d e d 1 . 7 4 5 . Mean s h o t s a g a i n s t p e r game r a n g e d f r o m 2 5 . 7 3 7 t o 5 . 8 7 5 . T h e d i f f e r e n c e b e t w e e n t h e m o s t e x t r e m e a v e r a g e s was 1 0 . 0 7 8 s h o t s a g a i n s t p e r game o r 3 . 4 s h o t s a p e r i o d . T o t a l mean g o a l s p e r game r a n g e d f r o m 6 . 2 7 5 t o 8 . 9 1 3 . T o t a l  mean s h o t s p e r game r a n g e d f r o m 5 7 . 1 3 8 t o 6 5 . 2 2 5 . I n g e n e r a l , an NHL game h a d a b o u t 55 t o 65 s h o t s and 7 o r 8 g o a l s . T h e d a t a s u g g e s t teams were q u i t e h o m o g e n e o u s . The d a t a a r e s u m m a r i z e d i n T a b l e A42 ( p a g e s 1 8 2 - 1 8 3 ) . D ISCUSSION I. C o m p a r i s o n s W i t h O t h e r S t u d i e s T h e r e s u l t o f t h i s s t u d y d i d n o t d i f f e r g r e a t l y f r o m t h e r e s u l t s o f p r e v i o u s s t u d i e s . T h e r a n d o m s e q u e n c e s o f w i n s - l o s s e s , g o a l s a n d s h o t s f o r p e r i o d s , games and s e t s o f games a g r e e d w i t h 68 t h e r a n d o m n e s s q u a l i t y o f e l i t e b a s k e t b a l l s h o o t i n g f o u n d by G i l o v i c h , V a l l o n e and T v e r s k y ( 1 9 8 5 ) , t h e r a n d o m n e s s o f h o c k e y o u t c o m e s s u g g e s t e d b y S c h u t z a n d J a n s s e n ' s (1987) m u l t i p l e l i n e a r r e g r e s s i o n a n a l y s i s o f c o a c h e s ' r e c o r d s , a n d w i t h H a b e r m a n ' s (1977) f o o t b a l l a n a l y s i s . S e q u e n c e s o f s c o r e s , w i n s and o t h e r o u t c o m e s a p p e a r t o f o l l o w t h e l a w s o f c h a n c e . I n g e n e r a l , t h e r e s u l t s a g r e e d w i t h t h e r a n d o m n e s s o f o u t c o m e s i m p l i e d by t h e b a s e b a l l s t u d i e s o f S i m o n (1977) and Cook ( 1 9 6 4 ) . T h i s s t u d y ' s d a t a d i d n o t t i e i n w e l l w i t h M o s t e l l e r ' s r e s e a r c h o n f o o t b a l l s c o r e s . T h e r e a s o n s a r e p r o b a b l y t w o f o l d : (a) t h i s s t u d y d i d n o t l o o k d i r e c t l y a t t h e s t a t i s t i c a l l i k e l i h o o d o f w i n n i n g and l o s i n g s c o r e s ( M o s t e l l e r ' s d a t a s u g g e s t e d n o n r a n d o m n e s s ) , (b) M o s t e l l e r ' s r e s e a r c h d e a l t w i t h c o l l e g e f o o t b a l l , an e l i t e s p o r t w h e r e h o m o g e n e i t y o f o p p o n e n t s o f t e n d o e s n ' t o c c u r . T h e d a t a o n d i s t r i b u t i o n o f o u t c o m e s , b o t h i n t h e l i t e r a t u r e a n d i n t h i s s t u d y , may a t f i r s t a p p e a r c o n t r a d i c t o r y . On c l o s e r e x a m i n a t i o n , t h i s d o e s n o t seem t o be t h e c a s e . T h e v a r i a t i o n s come f r o m : (1) d i f f e r e n t o u t c o m e s b e i n g e x a m i n e d ; (2) d i f f e r e n t m e t h o d s b e i n g e m p l o y e d t o e x a m i n e t h e same o u t c o m e ; and (3) t h e same o u t c o m e b e i n g e x a m i n e d u n d e r d i f f e r e n t c o n d i t i o n s . M u l l e t (1977) f o u n d t h a t h o c k e y s c o r e s f i t t h e P o i s s o n f a m i l y o f d i s t r i b u t i o n s ; w h i l e P o l l a r d ( 1 9 7 3 , 1 9 8 5 , 1 9 8 6 ) , R e e p , P o l l a r d a n d B e n j a m i n (1971) f o u n d t h a t s o c c e r , h o c k e y , b a s e b a l l a n d f o o t b a l l s c o r e s m a t c h e d a n e g a t i v e b i n o m i a l d i s t r i b u t i o n . B u t t h e p r o c e d u r e M u l l e t u s e d l o o k e d a t mean g o a l s , w h i c h h a d t h e e f f e c t 69 o f i n s u r i n g t h a t t h e r a t e o f s c o r i n g w i l l be r e l a t i v e l y c o n s t a n t , t h e b a s i c r e q u i r e m e n t f o r t h e P o i s s o n d i s t r i b u t i o n . P o l l a r d e t a l . l o o k e d a t g o a l s p e r game and e x p l a i n e d t h e i r f i n d i n g s by n o t i n g t h a t i f t h e r a t e o f s c o r i n g , h e n c e t h e p a r a m e t e r o f t h e P o i s s o n d i s t r i b u t i o n , v a r i e s f r o m game t o game, a n e g a t i v e b i n o m i a l d i s t r i b u t i o n may a r i s e . P o l l a r d (1973) n o t e d t h a t t h e n e g a t i v e b i n o m i a l ( o r Compound P o i s s o n ) c a n a r i s e f r o m a m i x t u r e o f P o i s s o n d i s t r i b u t i o n s . E l d e r t o n (1909 , 1 9 1 5 , 1927) and Wood (1945) f o u n d c r i c k e t b a t s m a n ' s s c o r e s t o be p r i m a r i l y g e o m e t r i c . Wood n o t e d t h a t m e d i a n s c o r e s o f many o f t h e b a t s m e n were v e r y c l o s e , a c o n d i t i o n l i k e l y t o g e n e r a t e s i m i l a r g e o m e t r i c d i s t r i b u t i o n s . Wood ( 1 9 4 5 , p . 1) s t a t e d t h a t i f a b a t s m a n h a s made n r u n s he i s e q u a l l y l i k e l y t o i n c r e a s e h i s s c o r e t o n + l , w h a t e v e r t h e v a l u e o f n (some b a t s m e n i n c r e a s e d t h e i r s c o r e s by n+2, n+3 o r n + 4 ) . I t i s i m p o r t a n t t o r e a l i z e t h a t w i t h c r i c k e t s c o r e s , t h e s c o r i n g o f t h e n + l t f i r u n i s d e p e n d e n t on h a v i n g s c o r e d t h e n t h r u n . C r i c k e t b a t s m e n s c o r e r u n s , i n a s e n s e , i n s t r e a k s . T h i s s t u d y f o u n d t h a t s t r e a k s ( w i n n i n g , l o s i n g o r t o t a l ) f o l l o w e d a g e o m e t r i c d i s t r i b u t i o n . A g a i n , w i n n + l was d e p e n d e n t o n t h e o c c u r r e n c e o f n . T h i s s t u d y a l s o f o u n d t h a t g o a l s , t o t a l g o a l s , s h o t s f o r , t o t a l s h o t s a n d s h o t s a g a i n s t a l l f i t t e d a n e g a t i v e b i n o m i a l d i s t r i b u t i o n , a f i n d i n g i n a g r e e m e n t w i t h P o l l a r d e t a l . T h i s s t u d y f o u n d g o a l s ( f o r , a g a i n s t , t o t a l ) v i e w e d a c r o s s t i m e ( a t 1 , 2 , 5 o r 10 m i n u t e i n t e r v a l s ) a l l f i t t e d a u n i f o r m d i s t r i b u t i o n ( e x c e p t f o r t h e l a s t 1 o r 2 m i n u t e s ) . T h i s 70 h e l d f o r i n d i v i d u a l a s w e l l as t o t a l t e a m s , f o r g o a l s f o r o r g o a l s a g a i n s t , a n d f o r g o a l s f o r a n d a g a i n s t c o m b i n e d . T h i s d i d n o t c o n t r a d i c t t h e o t h e r f i n d i n g s s i n c e t h i s i n v o l v e d c o u n t i n g e v e n t s w i t h i n v e r y l o n g t i m e f r a m e s ( e . g . , g o a l s a c r o s s 60 m i n u t e s ) w h e r e a s a l l e a r l i e r m e n t i o n e d o u t c o m e s m e a s u r e d s e q u e n t i a l c o u n t i n g : s t r e a k s , s c o r e s (a f i n a l s c o r e o f 4 - 3 i s a r e s u l t o f a d d i n g 1+1+1+1 f o r t h e w i n n e r and 1+1+1 f o r t h e l o s e r ) . T h e "how many g o a l s i n e a c h m i n u t e o f a game" i s e v i d e n t l y a d i f f e r e n t k i n d o f c o u n t i n g o f p r o b a b i l i t i e s . F i n a l l y , t h i s s t u d y f o u n d t h e t i m e s p a c e s b e t w e e n g o a l s ( a d d i n g a l l t h e 1 m i n u t e o r l e s s t i m e s b e t w e e n g o a l s t h e r e w e r e , 2 m i n u t e s o r l e s s , 3 m i n u t e s o r l e s s , e t c . ) f o l l o w e d b o t h a g e o m e t r i c and a n e g a t i v e b i n o m i a l d i s t r i b u t i o n . As s u g g e s t e d o n p a g e s 2 4 - 2 5 , t h i s was m o s t l i k e l y t h e c a s e w h e r e t h e s e two d i s t r i b u t i o n s c o n v e r g e . H o c k e y o u t c o m e s seem t o f o l l o w t h e n e g a t i v e b i n o m i a l : t h e e v e n t - r a t e ( e . g . , g o a l s , s h o t s ) e i t h e r v a r y i n g f r o m game t o game o r v a r y i n g o n c e a game commences a s a r e s u l t o f w i t h i n game o u t c o m e s . I I . M o d e l B u i l d i n g S c h u t z ( 1 9 8 2 , p . 5) n o t e d t h a t t h e a s s u m p t i o n u n d e r l y i n g d e t e r m i n i s t i c m o d e l s i s t h a t " t h e e n t i r e f u t u r e b e h a v i o r o f a s y s t e m i s e x a c t l y d e t e r m i n e d by t h e p r e s e n t s t a t u s o f t h e s y s t e m a n d t h e f o r c e s a c t i n g o n i t . " C a l c u l u s , f o r e x a m p l e , p r o v i d e s a s e r i e s o f d i f f e r e n t i a l e q u a t i o n s , u s u a l l y p r e d i c t i n g p r e c i s e c h a n g e o v e r t i m e . The k e y w o r d s a r e ' p r e s e n t s t a t u s . ' I f t i m e i s c o n c e i v e d o f a s b e i n g a m o v i n g s p a c e , t h e n ' p r e s e n t s t a t u s ' i t s e l f i s c o n t i n u a l l y c h a n g i n g s e c o n d by s e c o n d . I t i s t h i s c o n t i n u a l c h a n g e t h a t t h e p r e s e n t s t u d y may be d e a l i n g w i t h . T h e r e a l d i f f e r e n c e b e t w e e n a d e t e r m i n i s t i c a n d a s t o c h a s t i c m o d e l ( S c h u t z , 1982) i s p r i m a r i l y p e r c e p t u a l : i f we know e v e r y t h i n g a b o u t X , t h e n we c a n d e s c r i b e X w i t h a d e t e r m i n i s t i c m o d e l ; i f we do n o t know e v e r y t h i n g ( e s p e c i a l l y w h e t h e r X i s i n f l u e n c e d by c h a n c e f a c t o r s o r n o t ) , we must u s e a s t o c h a s t i c m o d e l . I f we assume c h a n c e a n d d e t e r m i n i s m a r e r e a l l y p a r t s o f t h e same t h i n g , we may c o n c e i v e o f e v e n t s a s b e i n g on a c o n t i n u u m s t r e t c h i n g f r o m d e t e r m i n e d . . . p a r t l y d e t e r m i n e d . . . m o s t l y c h a n c e . . . p u r e c h a n c e w i t h t h e l a b e l ' c h a n c e ' b e i n g p l a c e d on t h e p a r t s we c a n show t o be d e t e r m i n e d . B u t i t c o u l d be t h a t c h a n c e o p e r a t e s i n d e t e r m i n i s t i c w a y s . I t i s , p e r h a p s , b e s t a t t h i s e a r l y s t a g e o f t h e o r e t i c a l d e v e l o p m e n t , t o u s e an a x i o m a t i c m o d e l ( d e f i n e d a s b e i n g ' d e s c r i p t i v e ' r a t h e r t h a n ' p r e d i c t i v e ' ) . T h e a x i o m a t i c m o d e l a s s u m e s a r e a s o n a b l y b a l a n c e d l e a g u e a n d w o u l d be i n a p p r o p r i a t e i f a p p l i e d t o t h e d e v e l o p m e n t a l s t a g e t h a t seems t o c h a r a c t e r i z e m o s t e l i t e s p o r t s : i n b a s k e t b a l l ' s e a r l i e s t d a y s , t h e 1895 B u f f a l o G e r m a n s , one o f t h e f i r s t p r o b a s k e t b a l l t e a m s , ( D i c k e y , 1982) won 111 s t r a i g h t . S t r e a k s a r e m o s t l i k e l y t o o c c u r w i t h e x t r e m e l y g o o d o r e x t r e m e l y p o o r t e a m s . The 1 9 7 4 - 7 5 W a s h i n g t o n C a p i t a l s ( K l e i n & R e i f , 1987) were c i t e d e a r l i e r . T h e y h a d t h e l o w e s t w i n n i n g p e r c e n t a g e e v e r i n NHL h i s t o r y ( .131 ) and t h e l o n g e s t r o a d w i n l e s s s t r e a k e v e r (37 g a m e s ) . T h e f o l l o w i n g i s t h e a x i o m a t i c m o d e l o f an e l i t e , w e l l -e s t a b l i s h e d h o c k e y l e a g u e . I t i s a l s o a summary o f t h i s s t u d y ' s m a i n f i n d i n g s . 1 . I n g e n e r a l , t h e p a t t e r n o f o u t c o m e s w i l l be p r i m a r i l y " r a n d o m " a c r o s s t i m e ( e . g . , t h e p a t t e r n o f o u t c o m e s f o r team A w i l l n o t e x c e e d t h a t e x p e c t e d f o r a team w i t h team A ' s w i n - l o s s p e r c e n t a g e ) . 2 . V a r i a t i o n s i n u n i t o f m e a s u r e m e n t w i l l n o t a p p r e c i a b l y a f f e c t t h e p a t t e r n o f o u t c o m e s ( r a n d o m n e s s o c c u r s a c r o s s p e r i o d s , g a m e s , a n d f o u r - g a m e s e t s ) . 3 . P a t t e r n s o f o u t c o m e s r e p e a t t h e m s e l v e s s u f f i c i e n t l y e n o u g h t h a t a s e a s o n o u t c o m e b a s e d o n 1 p e r i o d o u t c o m e s i s f a i r l y c o m p a r a b l e t o a r e a l s e a s o n o u t c o m e ( 2 p e r i o d a r t i f i c i a l a n d r e a l game o u t c o m e s w i l l be e v e n more c l o s e l y c o r r e l a t e d ) . 4 . E x c e p t f o r t h e f i n a l m i n u t e o f p l a y , t o t a l g o a l s f o r a l l t eams w i l l be d i s t r i b u t e d e v e n l y a c r o s s t i m e ( u n i f o r m l y d i s t r i b u t e d ) . 5 . T h e f r e q u e n c y d i s t r i b u t i o n o f o u t c o m e s ( e . g . , g o a l s and s h o t s ) , h o w e v e r , w i l l m o s t l i k e l y f i t t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n ( s u g g e s t i n g t h a t r a t e s o f o u t c o m e s f l u c t u a t e r a n d o m l y ) . 6 . When t i m e b e t w e e n o u t c o m e s ( e . g . , f r o m one g o a l t o t h e n e x t ) i s e x a m i n e d , a n e g a t i v e b i n o m i a l d i s t r i b u t i o n w i l l emerge and i f r = 1 , a g e o m e t r i c d i s t r i b u t i o n w i l l a l s o e m e r g e . 7 . T h e p r o b a b i l i t y o f a w i n a f t e r a w i n , o r a l o s s a f t e r a l o s s i s no b e t t e r t h a n t h a t e x p e c t e d b y a t e a m ' s w o n - l o s s r e c o r d . 73 G a m e - t o - g a m e o u t c o m e s w i l l be i n d e p e n d e n t v a r i a b l e s . 8 . U p s e t s seem t o be no more p r o b a b l e i n a s m a l l t h a n a l a r g e l e a g u e (a s i x - t e a m o r e i g h t - t e a m t h a n a 2 1 - t e a m N H L ) . 9 . T h e r e may be a d i f f e r e n c e b e t w e e n s p o r t s i n r a t e o f u p s e t s : h o c k e y a p p e a r s t o h a v e more u p s e t s t h a n s o c c e r and b a s k e t b a l l , b u t s o c c e r and b a s k e t b a l l do n o t a p p e a r t o d i f f e r i n r a t e o f u p s e t s . T h e s e n i n e p o s t u l a t e s may h o p e f u l l y be a d a p t e d t o a t o t a l l y d i f f e r e n t s p o r t t h a n h o c k e y ( e . g . , f o o t b a l l o r b a s e b a l l ) . I I I . F i n a l Comments A number o f more c o m p l e x s t a t i s t i c a l p r o c e d u r e s may n e e d t o be a p p l i e d t o f u t u r e d a t a b a s e s . S n e d e c o r and C o c h r a n (1967 , p . 237) n o t e d t h a t t h e r e a r e two m e t h o d s o f t e s t i n g f o r a P o i s s o n d i s t r i b u t i o n : t h e g o o d n e s s o f f i t t e s t and t h e v a r i a n c e t e s t ( u s e d b y M u l l e t , 1 9 7 7 ) . I f members o f t h e p o p u l a t i o n a c t u a l l y f o l l o w P o i s s o n d i s t r i b u t i o n s w i t h d i f f e r e n t m e a n s , t h e v a r i a n c e t e s t i s more s e n s i t i v e i n d e t e c t i n g t h i s t h a n t h e g o o d n e s s o f f i t t e s t u s e d i n t h i s s t u d y . Z a r (1984) n o t e s t h a t t h e v a r i a n c e e q u a l s t h e mean i n a P o i s s o n d i s t r i b u t i o n . Q u e s t i o n a b l e d i s t r i b u t i o n s c o u l d be e x a m i n e d t o s e e i f s u c h were t h e c a s e . Z a r a l s o n o t e s t h a t a n o t h e r p r o c e d u r e u s i n g c h i - s q u a r e may be e m p l o y e d t o t e s t f o r a P o i s s o n d i s t r i b u t i o n f i t . T h e p r e c i s e c o n d i t i o n s f o r t h e P o i s s o n d i s t r i b u t i o n may n o t h a v e b e e n met i n t h i s s t u d y . Z a r (1984) n o t e s t h a t a P o i s s o n d i s t r i b u t i o n a s s u m e s : 74 1) a r a n d o m d i s t r i b u t i o n o f o b j e c t s i n s p a c e / t i m e i n w h i c h e a c h p o r t i o n o f t h e s p a c e h a s t h e same p r o b a b i l i t i t y o f c o n t a i n i n g an o b j e c t 2) t h e o c c u r r e n c e o f an o b j e c t i n any p o r t i o n o f s p a c e / t i m e i n no way i n f l u e n c e s t h e o c c u r r e n c e o f any o t h e r o f t h e o b j e c t s i n any p o r t i o n o f s p a c e / t i m e b u t : (a) t h e p r o b a b i l i t y o f a g o a l i n t h e f i r s t few s e c o n d s a f t e r a f a c e - o f f a t c e n t e r i c e i s p r a c t i c a l l y z e r o a n d (b) i t i s unknown i f t h e s c o r i n g o f one g o a l h a s any i n f l u e n c e on t h e s c o r i n g o f a n o t h e r . P o p u l a r t h i n k i n g h a s a team b e i n g ' f i r e d ' up by a g o a l o r ' c o m i n g a p a r t ' when down by two o r t h r e e . 3) t h e P o i s s o n d i s t r i b u t i o n a p p r o a c h e s t h e b i n o m i a l d i s t r i b u t i o n when n i s l a r g e and p i s v e r y s m a l l . T h u s , t h e P o i s s o n d i s t r i b u t i o n i s i m p o r t a n t i n d e s c r i b i n g b i n o m i a l l y d i s t r i b u t e d e v e n t s h a v i n g low p r o b a b i l i t y Z a r ' s e x a m p l e i s o f a n o f 100 w i t h a p o f 0 . 0 1 . T h e p r e s e n t d a t a t e n d t o a v e r a g e a b o u t 4 g o a l s i n 12 u n i t s o f 5 m i n u t e s e a c h , w h i c h may be a d i s t o r t i o n o f t h e r e q u i r e m e n t s f o r an a d e q u a t e P o i s s o n d i s t r i b u t i o n . P e r h a p s , m o s t i m p o r t a n t l y , i t ( L i n d g r e n , 1 9 6 8 , p . 147) s h o u l d be r e m e m b e r e d t h a t : p r o b a b i l i t y d i s t r i b u t i o n s a r e i d e a l i z a t i o n s . . . i m p l y i n g c o n d i t i o n s t h a t c a n n o t r e a l l y be met i n p r a c t i c e . . . one s i m p l y c a n n o t know w h e t h e r t h e c o n d i t i o n s a r e o r a r e n o t met i n p r a c t i c e . Z a r ( 1984 , p . 419) s u g g e s t s t h a t t h e mean s q u a r e s u c c e s s i v e d i f f e r e n c e t e s t may be u s e d t o t e s t f o r r u n s . S c h u t z (1970) l i s t s a number o f ways o f a n a l y z i n g t i m e s e r i e s d a t a : u s i n g a n a l y s i s o f v a r i a n c e f o r r e p e a t e d m e a s u r e s and t h e a s s o c i a t e d t r e n d a n a l y s i s ; a u t o c o r r e l a t i o n p r o c e d u r e s ( c o r r e l a t i n g a t i m e s e r i e s w i t h i t s e l f ) a n d a n a l y s i s b a s e d on M a r k o v c h a i n s . G i l o v i c h , V a l l o n e a n d T v e r s k y ( 1 9 8 5 , p . 302) u s e d t h e L e x i s r a t i o o f p l a y e r ' s s h o o t i n g p e r c e n t a g e s t o d e t e r m i n e d e v i a t i o n s f r o m c h a n c e ( c o m p a r i n g two e s t i m a t e s o f t h e s t a n d a r d e r r o r o f e a c h p l a y e r ' s p e r game s h o o t i n g p e r c e n t a g e : one b a s e d o n t h e s t a n d a r d d e v i a t i o n o f t h e p l a y e r ' s s h o o t i n g p e r c e n t a g e s f o r e a c h game, and one d e r i v e d f r o m t h e p l a y e r ' s o v e r a l l s h o o t i n g p e r c e n t a g e a c r o s s a l l g a m e s ) . A b e l s o n (1989) s u g g e s t s a means o f e s t i m a t i n g t h e number o f r u n s s c o r e d i n a h a l f - i n n i n g o f a b a s e b a l l game f r o m e l a p s e d r e a l t i m e . S i n c e b a s e b a l l i s a n o n - t i m e d . s p o r t , A b e l s o n ' s a n a l y s i s c o u l d be u s e d t o c o m p a r e ' n o n - t i m e d ' t o t i m e d s p o r t s . F i s k e a n d J o n e s (1954) s u g g e s t a means o f t e s t i n g s e q u e n t i a l -o r i e n t e d h y p o t h e s e s w i t h an economy o f s a m p l i n g . S u c h p r o c e d u r e s c o u l d be o f v a l u e i n p r e - t e s t i n g c o m p l e x t i m e - r e l a t e d p r o b l e m s r e q u i r i n g e x t e n s i v e d a t a i n p u t (as i n t h i s s t u d y ) . U s i n g F i s k e a n d J o n e s ' p r o c e d u r e o f s a m p l i n g o v e r a 2 0 - y e a r s p a n w o u l d o b v i o u s l y f a c i l i t a t e r e s e a r c h a n d , p e r h a p s , r e m o v e t h e n e e d t o i n p u t 20 t o t a l y e a r s o f g o a l a n d p e n a l t y t i m e s , e t c . T h e r e s p o n s e u n i t s m e a s u r e d i n t h i s s t u d y were w i n s and l o s s e s , g o a l s a n d s h o t s , w h i l e t i m e was d i v i d e d i n t o s e g m e n t s o f one p e r i o d , two p e r i o d s , one game a n d f o u r g a m e s . I t i s q u i t e c o n c e i v a b l e t h a t o t h e r u n i t s o f m e a s u r e m e n t c o u l d be o p e r a t i o n a l l y d e f i n e d and e m p l o y e d . S u c h m e a s u r e s a s s e c o n d s o f p o s s e s s i o n , c o m p l e t e d p a s s e s , r e b o u n d s r e c o v e r e d o f f b o a r d s , s e c o n d s c o n t r o l l i n g p u c k i n o p p o n e n t ' s e n d , r a t i o o f c h e c k s c o m p l e t e d t o c h e c k s a t t e m p t e d , l o o s e p u c k s r e c o v e r e d , i n t e r c e p t e d p a s s e s , e t c . , 76 c o u l d be u s e d . A c o m p l e x o f m e a s u r e s c o u l d p o s s i b l y be u n c o v e r e d t h a t r e f l e c t e d team a b i l i t y t o t h e e x t e n t t h a t p r e d i c t i o n o f o u t c o m e w o u l d be f a i r l y a c c u r a t e . U n i t s w i t h i n s u c h a c o m p l e x w o u l d s t i l l be e x p e c t e d t o f o l l o w t h e same l i m i t i n g " c h a n c e " e f f e c t s o u t l i n e d i n t h i s i n v e s t i g a t i o n : i f t eam X c o m p l e t e d 60% o f i t s p a s s e s , t h e s e q u e n c e o f c o m p l e t e / i n c o m p l e t e p a s s e s w o u l d n o t e x c e e d t h e p a t t e r n e x p e c t e d by a team w i t h a . 600 p a s s c o m p l e t i o n r e c o r d . I n o t h e r w o r d s , t h e same p o s t u l a t e s t h a t a p p l i e d t o g o a l s a n d s h o t s , w i n s and l o s s e s w o u l d a p p l y t o t h e s e m e a s u r e s . T h e b a s i c c o n c e p t b e h i n d t h i s s t u d y was t h a t t h e r e i s an u n d e r l y i n g m a t h e m a t i c a l s t r u c t u r e t o a p e r i o d , a game, a s e r i e s o f games a n d a s e a s o n t h a t i s p a r t l y d e t e r m i n i s t i c a n d p a r t l y rando m i n i t s o p e r a t i n g c h a r a c t e r i s t i c s . The r a n d o m a s p e c t i s c o n s i d e r e d a s f u n d a m e n t a l t o t h e f u n c t i o n i n g o f t h e s t r u c t u r e a s t h e d e t e r m i n i s t i c a s p e c t . I n a c c o r d a n c e w i t h H e i s e n b e r g ' s d i s c o v e r y ( O l i n i c k , 1 9 7 8 , p . 11) t h a t p u r e l y d e t e r m i n i s t i c m o d e l s a r e i n s u f f i c i e n t e v e n t o s t u d y p h y s i c a l p r o c e s s , "we c a n c o n c l u d e f r o m d e t e r m i n i s t i c m o d e l s o n l y t h e a v e r a g e b e h a v i o r o f a g r o u p o f a t o m s , b u t we c a n n o t a s s e r t w i t h c e r t a i n t y a n y t h i n g a b o u t t h e f u t u r e c o u r s e o f an i n d i v i d u a l a t o m . " I n t h e p r e s e n t r e s e a r c h , h o c k e y teams a r e a t o m s . I r o n i c a l l y , b o t h o r d e r and d i s o r d e r seem t o h a v e " p u r p o s e " . E i n s t e i n may h a v e b e e n o n l y p a r t l y c o r r e c t . S o m e t i m e s God p l a y s d i c e w i t h t h e u n i v e r s e ; a t o t h e r t i m e s , God d o e s n o t . 77 APPENDIX T a b l e A l C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s o f A l l Team G o a l s f o r A c t u a l P l a y i n g T i m e (One M i n u t e D i v i s i o n s ) 1 s t P e r i o d 2nd P e r i o d M i n Obs ND UD PD M i n Obs ND UD PD 1 70 32 98 0 1 85 37 112 1 2 109 42 98 3 2 117 49 112 3 3 77 54 98 9 3 99 64 112 11 4 92 68 98 24 4 109 80 112 29 5 109 83 98 52 5 100 96 112 62 6 101 98 98 92 6 130 113 112 110 7 80 112 98 141 7 115 129 112 166 8 105 123 98 188 8 130 142 112 220 9 97 132 98 223 9 106 15.2 112 259 10 98 137 98 239 10 105 157 112 274 11 123 138 98 232 11 126 157 112 264 12 102 135 98 207 12 115 153 112 233 13 99 127 98 170 13 117 144 112 190 14 88 116 98 130 14 113 131 112 144 15 110 103 98 93 15 122 116 112 102 16 106 89 98 62 16 97 100 112 67 17 98 74 98 39 17 108 83 112 42 18 96 60 98 23 18 127 67 112 25 19 109 47 98 13 19 105 52 112 14 20 85 35 98 7 20 104 39 112 7 T 1954 T 2230 2 2 X d f I s X d f I s ND UD PD 1 3 9 . 5 9 1 19 1 6 . 6 3 3 19 7 5 4 . 2 8 4 19 .001 n . s . . 001 ND UD PD i 1 5 5 . 7 8 5 1 2 . 4 3 4 8 5 8 . 6 4 1 19 19 19 .001 n . s . .001 T a b l e A l C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n o f A l l Team G o a l s F o r A c t u a l P l a y i n g T i m e (One M i n u t e D i v i s i o n s ) ( c o n t i n u e d ) 20 M i n u t e s : 3 r d P e r i o d F i r s t 19 M i n u t e s : 3 r d P e r i o d M i n Obs ND UD PD M i n Obs ND UD PD 1 70 33 103 0 1 70 33 97 1 2 92 43 103 2 2 92 44 97 4 3 90 54 103 7 3 90 58 97 13 4 108 68 103 20 4 108 72 97 34 5 110 81 103 44 5 110 88 97 68 6 101 95 103 81 6 101 102 97 113 7 98 109 103 128 7 98 116 97 163 8 122 120 103 178 8 122 126 97 205 9 91 130 103 219 9 91 133 97 229 10 94 135 103 243 10 94 136 97 230 11 101 138 103 245 11 101 134 97 210 12 91 136 103 226 12 91 127 97 176 13 90 131 103 . 193 13 90 117 97 136 14 101 122 103 153 14 101 104 97 98 15 89 111 103 113 15 89 89 97 66 16 95 98 103 78 16 95 74 97 41 17 103 84 103 51 17 103 59 97 24 18 83 70 103 31 18 83 46 97 14 19 108 57 103 18 19 108 34 97 7 20 213 45 103 10 T 1837 T 2050 2 2 X d f l s X d f l s ND UD PD 214 . ' 5 4 . 9 1 1 . 746 19 579 19 756 19 i . i I . i i . i 001 001 001 ND UD PD 136 12 694 . 076 .174 .734 18 18 18 .001 n . s . .001 80 T a b l e A2 C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n , N e g a t i v e B i n o m i a l and N o r m a l D i s t r i b u t i o n s o f A l l Team G o a l s f o r A c t u a l P l a y i n g T i m e (Two M i n u t e D i v i s i o n s ) 1 s t P e r i o d 2nd P e r i o d M i n Sc Obs ND UD PD Sc NBD Sc Obs ND UD PD Sc NBD 1-2 1 179 73 195 41 0 62 1 202 86 223 48 0 75 3 -4 2 169 123 195 114 1 165 2 208 143 223 134 1 196 5 -6 3 210 181 195 212 2 254 3 230 210 223 248 2 295 7 -8 4 185 235 195 296 3 296 4 245 272 223 344 3 340 9 - 1 0 5 195 270 195 332 4 290 5 211 310 223 381 4 330 1 1 - 1 2 6 225 273 195 309 5 252 6 241 311 223 352 5 284 1 3 - 1 4 7 187 244 195 247 6 200 7 230 276 223 278 6 224 1 5 - 1 6 8 216 192 195 173 7 151 8 219 216 223 193 7 169 1 7 - 1 8 9 194 134 195 108 8 106 9 235 149 223 119 8 118 1 9 - 2 0 10 194 82 195 60 9 71 10 209 90 223 66 9 79 T 1954 2230 X 2 d f l s X 2 d f l s ND 1 3 6 . 4 9 8 UD 6 . 7 4 5 PD 2 7 9 . 0 1 0 NBD 1 9 8 . 7 7 1 9 .001 9 n . s . 9 .001 9 .001 ND 1 5 3 . 3 2 7 9 .001 UD 4 . 6 4 5 9 n . s . PD 3 1 5 . 3 8 8 9 .001 NBD 2 1 2 . 5 8 3 9 .001 3 r d P e r i o d : 20 M i n u t e s 3 r d P e r i o d : 1 s t 18 M i n u t e s M i n Sc Obs ND UD PD Sc NBD Sc Obs ND UD PD Sc NBD 1-2 1 162 75 205 37 0 63 1 162 77 190 57 0 75 3 -4 2 198 122 205 106 1 164 2 198 133 190 144 1 186 5 -6 3 211 178 205 204 2 253 3 211 197 190 240 2 267 7 -8 4 220 231 205 295 3 297 4 220 248 190 300 3 289 9 - 1 0 5 185 268 205 341 4 295 5 185 269 190 300 4 262 1 1 - 1 2 6 192 277 205 328 5 261 6 192 249 190 250 5 210 1 3 - 1 4 7 191 254 205 271 6 212 7 191 197 190 179 6 153 1 5 - 1 6 8 184 209 205 195 7 165 8 184 134 190 112 7 106 1 7 - 1 8 9 186 153 205 125 8 119 9 186 78 190 62 8 68 1 9 - 2 0 10 321 99 205 72 9 82 T 1729 T 2050 2 2 X d f I s X d f l s ND 2 0 6 . 5 4 2 UD 3 5 . 3 6 1 PD 3 8 1 . 8 3 6 NBD 2 5 1 . 7 6 0 9 .001 9 .001 9 . 001 9 .001 ND 1 1 8 . 9 3 9 8 .001 UD 3 . 9 6 7 8 n . s . PD 1 9 4 . 3 2 8 8 .001 NBD 1 7 . 9 4 9 8 .001 81 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s B o s PPG UD PD ND 3 . 0 0 1 77 n s 7 . 7 6 6 74 n s 8 . 5 3 0 151 n s 1 1 . 3 1 6 — n s 1 8 . 7 9 8 — n s 2 6 . 5 7 2 .01 5 . 8 5 2 " — n s 1 1 . 9 5 9 " " — n s 1 2 . 6 6 5 n s NPPG UD PD ND 7 . 8 3 3 5 5 . 7 0 3 2 5 . 2 1 7 195 n s .001 .01 5 . 1 0 6 194 n s 4 5 . 4 2 2 .001 1 9 . 0 5 3 n s 7 . 3 4 3 389 n s 9 7 . 7 0 9 .001 3 8 . 5 3 7 .001 TG UD 6 . 5 0 5 272 n s 4 . 2 5 8 268 n s 5 . 7 7 1 540 ns PD 6 4 . 9 0 7 .001 6 4 . 3 1 9 .001 1 2 6 . 6 4 4 .001 ND 2 6 . 3 1 7 .01 2 5 . 6 5 9 .01 4 8 . 2 7 3 .001 B u f PPG UD PD ND 2 . 2 5 7 98 n s 7.943<< 65 n s 1 8 . 8 1 1 — ns 5.378@@ — n s 7 . 5 4 8 — n s 3.027>> — n s 5 . 8 7 3 163 ns 2 0 . 7 2 7 . 0 5 8 .241 n s NPPG UD PD ND 2 . 1 8 3 3 4 . 8 5 6 1 1 . 7 7 6 207 n s .001 n s 8 . 0 9 6 214 n s 4 6 . 5 0 7 .001 2 2 . 5 1 1 . 0 5 5 . 2 6 0 421 n s 7 7 . 5 8 8 .001 3 0 . 8 9 1 .01 TG UD 2 . 2 1 7 305 n s 7 . 7 8 3 279 n s PD 5 1 . 6 9 9 .001 4 9 . 7 9 9 .001 ND 1 7 . 8 2 4 ns 2 1 . 2 0 3 . 0 5 6 . 4 3 5 584 ns 9 9 . 1 2 7 .001 3 6 . 3 8 8 .001 C a i PPG UD PD ND 8 . 4 9 8 95 n s 1 7 . 2 8 7 * * — n s 1 0 . 5 4 3 + — n s 13 .467< 85 n s 11.276>> — n s 9.848++ — n s 1 7 . 6 8 6 180 n s 2 4 . 9 2 1 " .01 1 4 . 2 8 3 * n s NPPG UD 5 . 0 7 1 205 ns PD 5 0 . 0 3 4 .001 ND 1 9 . 6 1 3 n s 3 . 5 9 1 187 n s 4 0 . 8 4 8 .001 1 5 . 3 9 3 n s 5 . 9 9 5 389 n s 8 6 . 2 1 4 .001 3 5 . 5 8 4 .001 TG UD 7 . 0 6 8 300 n s PD 6 2 . 3 4 6 .001 ND 2 9 . 6 8 4 .01 4 . 5 3 1 272 4 5 . 7 4 6 1 8 . 1 8 7 n s 8 . 1 8 5 572 ns 001 1 0 5 . 6 9 2 .001 ns 4 4 . 4 7 1 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 , + = 4 , < = 5 , " " = 7 , ++ = 9 , << = 1 0 , @@ = 12 >> = 1 1 , 82 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s C h i PPG UD PD ND 9 . 4 8 3 100 n s 1 7 . 5 8 5 * * — n s 9 .945< — n s 8 .321 113 2 4 . 8 2 3 + 1 2 . 2 8 7 " n s 1 1 . 0 0 5 213 n s .01 3 5 . 2 9 8 .001 n s 1 5 . 9 6 9 n s NPPG UD 9 . 6 4 5 217 n s 1 2 . 6 6 1 197 PD 2 6 . 4 0 9 " .01 6 0 . 7 7 6 ND 9 . 8 7 2 ns 3 0 . 7 8 5 n s 8 . 6 8 4 001 7 7 . 5 6 8 L 01 3 1 . 2 8 3 414 ns 001 001 TG UD 1 3 . 1 1 2 317 n s PD 3 5 . 9 3 5 .001 ND 1 4 . 5 5 6 n s 1 0 . 6 5 3 314 7 6 . 0 3 0 3 3 . 1 3 0 n s 1 2 . 1 5 5 627 n s 001 1 0 8 . 9 1 8 .001 001 4 0 . 4 5 1 .001 D e t PPG UD PD ND 6 . 6 4 7 " 73 ns 1 1 . 4 5 8 + 2 2 . 2 6 8 ! ! - - . 0 5 1 7 . 7 1 9 " ' 1 2 . 0 2 5 " " — n s 2 3 . 3 0 4 78 n s 1 5 . 0 2 6 151 n s n s 3 7 . 7 7 3 " .001 — .02 2 1 . 7 3 1 * . 0 5 NPPG UD 1 2 . 0 9 9 223 n s PD 4 5 . 8 2 1 .001 ND 1 2 . 3 1 7 * * — n s 4 . 3 1 4 251 5 4 . 1 8 9 2 1 . 9 0 2 n s 1 0 . 8 0 9 474 ns 001 9 5 . 2 8 3 .001 . 0 5 4 1 . 0 3 7 .001 TG UD 5 . 9 2 2 296 n s PD 5 9 . 1 5 9 .001 ND 2 3 . 7 0 6 .02 4 . 8 2 4 329 6 0 . 1 8 8 2 4 . 5 9 3 — n s 7 . 0 5 6 625 n s .001 1 1 4 . 3 0 3 .001 .02 4 4 . 8 8 4 .001 Edm PPG UD PD ND 1 0 . 5 8 5 * * 73 n s 10.989<< — n s 9.016++ — n s 6 . 7 6 6 * 86 16.228>> — 9 . 8 1 0 " " — n s 1 3 . 4 7 2 159 n s n s 2 3 . 8 3 5 .02 n s 1 3 . 8 5 0 ns NPPG UD 2 . 9 6 1 247 n s PD 5 0 . 3 5 4 .001 ND 1 8 . 2 0 1 n s 7 . 7 6 2 218 6 2 . 8 0 8 2 7 . 5 2 2 n s 7 . 3 3 6 465 ns .001 1 1 0 . 1 3 3 .001 .01 4 3 . 3 3 4 .001 TG UD 4 . 9 4 4 320 n s PD 5 7 . 2 2 6 .001 ND 2 0 . 7 5 8 . 0 5 6 . 9 5 5 304 7 3 . 5 5 4 2 9 . 6 6 7 n s 8 . 0 3 8 624 n s 001 1 2 8 . 3 2 2 .001 .01 4 9 . 2 3 0 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s o f l e s s t h a n 5: * = 1, " = 2 , + = 4 , < = 5 , * * = 6 , " " = 7 , ! ! = 8 , ++ = 9 , << = 1 0 , >> = 11 83 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s Home Games Away Games T o t a l Games G o a l 2 2 2 T y p e D i s t ' n X n l s X n l s X n H a r PPG UD 4 . 1 8 8 81 n s PD 2 0 . 2 5 0 I I — . 0 5 ND 9 .043+ — n s NPPG UD 3 . 7 2 3 217 n s PD 4 6 . 1 8 7 .001 ND 1 8 . 2 3 0 n s TG UD 3 . 2 8 0 298 n s PD 6 5 . 1 1 8 .001 ND 2 3 . 4 4 3 .02 LA PPG UD 7 . 5 0 9 ~ 85 ns PD 1 3 . 8 7 6 * * — n s ND 6 . 3 3 8 * * — n s NPPG UD 8 . 7 0 0 293 n s PD 7 2 . 9 9 9 .001 ND 3 3 . 2 5 6 .001 TG UD 5 . 1 4 4 378 n s PD 8 4 . 8 5 0 .001 ND 3 4 . 7 5 9 .001 M i n PPG UD 3 . 9 1 4 84 n s PD 1 9 . 9 8 4 | | ~ . 0 5 ND 9 .767+ — n s NPPG UD 7 . 1 6 8 191 n s PD 3 4 . 7 6 6 ~ .001 ND 1 5 . 3 0 1 * n s TG UD 7 . 1 2 3 275 n s PD 5 1 . 7 2 4 .001 ND 2 3 . 5 3 3 .02 l s 5 .060< 76 n s 4 . 0 7 6 157 n s 2 1 . 9 7 1 ! ! — . 0 5 3 8 . 0 3 8 .001 11 .228< — n s 1 5 . 4 7 2 n s 5 . 4 4 8 210 n s 4 . 9 0 4 427 n s 5 1 . 4 0 6 .001 9 6 . 0 9 1 .001 20.654>> — . 0 5 3 6 . 2 5 6 .001 5 . 2 1 1 286 n s 3 . 3 4 8 584 n s 7 0 . 0 7 7 .001 1 3 1 . 4 3 0 .001 2 6 . 2 7 7 .01 4 7 . 9 5 7 .001 6 . 6 5 6 ~ ~ 73 n s 7 . 2 2 7 158 n s 17 .123++ — ns 2 6 . 7 5 3 * .01 9.088<< — n s 1 3 . 0 7 5 n s 7 . 3 8 0 253 n s 1 4 . 5 8 1 546 n s 5 1 . 1 5 3 .001 1 2 2 . 6 3 3 .001 2 2 . 2 5 4 . 0 5 5 3 . 5 7 0 .001 7 . 8 4 0 326 n s 1 1 . 8 1 5 704 ns 6 1 . 4 9 5 .001 1 4 8 . 3 4 4 .001 2 5 . 2 5 7 .01 5 9 . 2 3 3 .001 7 .728< 76 n s 4 . 6 0 7 160 ns 2 6 . 3 8 7 ! — .01 4 3 . 7 3 9 * .001 1 3 . 8 0 2 + — n s 1 7 . 8 8 8 n s 3 . 9 6 1 184 n s 6 . 4 2 0 375 n s 5 0 . 8 0 8 .001 8 0 . 3 1 2 .001 2 0 . 2 3 6 . 0 5 3 2 . 4 7 4 .001 5 . 0 3 4 260 n s 8 . 4 9 0 535 ns 7 1 . 4 0 5 .001 1 2 1 . 3 2 9 .001 2 8 . 2 2 0 .01 4 8 . 2 0 0 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , - = 2 , ! = 3 , + = 4 , < = 5 , * * = 6 , ~~ = 7 , ! ! = 8, ++ = 9 , << = 1 0 , >> = 11 84 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s Mon PPG UD PD ND 9 . 2 8 3 x x 65 n s 2 1 . 9 8 7 0 ® — . 0 5 1 3 . 6 4 5 x x — n s 9 . 1 6 6 " " 73 n s 9.605@@ — n s 7.040>> — n s 1 0 . 3 8 6 138 ns 22 .3751 . 0 5 1 2 . 5 5 8 " n s NPPG UD 8 . 3 1 9 199 n s PD 4 2 . 5 0 6 .001 ND 2 0 . 0 8 8 . 0 5 2 . 5 0 0 194 n s 5 . 2 7 3 3 5 . 8 4 8 .001 7 3 . 9 0 8 1 3 . 3 6 9 n s 2 7 . 5 6 0 393 ns 001 .01 TG UD 5 . 3 0 3 264 n s PD 5 3 . 3 7 0 .001 ND 2 2 . 5 2 3 . 0 5 4 . 4 4 5 267 n s 5 . 1 9 0 3 9 . 4 8 1 .001 9 0 . 6 2 5 1 4 . 2 8 7 n s 3 1 . 2 3 7 531 ns 001 .01 N J PPG UD 7 . 9 6 3 102 n s 5 . 0 9 9 PD 2 5 . 5 4 4 .01 2 6 . 6 0 1 ND 1 2 . 8 9 0 * n s 1 8 . 2 9 5 96 n s .01 n s 8 . 5 6 4 4 9 . 1 5 5 2 0 . 3 6 9 198 ns 001 . 0 5 NPPG UD 5 . 1 3 3 188 n s PD 4 0 . 1 8 4 .001 ND 1 8 . 2 9 5 n s 5 . 1 3 3 215 n s 9 . 5 8 9 6 3 . 4 1 3 .001 9 9 . 9 4 5 2 7 . 0 4 7 .01 4 2 . 8 3 5 403 ns ,001 .001 TG UD 5 . 7 3 8 290 ns PD 5 8 . 9 0 5 .001 ND 2 4 . 2 8 7 .02 8 .168 311 7 1 . 0 5 2 2 7 . 4 7 3 n s 8 . 9 2 2 001 1 3 9 . 7 6 5 .01 5 5 . 0 9 2 601 n s 001 001 NYI PPG UD PD ND 5 . 3 6 5 * 92 n s 8 . 1 0 8 * 95 n s 5 . 1 6 6 1 0 . 4 4 3 — n s 1 7 . 3 6 0 — n s 2 5 . 7 5 7 4 . 6 2 4 " — n s 8 .473+ — n s 8 . 6 0 0 187 n s .01 ns NPPG UD 9 . 7 6 2 196 n s PD 6 2 . 6 3 0 .001 ND 2 8 . 5 0 4 .01 4 . 9 6 0 201 n s 1 0 . 3 7 7 3 5 . 2 8 2 .001 9 4 . 9 7 0 1 4 . 0 5 5 n s 4 0 . 6 6 1 397 n s 001 .001 TG UD 6 . 4 5 3 288 n s 7 . 8 4 9 PD 6 9 . 1 8 4 .001 5 0 . 2 4 2 ND 2 8 . 7 5 6 .01 1 8 . 8 9 0 296 n s 9 . 0 4 6 .001 1 1 7 . 0 2 7 n s 4 5 . 7 9 3 584 n s 001 001 NB. = 1 , = 2 , 3 = 7 = 11 = >>, 12 = 13 = x x 85 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n l s T o t a l Games 2 X n l s NYR PPG UD PD ND 9 . 1 2 8 ~ ~ 76 n s 11.336@@ — n s 8.046@@ — n s 7 . 0 5 0 93 n s 9 . 5 8 3 17 .175++ — n s 2 7 . 6 9 2 9 . 6 7 8 * * — n s 1 4 . 4 8 4 169 ns .01 n s NPPG UD 8 . 8 8 0 230 n s PD 5 8 . 3 6 6 .001 ND 2 6 . 5 6 8 .01 3 . 7 8 4 216 n s 5 . 9 9 5 4 2 . 1 6 7 — .001 9 4 . 9 7 6 1 6 . 4 9 0 n s 3 6 . 8 1 7 446 ns 001 001 TG UD 6 . 8 5 2 306 n s PD 6 4 . 7 8 9 .001 ND 2 6 . 0 6 2 .01 3 . 7 5 3 309 n s 5 . 7 9 1 5 3 . 9 0 0 .001 1 1 3 . 5 1 6 2 0 . 3 2 4 05 4 1 . 5 3 4 615 n s 001 001 P h i PPG UD PD ND 8 . 4 0 9 * 92 n s 2 5 . 4 0 2 ~ ~ — .01 1 4 . 0 5 5 * * — n s 4 . 2 2 1 87 n s 8 . 4 4 9 13 .866++ — n s 3 5 . 5 1 4 6 . 9 5 2 ~ ~ — n s 1 6 . 6 6 6 179 n s 001 n s NPPG UD 2 . 5 6 8 203 ns 1 0 . 7 9 0 PD 3 9 . 2 2 5 .001 5 6 . 5 4 8 ND 1 3 . 9 8 2 n s 2 7 . 9 5 1 204 n s .001 .01 5 . 4 7 0 8 7 . 5 8 8 3 5 . 0 0 6 407 n s 001 .01 TG UD 2 . 8 5 2 295 n s 1 1 . 3 1 2 PD 6 1 . 3 2 1 .001 6 8 . 5 4 6 ND 2 2 . 7 4 2 .02 3 2 . 3 9 3 291 n s 8 . 0 0 3 .001 1 2 7 . 2 1 6 .001 3 0 . 2 3 3 586 n s 001 ,001 P i t PPG UD PD ND 8 . 2 3 6 115 n s 3 . 8 0 9 2 5 . 8 1 1 ! .01 2 5 . 3 8 9 1 3 . 7 4 9 " n s 1 0 . 3 2 2 116 n s .01 n s 7 . 6 0 3 4 8 . 8 7 5 1 0 . 7 2 7 ' 231 ns 001 ns NPPG UD 8 . 9 9 7 229 n s 3 . 4 3 3 PD 6 3 . 6 9 5 .001 4 7 . 7 8 4 ND 2 8 . 7 7 7 .01 1 9 . 7 1 5 233 n s 7 . 5 7 5 .01 1 0 6 . 9 1 0 . 0 5 4 3 . 7 8 4 462 ns 001 001 TG UD 6 . 0 8 1 344 n s 4 . 5 6 4 PD 7 9 . 0 0 7 .001 6 9 . 3 5 1 ND 3 2 . 0 0 2 .001 2 6 . 9 1 9 349 n s 7 . 2 8 0 — .001 1 4 6 . 9 0 0 .01 5 7 . 1 5 9 693 ns 001 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , ~ = 2 , ! = 3 , * * = 6 , ~~ = 7 , ++ = 9 , @@ = 12 86 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games T o t a l Games 2 2 X n I s X n I s Que PPG UD PD ND 11.839<< 77 n s 18.279<< — n s 12.522++ — n s 8 . 2 9 0 * 89 n s 1 4 . 6 1 5 1 8 . 8 4 0 * * — n s 3 3 . 3 7 0 ! 1 2 . 7 2 9 — n s 1 8 . 7 3 7 166 n s .001 n s NPPG UD 6 . 5 1 2 232 n s 3 . 3 5 1 210 n s 5 . 5 3 9 PD 4 9 . 2 3 6 .001 5 5 . 6 5 9 .001 1 0 3 . 7 0 7 ND 2 1 . 2 6 6 .02 2 1 . 0 4 9 . 0 5 3 9 . 3 5 8 442 n s 001 001 TG UD 8 . 5 4 1 309 n s 2 . 5 9 8 299 n s 7 . 1 5 2 PD 6 3 . 7 2 8 .001 6 9 . 4 6 1 .001 1 3 0 . 6 5 8 ND 2 7 . 5 7 6 .01 2 6 . 4 7 7 .01 4 9 . 5 6 8 608 n s 001 001 S t L PPG UD PD ND 4 . 8 3 2 x x 63 n s 8 .507< — n s 4.401>> — n s 1 2 . 6 0 3 * 1 2 . 6 8 5 ! ! 8 . 4 4 4 ! ! 94 n s n s n s 1 4 . 4 0 8 19 .955< 1 0 . 7 6 7 " 157 n s . 0 5 n s NPPG UD 7 . 9 1 5 199 n s 5 . 5 3 3 200 n s 8 . 5 9 1 PD 6 2 . 5 3 0 .001 4 6 . 2 6 1 .001 1 0 7 . 2 8 7 ND 2 6 . 2 6 8 .01 1 9 . 5 5 6 n s 4 2 . 3 4 7 399 ns 001 001 TG UD 5 . 9 1 6 262 n s 5 . 0 3 9 294 n s 8 . 3 4 2 PD 6 7 . 3 6 3 .001 5 7 . 2 5 4 .001 1 2 2 . 1 8 1 ND 2 6 . 1 1 0 .01 2 2 . 5 7 2 . 0 5 4 6 . 9 9 4 556 n s 001 ,001 T o r PPG UD PD ND 1 3 . 6 0 6 ! 1 2 . 9 9 3 1 0 . 3 8 2 ! ! 81 n s n s — n s 7 . 7 3 4 * 14 .865< 8 . 9 6 5 * * 72 n s n s n s 1 6 . 9 4 5 2 1 . 2 5 2 ' 1 4 . 7 8 3 ' 153 ns .01 ns NPPG UD 7 . 2 4 9 216 n s 1 2 . 0 7 5 227 n s 8 . 5 1 7 PD 6 5 . 8 7 7 .001 3 3 . 5 1 3 .001 9 2 . 2 1 9 ND 2 8 . 3 1 5 .01 1 6 . 4 2 6 n s 3 6 . 1 4 0 443 ns .001 .001 TG UD PD ND 9 . 0 0 7 7 3 . 6 9 4 3 0 . 2 9 9 297 n s .001 .01 1 2 . 6 7 3 299 4 2 . 1 8 7 1 7 . 3 2 3 n s 1 3 . 2 5 5 001 1 0 4 . 3 4 5 n s 4 1 . 7 0 3 596 ns .001 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 * = 1 , " = 2 , i = 3 , < = 5 , * * = 6 , x x = 13 ! ! = 8 , ++ = 9 , << = 10 87 T a b l e A3 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) B o t h Team G o a l s Home Games Away Games T o t a l Games G o a l 2 2 2 T y p e D i s t ' n X n l s X n l s X n l s V a n PPG UD PD ND NPPG UD PD ND TG UD PD ND Was PPG UD PD ND NPPG UD PD ND TG UD PD ND Win PPG UD PD ND NPPG UD PD ND 6 .238< 73 6 . 3 9 3 ! ! — 4 . 1 6 7 ! | — 1 .974 156 4 4 . 0 5 2 1 6 . 8 4 9 2 . 6 2 2 229 4 7 . 3 4 4 1 9 . 2 7 8 2 . 1 3 3 86 1 6 . 7 7 5 * * — 6 .102< — 7 . 9 9 0 188 4 2 . 5 2 5 1 9 . 7 7 0 6 . 7 2 4 274 5 5 . 1 4 2 2 3 . 9 0 1 7 . 8 2 2 * 83 15.673>> — 9.381++ — 6 . 9 6 6 232 5 7 . 6 0 6 2 3 . 8 5 2 n s ns n s n s n s n s n s n s n s 9 . 8 4 6 | ! 70 17.846<< — 11.948++ — 199 269 12 .713< 82 8.768++ — 7 . 2 9 7 ! I — 202 284 5 . 4 9 8 * 78 14.927<< - -6 . 5 5 3 ! 1 — 254 n s 1 1 . 8 7 1 n s 2 4 . 0 8 0 + n s 12 .7661 n s 5 . 4 7 9 .001 8 5 . 7 4 9 . 0 5 3 5 . 8 3 4 n s 3 . 0 5 0 .001 1 0 3 . 5 6 1 .02 3 7 . 7 3 8 n s 7 . 4 1 9 n s 2 1 . 3 2 1 " n s 7 . 9 9 4 n s - 5 . 1 7 7 .001 8 4 . 4 4 6 n s 3 4 . 2 1 9 n s 6 . 8 3 6 .001 1 0 0 . 2 8 0 . 0 5 3 8 . 0 9 9 n s 9 . 7 6 9 ns 2 7 . 1 7 8 n s 1 4 . 4 4 0 n s 7 . 3 8 7 .001 1 2 0 . 7 7 0 .01 4 5 . 8 4 0 n s 5 . 2 7 8 .001 1 4 7 . 1 0 3 .01 5 4 . 6 0 4 143 ns .02 ns 355 ns .001 .001 498 n s .001 .001 168 n s . 0 5 n s 390 n s .001 .001 558 n s .001 .001 161 n s .01 ns 486 ns .001 .001 647 n s .001 .001 n s .001 .02 5 . 091 6 6 . 6 0 9 2 6 . 2 6 7 TG UD 4 . 9 9 1 315 n s PD 6 8 . 4 4 2 .001 ND 2 8 . 6 7 8 n s 5 . 0 1 5 332 8 2 . 8 5 1 3 1 . 1 5 7 n s 6 . 2 6 9 .001 4 6 . 0 0 8 n s 1 9 . 7 3 4 n s 3 . 2 7 3 .001 5 8 . 1 6 9 n s 2 3 . 3 7 1 n s 3 . 3 2 6 .001 4 6 . 4 9 9 n s 1 9 . 1 7 7 n s 5 . 4 6 8 .001 5 0 . 9 1 7 .02 1 9 . 9 1 4 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1, ~ = 2 , i = 3 , + = 4 , < = 5 , * * = 6 , II = 8 , ++ = 9 , << = 1 0 , >> = 11 88 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s B o s PPG UD PD ND 41 44 n s 4 . 8 9 2 85 n s — 1 9 . 8 4 0 ! ! — . 0 5 9 . 0 9 1 * * — n s NPPG UD 5 . 4 8 7 98 n s PD 28 .715< - - .01 ND 1 3 . 5 9 8 * — n s 7 . 2 6 9 103 n s 5 . 9 4 7 2 8 . 1 4 9 .01 5 3 . 7 5 0 1 3 . 0 3 2 " n s 2 3 . 8 5 1 201 ns 001 .02 TG UD 5 . 2 3 2 139 n s PD 3 7 . 5 3 5 * .001 ND 1 7 . 1 3 0 ns 6 . 7 4 0 147 n s 4 . 6 1 4 4 1 . 3 3 3 * .001 7 6 . 9 0 8 1 9 . 3 9 2 n s 3 1 . 2 4 6 286 n s 001 .01 B u f PPG UD PD ND 52 25 77 NPPG UD 5 . 4 5 0 114 n s PD 1 7 . 2 3 7 + n s ND 7 .223+ 8 . 5 2 4 98 n s 8 . 8 3 3 14 .177++ — n s 2 3 . 8 7 6 8 . 1 3 4 ! ! — n s 9 . 8 8 0 212 ns .02 n s TG UD 6 . 0 4 3 166 n s PD 2 3 . 0 4 7 .02 ND 9 . 1 4 3 n s 7 . 3 8 1 123 n s 9 . 5 6 0 1 6 . 5 3 5 + ns 3 5 . 5 1 9 7 . 6 5 9 " n s 1 3 . 3 6 1 289 n s 001 ns C a i PPG UD PD ND 61 41 — 1 4 . 4 3 1 ' 102 n s NPPG UD 4 . 6 0 0 137 n s PD 3 4 . 0 0 4 * .001 ND 1 4 . 5 8 2 n s 6 . 6 6 9 110 n s 7 . 8 6 2 2 6 . 1 7 8 " .01 6 0 . 8 1 1 1 5 . 4 7 6 " n s 2 7 . 2 3 1 247 n s 001 .01 TG UD 4 . 5 1 4 198 n s PD 4 6 . 4 2 5 .001 ND 1 9 . 8 6 8 . 0 5 7 . 8 9 4 151 n s 8 . 7 3 6 3 4 . 0 2 7 .001 7 7 . 3 4 2 1 7 . 3 5 0 n s 3 3 . 8 1 2 349 ns 001 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 , + = 4 , < = 5 , * * = 6 , i i = 8 , ++ = 9 89 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s ns n s 001 . 0 5 C h i PPG UD PD ND NPPG UD PD ND 7 . 0 5 8 1 6 . 9 5 8 ! 8 . 1 5 9 " 49 115 n s n s n s 42 1 0 . 2 6 0 " 89 3 3 . 0 0 0 + — 1 6 . 2 8 1 " - -ns 1 3 . 4 0 5 ns 001 n s 8 . 9 0 9 4 2 . 3 5 6 2 0 . 8 6 7 91 204 TG UD 1 3 . 6 8 6 164 n s 8 . 1 3 9 131 n s 1 4 . 8 1 0 PD 2 6 . 0 6 8 " .01 3 7 . 6 9 9 ! — .001 5 7 . 7 2 0 ND 1 3 . 4 6 1 " n s 1 9 . 3 2 2 — n s 2 6 . 1 4 2 295 ns 001 .01 D e t PPG UD PD ND 41 32 - - - 73 — 2 2 . 4 5 5 — . 0 5 — 1 2 . 7 4 4 | | ~ ns NPPG UD 7 . 6 5 4 115 n s PD 2 4 . 8 5 3 * .01 ND 1 4 . 2 2 7 n s 4 . 6 4 0 122 n s 2 9 . 9 0 5 * - - .01 1 2 . 5 8 8 — n s 6 . 3 8 4 237 n s 4 9 . 9 2 1 .001 2 1 . 3 0 0 . 0 5 TG UD 4 . 3 9 7 156 n s PD 3 6 . 1 9 0 .001 ND 1 4 . 8 5 9 n s 5 . 0 0 8 154 n s 3 0 . 0 2 3 .01 1 3 . 6 8 8 n s 4 . 7 4 0 310 n s 6 2 . 3 8 1 .001 2 5 . 3 0 3 .01 Edm PPG UD PD ND NPPG UD PD ND 43 39 3 . 8 7 3 128 ns 2 6 . 2 9 2 .01 1 0 . 6 8 6 n s 1 0 . 7 8 7 111 33 .6431 1 9 . 0 3 9 — 11 .720< 82 n s n s 7 . 7 4 5 239 n s 001 5 3 . 8 2 2 .001 n s 2 4 . 3 2 3 .02 TG UD 4 . 6 4 5 171 ns PD 2 8 . 1 9 5 .01 ND 1 1 . 9 3 9 n s 9 . 3 1 5 150 n s 3 7 . 7 6 4 " .001 1 7 . 9 4 8 n s 8 . 8 9 8 321 n s 6 2 . 9 4 9 .001 2 6 . 2 1 2 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : = 2 , = 3 , < = 5 , 90 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s H a r PPG UD PD ND NPPG UD PD ND 41 6 . 8 3 6 115 3 1 . 2 5 3 " 1 6 . 3 0 5 " n s .01 n s 46 96 2 3 . 9 7 5 * — 1 1 . 0 0 9 n s .02 n s 6 . 2 5 0 * 87 n s 4 . 7 2 1 211 n s 5 4 . 1 0 9 .001 2 3 . 3 1 2 .02 TG UD 5 . 6 8 4 156 n s PD 4 2 . 5 3 2 .001 ND 1 7 . 9 4 2 n s 4 . 8 2 1 142 n s 5 . 4 8 9 3 5 . 1 2 7 .001 7 2 . 2 7 5 1 5 . 5 0 9 n s 2 9 . 3 9 9 298 n s 001 .01 LA PPG UD PD ND NPPG UD PD ND 51 28 6 . 5 8 9 159 n s 3 7 . 3 0 4 .001 1 7 . 0 5 1 n s 1 . 9 2 7 132 n s 2 3 . 4 2 2 " .02 8 . 7 0 5 n s 7 . 1 3 2 ' 5 . 7 4 4 6 1 . 0 1 8 2 5 . 3 3 7 79 291 n s n s 001 .01 TG UD 5 . 9 0 5 210 ns PD 4 5 . 3 6 8 .001 ND 1 8 . 0 9 1 n s 3 . 5 7 3 160 n s 6 . 5 6 0 2 9 . 5 0 0 * .01 7 2 . 2 7 5 1 2 . 3 8 8 n s 2 8 . 5 7 6 370 n s 001 .01 M i n PPG UD PD ND NPPG UD PD ND 4 . 4 4 1 2 1 . 8 6 8 ! 9 . 5 3 2 " 45 96 n s . 0 5 n s 4 . 9 7 9 " 2 9 . 6 2 0 * 1 4 . 8 6 3 " 33 84 n s .01 ns 6 . 5 7 3 5 2 . 7 4 9 2 1 . 8 2 8 78 180 n s 001 .05 TG UD 3 . 6 4 9 141 n s PD 3 0 . 1 9 2 .01 ND 1 2 . 1 8 9 n s 6 . 0 1 1 117 ns 6 . 3 9 8 4 0 . 5 9 5 .001 6 8 . 4 4 5 1 8 . 0 3 6 n s 2 7 . 8 0 2 258 ns 001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: = 1 , " = 2 , 1 = 3 91 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n l s T o t a l Games 2 X n l s Mon PPG UD PD ND NPPG UD PD ND 37 4 . 7 2 0 123 2 9 . 3 9 0 1 2 . 8 7 1 n s .01 n s 43 4 . 0 5 3 110 2 1 . 7 2 7 ! 1 1 . 3 4 6 * n s ,05 n s 6 . 6 4 1 * 80 ns 5 . 2 8 7 233 n s 4 9 . 9 1 5 .001 2 0 . 3 2 6 . 0 5 TG UD 2 . 6 3 8 160 n s PD 3 4 . 8 6 4 .001 ND 1 4 . 5 4 2 n s 4 . 1 3 4 153 n s 2 1 . 1 5 3 * . 0 5 1 0 . 1 8 2 n s 4 . 4 4 2 313 n s 5 8 . 8 7 1 .001 2 1 . 8 2 7 . 0 5 N J PPG UD PD ND NPPG UD PD ND 4 . 2 1 7 9 . 7 6 5 ' 46 97 n s n s 39 98 7 . 0 2 4 2 8 . 4 2 4 " — 1 4 . 6 0 9 * — n s n s ,01 n s 9 . 1 9 2 ' 85 7 . 3 2 6 195 4 9 . 4 5 3 2 0 . 9 1 1 ns n s 001 . 0 5 TG UD 5 . 1 5 2 143 ns PD 3 0 . 1 9 2 .01 ND 1 2 . 6 2 1 n s 6 . 1 8 2 137 n s 4 1 . 8 3 2 .001 1 8 . 4 3 5 n s 7 . 9 9 0 280 n s 6 9 . 6 2 1 .001 2 9 . 8 4 7 .01 NYI PPG UD PD ND 43 41 1 0 . 7 0 9 " 84 n s 6 . 7 8 5 — n s 4 . 9 1 1 — n s NPPG UD 5 . 3 2 5 96 n s 3 . 3 4 0 PD 3 2 . 6 0 6 — .001 1 7 . 2 0 9 ND 1 7 . 2 2 0 — n s 7 . 2 7 3 82 n s 6 . 9 5 4 178 n s n s 4 4 . 0 4 2 .001 ns 2 0 . 6 0 1 . 0 5 TG UD 6 . 3 8 6 139 n s PD 2 9 . 1 8 9 .01 ND 1 3 . 2 8 8 n s 4 . 8 7 1 123 n s 1 7 . 0 2 9 n s 7 . 4 5 5 n s 7 . 2 2 3 262 n s 4 5 . 4 4 3 .001 1 8 . 6 9 7 n s N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 , ! = 3 92 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games X n l s Away Games 2 X n l s T o t a l Games 2 X n l s NYR PPG UD PD ND NPPG UD PD ND TG UD PD ND 39 1 1 . 3 1 8 121 1 6 . 3 0 4 6 . 6 9 3 160 3 3 . 2 4 1 1 5 . 8 7 6 — 45 n s • 2 . 6 3 0 104 n s 1 0 . 2 0 1 * n s 2 . 7 1 7 149 001 3 0 . 8 5 0 n s 1 2 . 2 5 7 9 . 4 2 8 ' n s 7 . 2 8 4 — 5 0 . 0 9 8 n s 2 1 . 8 8 7 n s .02 n s 5 . 3 9 9 6 1 . 2 2 2 2 3 . 1 3 0 84 225 309 n s ns 001 . 0 5 n s 001 .02 P h i PPG UD PD ND NPPG UD PD ND 54 3 . 7 6 2 111 2 8 . 0 9 1 " 1 1 . 8 7 6 * n s n s .01 n s 9 . 4 2 2 2 0 . 3 0 6 45 96 n s n s n s 7 . 6 8 5 1 2 . 1 4 9 4 . 3 2 5 5 7 . 2 6 3 2 3 . 4 4 8 99 207 n s n s n s 001 .02 TG UD 3 . 5 2 8 165 n s 8 .111 PD 3 4 . 9 1 9 .001 3 7 . 0 2 6 ND 1 3 . 7 9 9 n s 1 9 . 9 9 7 141 n s 001 . 0 5 6 . 3 7 0 6 8 . 9 5 3 2 8 . 6 8 7 306 n s 001 .01 P i t PPG UD PD — — ND 60 59 9 . 6 5 5 1 4 . 1 9 0 119 n s n s NPPG UD 8 .991 125 n s 4 . 9 4 2 101 n s 4 . 1 5 7 PD 3 9 . 9 1 4 — .001 n s 5 1 . 8 1 1 ND 2 0 . 9 1 9 — . 0 5 1 0 . 6 9 1 " n s 2 1 . 0 4 2 226 ns 001 . 0 5 TG UD 5 . 3 1 1 185 ns 5 . 1 1 8 PD 4 5 . 6 8 2 .001 2 8 . 9 5 8 ND 1 8 . 2 8 1 n s 1 2 . 9 5 4 160 n s 5 . 6 0 0 .01 7 0 . 3 8 6 n s 2 7 . 9 4 0 345 n s 001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1 , = 2 93 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games X n I s Away Games 2 X n I s T o t a l Games 2 X n I s Que PPG UD PD ND NPPG UD PD ND 44 6 . 8 3 1 105 1 0 . 0 5 7 n s 6 . 1 9 2 2 4 . 5 7 3 41 77 — 1 9 . 6 5 4 n s ,02 n s 5 . 6 9 3 3 9 . 9 3 2 1 6 . 3 5 3 85 182 n s ns 001 ns TG UD 7 . 4 0 8 149 n s PD 2 5 . 6 6 0 * .01 ND 1 2 . 3 0 4 ns 6 . 0 4 3 118 n s 5 . 8 2 2 3 4 . 4 5 0 * .001 5 3 . 8 6 2 1 5 . 3 9 7 n s 2 1 . 3 1 9 267 n s 001 . 0 5 S t L PPG UD PD ND NPPG UD PD ND — 33 4 . 7 6 9 113 n s 3 1 . 0 6 4 * .01 1 3 . 3 7 5 n s 41 5 . 7 3 3 — 4 8 . 7 0 5 — 1 9 . 8 6 0 74 198 n s 001 . 0 5 TG UD 5 . 7 6 9 146 n s PD 3 3 . 3 7 3 * .001 ND 1 5 . 4 9 2 n s 5 . 8 9 0 126 3 7 . 0 2 6 " 3 6 . 3 6 5 ns MISSING .01 5 8 . 7 1 4 001 2 3 . 4 9 9 272 n s ,001 .02 T o r PPG UD PD ND NPPG UD PD ND 5 . 6 4 2 1 7 . 1 4 2 35 98 n s n s 20 8 . 9 1 4 105 n s 5 . 0 8 6 4 5 . 0 6 2 1 8 . 6 0 4 55 203 n s 001 n s TG UD 4 . 4 7 6 133 n s 7 . 8 8 4 125 n s 5 . 6 4 8 PD 3 4 . 8 3 1 .001 2 1 . 8 8 0 " . 0 5 4 8 . 4 2 3 ND 1 4 . 4 7 6 n s 1 1 . 2 5 3 n s 1 9 . 8 6 8 258 n s 001 . 0 5 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 94 T a b l e A4 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s V a n PPG UD PD ND 44 34 7 . 9 8 2 ' 78 n s NPPG UD PD ND 77 — 3 . 1 5 7 — 1 6 . 0 9 2 7 . 2 3 0 ' 94 n s 4 . 1 8 0 171 n s n s 3 7 . 2 7 4 .001 1 6 . 3 5 0 n s TG UD 4 . 5 9 6 121 n s PD 2 8 . 4 0 0 * - — .01 ND 1 2 . 0 1 2 n s 2 . 2 7 9 128 n s 2 . 4 1 1 2 7 . 1 3 3 " .01 5 0 . 7 6 8 1 0 . 2 8 5 n s 1 8 . 0 5 4 249 n s 001 n s Was PPG UD PD ND 55 40 4 . 8 8 1 95 ns NPPG UD 9 . 9 0 8 104 n s PD 3 1 . 5 3 2 .01 ND 1 6 . 8 8 2 " n s 5 . 9 1 0 104 n s 3 1 . 6 9 6 .001 1 5 . 6 6 5 — n s 5 . 0 2 6 208 n s 5 5 . 0 3 0 .001 2 2 . 0 5 3 . 0 5 TG UD 7 . 3 8 1 159 ns PD 4 3 . 0 6 4 .001 ND 1 9 . 2 9 6 n s 7 . 8 3 4 144 n s 3 7 . 7 5 2 .001 1 9 . 1 8 2 n s 6 . 3 1 7 303 n s 7 0 . 7 5 8 .001 2 9 . 8 4 7 .01 Win PPG UD PD ND 45 34 6 . 0 1 9 * 79 n s NPPG UD 7 . 9 8 7 104 n s PD 3 9 . 1 9 6 .001 ND 1 8 . 8 0 4 n s 3 . 9 6 0 111 n s 2 5 . 8 2 0 " .01 1 1 . 7 6 3 * 4 . 7 9 0 215 n s 5 9 . 0 5 3 .001 2 4 . 7 8 8 .01 TG UD 5 . 1 1 5 149 n s PD 4 4 . 2 5 4 .001 ND 1 9 . 5 4 7 n s 4 . 1 5 7 145 n s 2 9 . 4 2 2 * .01 1 2 . 1 3 7 n s 4 . 8 6 6 294 n s 6 9 . 4 1 2 ' .001 2 7 . 0 5 4 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 95 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games X n l s Away Games 2 X n l s T o t a l Games 2 X n l s B o s PPG UD PD ND 36 30 7 . 0 8 3 66 n s NPPG UD PD ND 8 . 3 7 9 ' 3 1 . 9 3 6 ' 1 6 . 9 4 5 97 n s 001 n s 3 . 0 0 1 91 n s 5 . 7 8 6 4 2 . 2 6 8 1 5 . 9 5 0 188 ns 001 n s TG UD 6 . 1 7 4 133 ns PD 3 2 . 4 3 5 " .001 ND 1 3 . 7 9 4 n s 3 . 3 2 2 121 n s 6 . 0 4 4 2 5 . 8 2 4 " .01 5 1 . 6 6 4 1 0 . 6 9 1 n s 2 1 . 5 7 6 254 ns 001 . 0 5 B u f PPG UD PD ND 46 40 3 . 4 4 9 86 ns NPPG UD 2 . 6 1 3 93 ns PD 2 2 . 6 7 7 — .02 ND 9 . 8 6 6 " — n s 8 .421 116 n s 3 . 4 4 9 4 2 . 8 5 8 .001 6 2 . 0 0 4 2 0 . 6 0 0 . 0 5 2 6 . 3 1 8 209 n s 001 .01 TG UD 5 . 2 6 0 139 n s 5 . 2 6 6 156 n s 6 . 3 5 2 PD 3 5 . 6 6 7 .001 3 7 . 4 1 1 .001 7 1 . 1 7 9 ND 1 7 . 4 6 4 n s 1 6 . 8 2 2 n s 2 8 . 0 7 7 295 n s 001 .01 C a l PPG UD PD ND 34 44 78 NPPG UD PD ND 68 3 . 6 4 0 " 77 n s 5 . 0 1 4 145 n s — — 3 2 . 8 8 5 " .001 — — 1 2 . 6 5 6 ns TG UD PD ND 5 . 1 8 1 102 ns 5 .318 121 n s 9 . 0 9 9 223 ns — 3 2 . 1 3 8 * .001 1 3 . 5 3 3 ns N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 * = 1 , " = 2 96 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n l s T o t a l Games 2 X n l s C h i PPG UD PD ND 51 71 6 . 0 1 0 122 n s 2 3 . 1 3 7 " .02 1 0 . 7 4 6 ns NPPG UD PD ND 8 . 1 1 4 102 n s 6 . 5 8 4 108 n s 1 4 . 2 2 2 * — n s 6 . 1 2 8 3 6 . 5 6 3 1 4 . 9 4 0 210 ns 001 ns TG UD PD ND 1 1 . 1 6 3 145 n s 9 . 8 7 4 n s 5 . 1 9 4 179 n s 7 . 3 2 5 4 3 . 3 1 9 .001 5 5 . 7 6 2 1 7 . 7 6 9 n s 2 1 . 9 9 2 332 n s 001 . 0 5 D e t PPG UD PD ND 32 46 78 NPPG UD 8 . 7 2 6 108 n s PD 2 5 . 7 9 5 " .01 ND 1 4 . 9 3 3 * n s 4 . 7 9 6 129 n s 9 . 2 6 5 2 7 . 9 9 3 " .01 5 0 . 3 6 4 1 3 . 6 1 4 * n s 2 3 . 5 2 7 237 ns 001 .02 TG UD 6 . 9 8 8 140 n s 8 . 5 9 4 PD 2 8 . 6 8 3 .01 3 5 . 2 1 0 ND 1 4 . 4 3 7 n s 1 7 . 7 3 1 175 n s 001 n s 9 . 5 9 5 58 .982 2 6 . 5 9 7 315 n s 001 .01 Edm PPG UD PD ND 30 47 77 NPPG UD 3 . 9 0 0 119 n s 3 . 0 5 0 PD 2 8 . 0 0 5 " .01 3 2 . 9 5 8 ND 1 2 . 8 2 8 n s 1 3 . 2 1 1 107 n s 001 n s 3 . 1 5 8 5 8 . 2 8 5 2 3 . 3 6 6 226 - n s 001 .02 TG UD 3 . 0 5 6 149 n s 2 . 8 6 5 PD 3 1 . 2 1 9 * .01 3 6 . 2 8 7 ND 1 2 . 3 4 6 n s 1 5 . 3 0 4 154 n s 001 n s 3 . 1 7 9 6 8 . 5 2 1 2 5 . 9 4 4 303 ns 001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5; * = 1, " = 2 97 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games X n I s Away Games 2 X n I s T o t a l Games 2 X n I s H a r PPG UD PD ND 40 30 70 NPPG UD PD ND 3 . 8 6 5 102 7 . 9 1 9 " n s n s 7 . 6 2 3 114 n s 7 . 9 5 8 3 0 . 9 9 9 " .01 4 7 . 5 2 7 1 5 . 8 5 0 " n s 2 1 . 5 4 5 216 n s 001 . 0 5 TG UD 4 . 4 9 0 142 n s PD 3 0 . 1 0 2 .01 ND 1 2 . 1 1 8 n s 4 . 2 8 7 144 n s 6 . 0 2 5 3 7 . 9 1 8 .001 6 5 . 2 2 2 1 5 . 2 9 2 ns 2 6 . 1 4 4 286 n s .01 .01 LA PPG UD PD ND 34 45 4 . 7 6 4 79 n s NPPG UD 6 . 3 6 5 134 n s PD 4 0 . 5 7 4 .001 ND 1 8 . 7 0 9 n s 9 . 8 5 4 121 n s 1 4 . 1 9 4 6 8 . 0 1 0 3 2 . 6 3 0 255 ns 001 001 TG UD 4 . 0 9 6 168 n s PD 4 4 . 8 0 8 .001 ND 1 9 . 8 7 1 . 0 5 8 . 4 4 9 166 n s 8 . 9 4 3 3 1 . 6 2 6 * .001 7 9 . 3 1 8 1 6 . 5 6 3 n s 3 5 . 0 3 3 334 n s 001 ,001 M i n PPG UD PD ND 39 43 6 . 6 3 4 82 ns NPPG UD PD ND 7 . 539' 95 n s 4 . 6 5 8 100 n s 5 . 936 3 4 . 7 1 8 1 6 . 0 4 9 195 ns 001 ns TG UD 7 . 5 5 3 134 n s PD 2 5 . 6 4 5 " .01 ND 1 3 . 5 7 4 " . n s 2 . 7 7 1 143 n s 6 . 6 9 4 3 4 . 5 0 2 .001 5 7 . 4 1 8 1 3 . 5 4 8 n s 2 4 . 1 0 7 277 n s 001 .02 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1 , " = 2 98 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n T o t a l Games 2 l s X n l s Mon PPG UD PD ND 28 30 58 NPPG UD PD ND 76 2 . 1 8 2 84 n s 6 . 0 7 3 2 8 . 7 8 4 1 2 . 1 4 3 160 ns .01 n s TG UD PD ND 1 3 . 9 5 4 9 . 0 9 2 104 n s 4 . 3 5 0 114 n s 7 . 3 8 7 " n s 6 . 6 6 4 — 3 6 . 7 5 2 n s 1 3 . 8 4 7 218 n s 001 n s N J PPG UD PD ND 56 57 - - 3 . 5 9 4 — 2 2 . 5 7 7 * 9 . 0 9 1 113 n s . 0 5 n s NPPG UD 6 . 8 4 0 91 n s 5 . 7 1 6 117 n s 8 . 0 7 7 PD — — 3 7 . 7 6 7 .001 5 7 . 7 0 8 ND — — 1 7 . 4 1 9 n s 2 6 . 1 1 7 208 ns 001 .01 TG UD 5 . 3 6 5 147 n s PD 3 3 . 3 3 2 .001 ND 1 4 . 5 5 1 n s 3 . 4 7 6 174 ns 7 . 1 4 7 4 5 . 3 7 3 .001 7 5 . 9 9 3 1 7 . 9 1 7 ns 3 1 . 3 8 0 321 n s 001 001 NYI PPG UD PD ND 49 54 — 3 . 4 9 8 — 1 0 . 6 3 1 ' 103 n s ns NPPG UD 5 . 7 4 7 100 n s PD 3 4 . 8 5 7 .001 ND 1 7 . 1 9 3 n s 6 . 4 0 4 119 n s 8 . 3 6 5 2 3 . 6 2 9 .02 5 5 . 0 9 4 1 1 . 7 1 8 n s 2 4 . 6 3 7 219 n s 001 .02 TG UD 4 . 2 7 3 149 n s PD 4 2 . 7 8 3 .001 ND 1 9 . 0 1 6 n s 6 . 7 5 2 173 ns 6 . 9 1 4 3 5 . 9 6 1 .001 7 8 . 2 2 5 1 5 . 5 8 5 ns 3 0 . 8 8 7 322 ns 001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * =1, " = 2 99 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s NYR PPG UD PD ND 37 45 85 NPPG UD 6 . 3 2 3 109 n s PD 3 7 . 0 7 4 * .001 ND 1 7 . 8 9 5 n s 4 . 0 7 2 112 n s 8 . 8 5 4 n s 4 . 6 0 8 5 0 . 0 6 3 2 1 . 2 6 8 221 ns .001 . 0 5 - T G UD PD ND 6 . 7 8 7 3 9 . 4 4 9 1 6 . 1 8 1 146 ns 001 n s 4 . 9 3 5 2 6 . 9 1 5 1 0 . 9 0 5 160 n s ,01 n s 6 .062 5 6 . 8 1 7 2 2 . 8 9 8 306 n s .001 .02 P h i PPG UD PD ND 38 42 6 . 9 6 9 ' 80 n s NPPG UD PD ND 4 . 2 7 7 92 n s 7 . 7 1 4 108 n s 1 3 . 7 9 2 " n s 4 . 1 5 1 3 1 . 0 9 8 1 3 . 2 9 9 200 n s .02 n s TG UD PD ND 4 . 6 0 3 3 1 . 6 5 0 1 3 . 7 6 0 130 n s 001 n s 7 . 4 6 0 3 4 . 4 8 9 1 7 . 1 1 7 150 n s 001 n s 3 . 4 6 9 5 9 . 8 7 5 2 2 . 4 8 4 280 ns .001 . 0 5 P i t PPG UD PD ND 55 57 — 2 . 8 9 2 - - 2 5 . 0 7 2 — 1 0 . 7 1 3 112 n s .01 n s NPPG UD PD ND 5 . 2 4 8 2 7 . 5 5 1 1 4 . 1 1 8 104 n s .01 n s 4 . 4 2 9 3 3 . 1 6 2 1 4 . 9 6 6 132 ns .001 n s 6 . 8 7 0 5 7 . 6 0 8 2 5 . 5 2 8 236 ns .001 .01 TG UD PD ND 5 . 4 2 3 3 6 . 7 2 2 1 7 . 0 4 2 159 n s 001 n s 4 . 6 8 3 4 4 . 9 7 4 1 8 . 8 4 4 189 ns .001 n s 7 . 1 5 9 7 8 . 0 8 7 3 1 . 8 4 2 348 ns .001 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1 , = 2 100 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s V a n PPG UD PD ND 29 n s 3 . 4 1 4 93 n s 9 . 5 8 3 169 n s NPPG UD PD ND 5 . 6 8 9 " 79 ns 6 . 6 7 2 105 _ _ 1 4 . 6 8 5 ns n s 6 . 4 1 1 5 2 . 8 7 3 2 3 . 0 7 9 184 ns .001 .02 TG UD 4 . 7 8 0 108 n s PD 2 5 . 9 0 1 " .01 ND 1 3 . 2 0 6 * n s 4 . 5 3 3 141 n s 4 . 5 9 7 249 n s 3 6 . 6 9 5 " .001 5 5 . 9 6 4 .001 1 4 . 9 9 9 n s 2 2 . 9 5 7 .02 Was PPG UD PD ND 31 42 73 NPPG UD PD ND 6 . 3 0 4 * 84 n s 3 . 3 5 9 98 9 . 3 9 8 " — n s .05 n s 4 . 7 5 8 3 2 . 0 1 8 1 4 . 1 1 1 182 n s .001 ns TG UD PD ND 4 . 3 3 5 115 1 7 . 5 7 4 " 6 . 6 6 0 " n s 5 . 3 3 4 140 n s 7 . 6 4 9 255 n s n s 1 8 . 8 1 3 " n s 3 2 . 9 4 1 .001 n s 7 . 5 8 6 * n s 1 2 . 9 3 5 n s Win PPG UD PD ND 38 44 6 . 9 6 7 82 n s NPPG UD 6 . 1 6 2 128 n s 5 . 3 3 4 PD 2 3 . 9 5 1 .02 4 4 . 1 6 0 ND 1 1 . 6 0 8 n s 1 7 . 9 3 4 143 n s 001 n s 6 . 8 4 7 6 6 . 0 4 8 2 7 . 0 7 4 271 n s .001 .01 TG UD 3 . 6 8 8 166 ns 4 . 8 1 4 PD 2 7 . 2 9 1 .01 5 6 . 0 2 8 ND 1 1 . 2 8 8 n s 2 2 . 0 6 0 187 ns .001 n s 3 . 6 0 9 7 8 . 6 3 1 2 9 . 1 2 4 353 ns .001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1, " = 2 101 T a b l e A5 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e ( F i v e M i n u t e D i v i s i o n s : d f = 11) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n T o t a l Games 2 l s X n l s V a n PPG UD PD ND 29 n s 3 . 4 1 4 93 n s 9 . 5 8 3 169 ns NPPG UD PD ND TG UD PD ND 5 . 6 8 9 ' 4 . 7 8 0 2 5 . 9 0 1 " 1 3 . 2 0 6 * 79 108 n s n s .01 n s 6 . 6 7 2 105 1 4 . 6 8 5 4 . 5 3 3 141 3 6 . 6 9 5 " 1 4 . 9 9 9 ns n s n s 001 n s 6 . 4 1 1 5 2 . 8 7 3 2 3 . 0 7 9 4 . 5 9 7 5 5 . 9 6 4 2 2 . 9 5 7 184 249 n s ,001 . 02 n s ,001 .02 Was PPG UD PD ND 31 42 73 NPPG UD PD ND 6 . 3 0 4 * 84 n s 3 . 3 5 9 98 9 . 3 9 8 " — n s .05 n s 4 . 7 5 8 3 2 . 0 1 8 1 4 . 1 1 1 182 n s 001 ns TG UD 4 . 3 3 5 115 ns 5 . 3 3 4 140 n s 7 . 6 4 9 PD 1 7 . 5 7 4 " ns 1 8 . 8 1 3 " n s 3 2 . 9 4 1 ND 6 . 6 6 0 " n s 7 . 5 8 6 * n s 1 2 . 9 3 5 255 ns 001 ns Win PPG UD PD ND 38 44 6 . 9 6 7 82 ns NPPG UD 6 . 1 6 2 128 n s PD 2 3 . 9 5 1 .02 ND 1 1 . 6 0 8 ns 5 . 3 3 4 143 n s 6 . 8 4 7 4 4 . 1 6 0 .001 6 6 . 0 4 8 1 7 . 9 3 4 n s 2 7 . 0 7 4 271 ns 001 .01 TG UD 3 . 6 8 8 166 n s PD 2 7 . 2 9 1 .01 ND 1 1 . 2 8 8 n s 4 . 8 1 4 5 6 . 0 2 8 -2 2 . 0 6 0 -187 n s 001 n s 3 . 6 0 9 7 8 . 6 3 1 2 9 . 1 2 4 353 ns 001 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 102 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s B o s PPG UD PD ND 3 . 0 2 8 1 . 5 3 5 3 . 5 3 5 77 n s n s n s 5 . 9 3 3 6 . 5 3 5 9 . 1 9 9 74 n s n s n s 7 . 3 8 3 5 . 5 3 8 1 2 . 0 0 7 151 n s ns .05 NPPG UD PD ND 4 . 4 6 8 1 8 . 1 4 8 2 1 . 4 4 0 195 n s .01 001 2 . 5 4 0 1 2 . 6 4 7 1 5 . 7 1 5 194 n s .05 .01 4 . 1 5 1 2 7 . 5 8 3 3 4 . 6 2 0 389 n s .001 001 TG UD PD ND 3 . 6 1 4 1 6 . 0 7 4 2 3 . 8 1 5 272 ns .01 001 1 . 6 9 6 1 5 . 4 7 1 2 2 . 1 1 5 268 ns .01 001 3 . 4 1 7 3 1 . 4 6 1 4 3 . 8 8 2 540 n s 001 001 B u f PPG UD PD ND 1 . 3 1 9 3 . 3 0 9 6 . 5 2 6 98 n s n s n s 7 . 1 3 9 0 . 4 3 7 1 .162 65 n s n s n s 5 . 1 6 1 1 . 9 5 7 6 . 2 7 5 163 n s ns ns NPPG UD PD ND 1 . 5 7 4 8 . 4 0 9 1 2 . 2 9 7 207 n s n s .05 0 . 8 9 5 1 1 . 3 9 2 1 5 . 3 3 6 214 n s ,05 ,01 1 .537 1 9 . 0 9 5 2 6 . 4 1 0 421 n s .01 001 TG UD PD ND 1 . 4 9 3 1 0 . 5 5 3 1 7 . 2 7 7 305 n s ns .01 1 .448 8 .988 1 5 . 0 1 7 279 n s ns ,02 2 . 6 5 1 1 8 . 8 4 7 3 2 . 7 2 1 584 n s .01 001 C a i PPG UD PD ND 4 . 3 2 6 3 . 2 7 9 6 . 4 4 6 95 ns n s n s 8 . 7 6 3 1 . 5 7 6 4 . 1 8 6 85 n s n s n s 1 0 . 7 0 3 3 . 4 9 9 6 . 9 1 8 180 n s ns n s NPPG UD PD ND 1 .324 1 5 . 0 0 6 1 8 . 9 3 6 205 n s ,02 .01 2 . 6 1 4 1 1 . 5 1 4 1 4 . 1 9 6 187 n s .05 ,02 3 . 2 6 8 2 7 . 0 5 3 3 2 . 6 9 0 392 n s ,001 001 TG UD PD ND 1 .807 1 5 . 5 0 4 2 1 . 1 2 5 300 n s .01 001 2 . 9 8 9 8 . 9 7 1 1 5 . 3 6 5 272 ns n s .01 4 . 1 9 7 2 4 . 6 1 4 3 7 . 9 8 2 572 n s 001 001 103 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s C h i PPG UD PD ND 4 . 6 0 4 2 . 6 6 9 4 . 2 5 4 100 n s n s ns 3 . 0 2 3 4 . 4 9 3 7 . 1 6 6 113 n s n s n s 4 . 8 4 4 4 . 9 6 5 9 . 3 6 0 213 n s ns ns NPPG UD PD ND 5 . 597 5 . 817 8 . 855 217 n s ns n s 1 . 2 1 7 1 4 . 9 6 6 1 7 . 7 5 2 197 n s .02 .01 2 . 6 3 4 1 7 . 5 2 4 2 5 . 1 2 5 414 n s .01 001 TG UD PD ND 7 . 6 5 9 7 . 4 6 0 1 2 . 5 2 3 317 n s n s n s 1 .350 1 7 . 4 5 2 2 3 . 9 8 9 310 n s .01 001 4 . 5 4 4 2 0 . 6 1 3 3 2 . 9 2 9 627 n s ,001 ,001 D e t PPG UD PD ND 4 . 1 3 1 6 . 6 4 9 8 . 2 0 6 73 n s n s n s 4 . 0 8 5 3 . 0 0 9 4 . 7 9 4 78 n s n s n s 6 . 7 0 0 8 . 5 8 0 1 3 . 3 1 3 151 ns n s ,05 NPPG UD PD ND 8 . 1 8 6 1 6 . 0 6 3 2 1 . 0 9 9 223 n s .01 001 2 . 3 8 1 1 2 . 9 4 7 1 6 . 9 0 6 251 n s .05 ,01 8 . 4 6 7 2 7 . 2 2 8 3 5 . 5 8 7 474 n s ,001 ,001 TG UD PD ND 4 . 0 6 0 1 4 . 7 6 6 2 1 . 5 1 3 296 n s .02 001 2 . 5 2 9 1 1 . 9 6 3 1 8 . 9 7 1 329 n s ,05 ,01 5 . 7 8 7 26 .713 4 0 . 8 0 4 625 n s 001 001 Edm PPG UD PD ND 4 . 0 3 3 0 . 5 2 5 2 . 5 0 5 73 ns n s n s 4 . 6 5 7 4 . 0 3 8 6 . 3 2 2 86 n s n s n s 7 . 7 4 2 3 . 4 2 5 8 . 8 7 9 159 ns n s ns NPPG UD PD ND 1 . 3 5 9 1 1 . 9 2 1 1 5 . 8 8 5 247 n s .05 .01 3 . 9 1 6 2 1 . 0 7 3 2 3 . 0 8 9 218 n s ,001 001 2 . 8 6 1 3 1 . 4 0 3 3 7 . 7 2 8 465 ns ,001 ,001 TG UD PD ND 1 .708 1 0 . 0 9 2 1 7 . 5 3 2 320 n s n s ,01 3 . 2 7 8 2 1 . 3 3 6 1 5 . 4 6 7 304 n s ,001 .01 2 . 4 3 6 3 0 . 5 5 0 4 3 . 7 4 5 624 n s 001 ,001 104 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games T o t a l Games 2 2 X n I s X n I s H a r PPG UD PD ND 2 . 0 5 5 5 . 0 4 4 7 . 3 6 0 81 n s n s ns 0 . 5 5 7 4 . 3 5 4 4 . 8 3 7 76 n s ns n s 1 . 4 3 2 7 . 7 5 8 1 1 . 7 1 8 157 n s ns .05 NPPG UD PD ND 1 .742 1 3 . 6 5 2 1 6 . 6 2 8 217 n s 02 01 1 . 5 7 3 1 5 . 2 1 6 1 7 . 6 2 8 210 n s .01 .01 2 . 2 6 5 2 8 . 3 9 4 3 3 . 4 4 0 427 n s 001 001 TG UD PD ND 0 . 8 1 2 1 7 . 0 8 3 2 2 . 6 6 0 298 n s .01 .001 1 .844 1 8 . 9 4 1 2 3 . 1 0 7 286 n s .01 001 0 . 9 8 9 3 3 . 8 7 2 4 4 . 1 8 8 584 n s 001 001 LA PPG UD PD ND 4 . 3 9 8 3 . 2 1 5 5 . 8 8 6 85 n s n s n s 1 .131 2 . 9 4 8 3 . 960 73 n s ns n s 2 . 9 2 3 4 . 2 0 3 8 . 6 3 8 158 n s n s n s NPPG UD PD ND 7 . 1 1 5 2 6 . 9 7 3 3 0 . 1 4 3 293 n s .001 .001 5 . 7 6 0 1 5 . 8 7 0 2 0 . 2 4 4 253 n s ,01 ,01 1 2 . 0 4 2 4 2 . 4 7 5 5 0 . 3 1 6 546 . 0 5 001 001 TG UD PD ND 4 . 1 0 1 2 6 . 6 8 2 3 2 . 7 4 7 378 n s .001 .001 6 . 2 2 1 17 .776 2 3 . 9 9 6 326 n s .01 001 9 . 398 4 4 . 4 1 1 5 7 . 0 7 8 704 n s 001 001 M i n PPG UD PD ND 1 . 5 1 9 6 . 7 1 0 7 . 4 6 9 84 n s n s n s 1 . 7 9 2 7 . 871 9 . 8 6 8 76 n s n s n s 2 . 1 2 8 1 3 . 7 0 4 1 5 . 9 0 1 160 n s .02 .01 NPPG UD PD ND 3 . 6 1 8 8 . 9 1 2 1 2 . 6 9 9 191 ns n s . 0 5 3 . 1 6 7 1 7 . 7 9 0 1 9 . 7 0 4 184 n s .01 .01 4 . 4 1 2 2 4 . 7 6 7 3 0 . 6 7 1 375 n s ,001 ,001 TG UD PD ND 2 . 7 4 9 1 4 . 4 9 2 1 8 . 9 7 2 275 n s .02 .01 3 . 4 7 8 2 3 . 8 6 5 2 5 . 7 9 8 260 n s 001 001 4 . 9 4 9 3 7 . 3 1 4 4 5 . 3 7 0 535 n s 001 001 105 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s Mon PPG UD PD ND 4 . 2 5 6 6 . 4 6 7 8 . 6 8 3 65 n s n s ns 6 . 6 4 3 2 . 7 1 4 4 . 7 7 4 73 n s n s n s 7 . 1 2 2 5 . 0 0 3 9 . 2 7 0 138 ns n s ns NPPG UD PD ND 4 . 9 2 4 1 2 . 8 2 5 1 4 . 7 3 9 199 n s . 0 5 .02 1 . 1 2 7 7 . 4 4 1 1 1 . 8 8 9 194 n s n s .05 2 . 8 3 1 1 6 . 7 5 9 2 3 . 9 3 3 393 n s — .01 — .001 TG UD PD ND 1 .668 1 1 . 7 0 8 1 6 . 0 3 2 264 n s . 0 5 .01 2 . 9 5 9 5 . 6 4 7 1 2 . 6 7 1 267 n s n s . 0 5 3 . 1 9 2 1 6 . 1 3 7 2 8 . 0 4 5 531 ns .01 001 N J PPG UD PD ND 2 . 1 3 7 6 .010 7 . 4 3 3 102 n s n s ns 2 . 4 5 7 8 . 2 4 8 9 . 8 9 9 96 n s n s n s 2 . 847 1 2 . 5 1 6 1 6 . 2 9 0 198 n s . 0 5 .01 NPPG UD PD ND 3 . 2 1 7 1 1 . 0 3 9 1 5 . 7 7 7 188 n s ns .01 6 . 0 3 7 2 4 . 4 5 9 2 6 . 0 0 1 215 n s ,001 001 6 . 2 8 1 3 3 . 6 1 9 3 8 . 8 0 9 403 ns 001 001 TG UD PD ND 3 . 2 2 4 1 4 . 9 9 6 2 1 . 4 9 9 290 n s .02 001 4 . 802 3 0 . 6 9 0 3 2 . 1 5 3 311 n s 001 ,001 6 . 4 5 4 4 4 . 1 4 8 5 3 . 3 5 4 601 n s 001 001 NYI PPG UD PD ND 2 . 5 3 2 1 . 842 3 . 3 5 0 92 n s n s n s 3 . 9 5 3 2 . 2 6 9 4 . 9 8 8 95 n s n s ns 4 . 2 1 7 3 . 6 9 3 8 . 3 1 0 187 ns n s ns NPPG UD PD ND 4 . 894 2 0 . 3 6 7 2 3 . 3 3 1 196 ns .01 001 3 . 7 3 0 7 . 1 4 0 1 1 . 2 0 1 201 n s n s n s 5 . 6 9 1 2 5 . 4 5 8 3 3 . 4 8 2 397 n s 001 001 TG UD PD ND 3 . 6 1 8 1 8 . 9 8 5 2 5 . 1 0 3 288 n s .01 001 5 . 5 8 3 8 . 0 6 4 1 5 . 6 4 4 296 n s n s .01 5 . 8 8 8 2 6 . 7 2 5 3 9 . 6 1 0 584 ns 001 001 106 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games T o t a l Games 2 2 X n I s X n I s NYR PPG UD PD ND 7 . 2 8 1 76 n s 3 . 4 1 4 1 . 9 2 6 — n s 2 . 0 4 8 4 . 5 2 7 * — n s 5 . 2 0 8 93 n s n s n s 9 . 5 8 3 3 . 5 7 7 8 . 9 7 8 169 ns n s ns NPPG UD PD ND 1 .871 1 6 . 0 9 0 1 9 . 6 1 8 230 n s 01 .01 1 . 3 8 3 1 0 . 0 0 2 1 4 . 3 7 8 216 n s n s .02 2 . 2 7 1 2 5 . 4 0 2 3 3 . 1 4 8 446 n s ,001 ,001 TG UD PD ND 1 .278 1 3 . 0 1 2 1 9 . 7 7 8 306 n s .05 ,01 1 .532 1 0 . 0 6 8 1 6 . 7 9 3 309 n s n s ,01 1 . 8 7 9 2 1 . 5 1 0 3 5 . 7 1 2 615 n s ,001 001 P h i PPG UD PD ND 6 . 3 7 0 9 . 2 2 2 1 1 . 2 9 1 92 n s ns .05 4 . 0 5 5 3 . 1 9 9 6 . 9 2 3 87 n s n s n s 7 . 2 8 7 9 . 2 7 2 1 7 . 0 9 2 179 n s n s .01 NPPG UD PD ND 1 .197 6 . 4 5 8 1 1 . 2 9 6 203 ns n s . 0 5 3 . 5 0 8 1 7 . 3 9 4 2 1 . 6 0 0 204 n s .01 001 1 . 9 6 6 2 1 . 4 6 4 3 1 . 7 4 0 407 ns ,001 ,001 TG UD PD ND 1 . 7 7 5 1 2 . 4 1 2 2 0 . 8 8 5 295 ns . 0 5 .001 5 . 1 4 5 1 9 . 5 2 9 2 8 . 2 5 8 291 n s . 01 001 3 . 9 8 5 3 0 . 2 3 3 4 7 . 0 0 3 586 ns 001 001 P i t PPG UD PD ND 3 . 6 4 3 4 . 7 1 6 8 . 4 5 5 115 n s n s n s 1 . 3 7 7 5 . 1 9 2 7 . 4 8 5 116 n s ns n s 2 . 4 2 9 8 . 8 6 5 1 5 . 0 5 9 231 ns n s .02 NPPG UD PD ND 3 . 4 3 0 1 9 . 7 2 2 2 1 . 2 2 2 229 ns .01 .001 0 . 7 4 5 9 . 2 7 1 1 5 . 3 3 0 233 n s n s .01 1 . 2 8 3 2 6 . 9 4 0 3 5 . 2 6 4 462 n s 001 001 TG UD PD ND 1 .524 1 9 . 6 2 2 2 5 . 9 6 8 344 n s .01 .001 0 .903 1 4 . 3 0 0 2 2 . 1 4 6 349 n s .02 .001 1 . 2 9 6 3 3 . 7 6 1 4 8 . 1 4 1 693 ns 001 001 107 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e ' A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games T o t a l Games 2 2 X n I s X n I s Que PPG UD PD ND 6 . 9 2 0 77 ns 5 . 5 5 4 89 n s 9 . 8 5 6 166 n s 4 . 5 4 6 — n s 6 . 8 0 5 — n s 9 . 2 3 7 n s 7 . 7 6 7 — n s 9 . 3 0 6 — n s 1 6 . 1 4 5 .01 NPPG UD 2 . 3 1 9 232 n s PD 9 . 8 4 6 n s ND 1 4 . 5 6 0 .02 1 . 2 4 2 210 n s 1 5 . 9 0 1 .01 1 9 . 5 3 6 .01 1 .532 442 n s 2 4 . 4 3 5 .001 3 2 . 8 5 3 .001 TG UD 4 . 4 5 6 309 n s PD 1 1 . 7 4 4 . 0 5 ND 2 0 . 6 0 4 .001 1 .128 299 n s 1 7 . 4 5 5 .01 2 4 . 0 1 0 .001 4 . 1 6 3 608 n s 2 7 . 9 7 8 .001 4 4 . 4 1 9 .001 S t L PPG UD PD ND 3 . 6 2 8 63 n s • 6 . 7 3 5 0 . 7 1 1 * — n s 0 . 4 7 7 2 . 4 7 7 " — ns 3 . 2 0 0 94 n s 9 . 1 5 7 157 n s n s 2 . 6 5 6 n s n s 6 . 8 7 5 ns NPPG UD 2 . 1 4 3 199 n s PD 1 7 . 6 9 1 .01 ND 1 9 . 6 3 4 .01 3 . 0 2 3 200 n s 1 2 . 6 0 7 . 0 5 1 4 . 8 4 2 . 02 3 . 1 7 2 399 n s 3 0 . 2 9 1 .001 3 4 . 1 8 4 9 .001 TG UD 0 . 8 6 9 262 n s PD 1 4 . 6 0 2 .02 ND 9 . 8 0 6 n s 1 . 9 2 5 294 ns 1 0 . 9 0 3 n s 1 7 . 8 5 5 .01 1 . 8 2 4 556 ns 2 5 . 6 6 8 .001 3 7 . 3 8 6 .001 T o r PPG UD PD ND 6 . 6 8 2 81 n s 1 .557 72 n s 7 . 0 2 4 153 n s 0 . 9 3 2 — n s 1 .557 — n s 1 .502 n s 1 . 1 3 7 — n s 3 . 0 4 2 — n s 3 . 8 3 8 ns NPPG UD 3 . 6 9 8 216 n s 9 . 2 1 7 PD 2 1 . 8 6 0 .001 8 . 0 1 0 ND 2 4 . 9 6 3 .001 9 . 6 8 4 227 n s 3 . 2 6 8 443 ns n s 1 9 . 6 7 9 .01 n s 2 7 . 5 4 9 .001 TG UD 2 . 1 7 7 297 n s PD 1 4 . 9 1 4 .02 ND 2 2 . 6 1 6 .001 8 .991 299 n s 7 . 5 8 3 n s 1 1 . 7 5 1 . 0 5 5 . 5 1 3 596 ns 1 7 . 5 1 1 .01 2 8 . 8 3 1 .001 * 1 c e l l w i t h e x p e c t e d c o u n t l e s s t h a n 5 " 2 c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 108 T a b l e A6 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) B o t h Team G o a l s G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s V a n PPG UD PD ND 4 . 2 7 7 73 n s 3 . 0 4 7 0 . 8 7 4 - - n s 5 . 2 3 1 1 . 7 8 6 * — n s 6 . 7 3 4 70 n s ns n s 4 . 4 0 9 3 . 4 8 2 7 . 801 143 n s ns ns NPPG UD PD ND 1 .297 1 4 . 1 0 6 1 5 . 7 3 7 156 n s .02 .01 3 . 0 6 0 1 4 . 3 6 4 1 6 . 9 7 3 199 n s ,02 ,01 3 . 3 9 3 2 6 . 3 4 8 3 2 . 5 9 8 355 ns ,001 ,001 TG UD PD ND 0 . 9 7 9 1 0 . 9 3 3 1 5 . 5 2 5 229 n s n s .01 1 .632 1 5 . 2 0 8 2 1 . 8 6 3 269 n s .01 001 2 . 1 1 0 2 6 . 0 9 3 3 7 . 0 7 3 498 n s 001 001 Was PPG UD PD ND 0 . 7 7 3 3 . 2 2 5 5 . 3 5 8 86 n s n s n s 9 . 1 1 3 3 . 0 0 6 2 . 806 82 n s n s n s 5 . 4 5 9 2 . 3 2 4 5 . 1 4 6 168 ns n s ns NPPG UD PD ND 0 . 8 3 8 9 . 3 9 4 1 2 . 4 8 3 188 ns n s .05 1 .856 1 5 . 3 0 9 1 7 . 2 0 1 202 n s ,01 ,01 1 . 6 6 7 2 2 . 6 7 5 2 9 . 5 5 4 390 n s ,001 ,001 TG UD 1 . 2 6 2 274 n s 3 . 5 7 5 284 n s 2 . 6 5 5 558 n s PD 1 1 . 7 9 2 . 0 5 1 1 . 8 2 9 . 0 5 2 1 . 6 4 4 .001 ND 1 7 . 6 9 0 .01 1 6 . 2 4 9 .01 3 2 . 6 1 3 .001 Win PPG UD PD ND 5 . 842 3 . 5 8 8 7 . 5 8 6 83 ns n s n s 2 . 4 2 6 1 .471 4 . 4 8 1 78 n s n s n s 7 . 5 9 3 4 . 5 3 9 1 0 . 3 8 7 161 n s ns n s NPPG UD PD ND 4 . 6 3 2 1 9 . 3 2 2 2 1 . 3 3 9 232 ns .01 001 2 . 6 3 9 2 0 . 5 8 2 2 4 . 0 0 0 254 n s 001 001 4 . 8 8 0 3 8 . 1 6 5 4 3 . 0 8 7 486 n s 001 001 TG UD PD ND 1 . 9 5 9 1 7 . 0 4 8 2 3 . 8 2 0 325 n s .01 001 1 .858 2 0 . 4 3 6 2 7 . 3 3 2 332 n s .01 001 2 . 2 1 9 3 5 . 2 4 2 4 9 . 5 6 5 647 ns 001 001 * 1 c e l l w i t h e x p e c t e d c o u n t l e s s t h a n 5 109 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s B o s PPG UD PD ND 2 . 5 4 8 41 n s 3 . 4 9 9 44 n s 4 . 0 7 9 2 . 7 4 6 : — n s 5 .179+ - - n s 5 . 7 6 9 3 . 9 3 0 " — n s 6 .735+ — n s 8 . 3 0 5 85 n s n s ns NPPG UD 3 . 0 8 4 98 n s 3 . 4 2 4 103 n s 3 . 8 0 3 PD 9 . 2 2 9 ~ ns 9 . 1 5 9 n s 1 6 . 9 8 9 ND 1 1 . 6 0 1 — . 0 5 9 . 4 5 5 n s 1 9 . 4 9 2 201 ns .01 .01 TG UD 2 . 0 1 6 139 n s PD 1 0 . 3 7 5 n s ND 1 3 . 6 2 1 .02 1 .630 147 n s 2 . 4 2 9 286 ns 1 0 . 9 4 7 n s 2 1 . 5 5 3 .001 1 3 . 6 8 4 .01 2 7 . 4 7 2 .001 B u f PPG UD 1 .951 52 n s 2.354<< 25 n s 2 . 9 1 0 PD 1 .651 — n s 0 . 7 9 2 ! ! — n s 0 . 7 8 1 ND 2 . 6 6 5 " — n s 1 . 3 2 3 " " — n s 3 . 2 0 0 77 n s n s n s NPPG UD 2 . 5 6 4 114 n s 3 . 9 0 5 PD 3 . 3 4 1 n s 2 . 4 4 0 ND 4 . 7 9 3 n s 2 . 8 8 3 98 n s n s n s 5 . 855 4 . 8 8 2 4 . 7 9 3 212 n s ns ns TG UD 3 . 0 9 3 166 ns PD 3 . 6 2 1 n s ND 7 . 1 6 8 n s 3 . 6 6 0 123 n s 6 . 3 2 7 1 . 4 1 3 n s 5 . 3 1 6 3 . 8 9 4 n s 1 0 . 7 6 8 289 ns n s n s C a l PPG UD 2 . 3 3 3 61 ns . 5 . 8 3 3 " 41 n s 5 . 4 3 2 PD 3 . 8 6 3 — n s 3 . 5 2 K — n s 4 . 4 9 3 ND 4 . 9 3 6 - - ns 3 . 5 6 0 * * — n s 6 . 9 1 8 102 n s ns ns NPPG UD 2 . 5 5 3 137 n s PD 1 2 . 4 3 1 . 0 5 ND 1 5 . 2 0 0 .01 4 . 6 6 8 110 n s 5 . 7 5 8 247 n s 9 . 5 8 0 n s 2 1 . 0 4 6 .001 1 1 . 1 8 9 . 0 5 2 4 . 2 1 8 .001 TG UD 2 . 6 8 1 198 n s PD 1 4 . 4 3 0 .02 ND 1 7 . 7 7 4 .01 4 . 3 4 5 151 n s 6 . 1 8 9 349 ns 9 . 3 5 4 n s 2 2 . 7 7 7 .001 1 2 . 3 7 6 . 0 5 2 9 . 5 5 3 .001 N B . Number o f e x p e c t e d c o u n t s l e s s t h a n 5 i n d i c a t e d b y f o l l o w i n g : " = 2 , ! = 3 , + = 4 , < = 5 , * * = 6 , " " = 7 , ! ! = 8, << = 10 110 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games T o t a l Games 2 2 X n I s X n I s C h i PPG UD PD ND 4 . 2 9 9 1 . 8 7 9 ' 2 . 4 8 2 ' 49 n s n s n s 2 . 3 4 0 1 .664 i 42 n s n s n s 3 . 5 5 7 2 . 0 2 1 3 . 8 7 2 89 n s n s ns NPPG UD 2 . 0 2 9 115 n s 1 .512 PD 3 . 1 4 6 n s 6 . 6 9 1 ND 4 . 9 2 3 n s 4 . 9 2 3 89 n s n s n s 0 . 9 1 5 8 . 0 5 5 1 3 . 1 0 6 204 n s ns .05 TG UD 3 . 9 0 3 164 n s PD 4 . 0 2 9 n s ND 7 . 0 6 6 n s 0 . 8 1 5 131 n s 2 . 5 5 1 6 . 7 4 9 n s 8 . 6 2 8 9 . 8 8 8 n s 1 6 . 0 7 8 295 n s ns .01 D e t PPG UD PD ND 4 . 9 0 0 " 41 n s 7 . 5 7 1 ! — ns 8 . 4 9 9 ! — n s 1 . 7 7 6 * 32 n s 4 . 0 5 6 1 . 1 0 6 * * — n s 6 . 0 9 1 1 . 4 6 1 * * — n s 8 . 4 8 4 73 n s ns ns NPPG UD 5 . 2 1 7 115 ns PD 9 . 6 4 9 n s ND 1 0 . 2 1 6 n s 1 .027 5 . 1 4 3 -7 . 7 5 8 -122 n s n s n s 4 . 6 1 9 1 3 . 8 6 1 1 7 . 7 4 2 237 n s .02 .01 TG UD 3 . 0 8 6 156 n s PD 1 0 . 1 9 5 n s ND 1 2 . 9 6 7 . 0 5 1 .086 154 n s 2 . 7 9 4 4 . 8 3 7 — n s 1 3 . 8 5 8 8 . 8 1 2 n s 2 1 . 0 0 0 310 ns .02 001 Edm PPG UD PD ND 1 .938 43 n s 6 . 1 9 5 — n s 1 .215+ — n s 5 . 6 5 3 " 39 n s 4 . 6 7 7 5 . 9 4 6 : — n s 3 . 3 8 0 6 .879+ — n s 6 . 1 1 9 82 n s ns n s NPPG UD PD ND 1 .938 128 6 . 1 9 5 8 .698 n s 7 . 6 5 0 111 n s 3 . 6 7 6 n s 1 4 . 7 9 0 .02 1 6 . 9 4 5 n s 1 4 . 5 1 6 .02 1 8 . 8 6 4 239 n s .01 .01 TG UD PD ND 2 . 9 3 0 171 5 . 8 1 0 8 . 5 8 5 n s 5 . 5 5 0 150 n s 4 . 5 8 4 ns 1 4 . 2 1 7 .02 1 7 . 7 3 6 n s 1 5 . 4 6 7 .01 2 2 . 1 1 9 321 n s .01 001 N B . Number o f e x p e c t e d c o u n t s l e s s t h a n f i v e d e n o t e d b y : = 1, = 2 , 3 , + = 4 , * * = 6 I l l T a b l e A7 I n d i v i d u a l C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games T o t a l Games 2 2 X n l s X n l s H a r PPG UD PD ND 3 . 5 0 2 3 . 0 8 3 " 4 . 9 3 6 ! 41 n s n s n s 1 . 0 6 5 2 . 0 8 1 " 3 . 3 4 8 ! 46 n s ns n s 2 . 4 8 3 4 . 0 6 9 7 . 1 5 6 87 n s ns n s NPPG UD 3 . 9 8 0 115 n s 1 .811 PD 1 1 . 7 4 6 . 0 5 7 . 1 7 3 ND 1 3 . 6 0 3 .02 9 . 7 6 6 96 n s n s n s 2 . 6 3 1 1 6 . 9 4 5 1 9 . 7 5 2 211 ns .01 .01 TG UD 2 . 3 2 0 156 n s PD 1 2 . 6 9 5 . 0 5 ND 1 5 . 1 4 8 .01 1 .380 142 n s 1 . 1 7 2 8 . 4 0 7 n s 1 8 . 6 4 1 1 2 . 3 0 6 . 0 5 2 4 . 8 3 2 298 ns .01 001 LA PPG UD PD ND 4 . 6 8 9 4 . 1 2 8 ' 5 . 6 6 9 ' 51 ns n s n s 1 .468 28 1.612++ — n s n s ns 3 . 8 1 8 3 . 5 5 5 5 . 4 3 8 79 n s n s ns NPPG UD 4 . 7 4 1 159 n s PD 1 2 . 6 1 9 . 0 5 ND 1 4 . 4 1 9 .02 1 . 2 8 5 132 n s 3 . 8 6 2 5 . 2 1 3 n s 1 7 . 4 2 0 9 . 1 8 3 ns 2 2 . 5 5 2 291 n s .01 001 TG UD 3 . 5 2 5 210 n s PD 1 1 . 8 5 3 . 0 5 ND 1 5 . 7 9 3 .01 2 . 1 6 4 160 n s 4 . 2 8 7 7 . 3 0 4 n s 1 8 . 1 7 2 1 0 . 5 2 4 n s 2 5 . 0 8 7 370 ns .01 001 M i n PPG UD 1 . 5 4 9 45 n s PD 4 . 2 6 0 * — n s ND 4 . 6 2 1 * — n s i 33 n s 2 . 0 1 6 4 . 8 8 5 * * — n s 7 . 1 6 6 4 . 2 7 7 * * — n s 8 . 1 1 3 78 ns ns n s NPPG UD 1 .317 96 ns 2 . 3 2 6 PD 4 . 3 0 2 — n s 1 0 . 3 6 4 ND 6 . 2 4 9 — ns 1 1 . 3 2 1 84 n s n s ,01 1 . 9 8 4 1 5 . 3 8 1 1 7 . 0 1 1 180 n s .01 .01 TG UD 1 .241 141 n s PD 7 . 2 6 0 n s ND 9 . 4 2 9 n s 3 . 0 9 3 117 n s 2 . 7 3 5 1 5 . 3 2 8 .01 2 1 . 0 8 2 1 5 . 2 0 7 .01 2 3 . 2 7 1 258 ns 001 001 NB, Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1, = 2 , 3 , 6 = 8 = i i I I r 9 = + + 112 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s Mon PPG UD PD ND 3 . 8 3 9 " 37 n s 5 .365+ - - ns 6 .930+ — n s 5 . 6 6 2 " 43 n s 5 . 8 1 1 3 . 7 8 5 * * — ns 4 . 5 7 4 3 .759< — n s 7 . 5 2 0 80 n s n s ns NPPG UD PD ND 1 .594 123 n s 8 . 2 2 7 n s 9 . 0 0 3 n s 1 . 5 0 6 110 ns 0 . 8 6 7 5 . 1 5 9 n s 1 0 . 8 0 9 7 . 5 4 4 n s 1 4 . 6 5 7 233 ns n s .02 TG UD 0 . 2 5 0 160 n s PD 8 . 3 1 7 n s ND 1 0 . 9 6 9 n s 2 . 6 9 7 153 n s 1 . 5 5 4 3 . 2 0 0 n s 1 0 . 7 6 5 7 . 6 4 1 n s 1 8 . 0 2 8 313 n s n s .01 N J PPG UD PD ND 3 . 0 7 3 5 . 1 6 2 ' 5 . 4 4 7 ' 46 ns n s n s 2 . 1 0 5 6 . 8 3 8 " 6 . 8 2 3 * 39 n s n s n s 2 . 1 2 6 8 . 9 2 9 9 . 1 4 1 85 n s ns ns NPPG UD 1 .317 97 n s 3 . 4 8 3 98 n s 3 . 1 4 7 PD 4 . 1 2 3 — n s 1 1 . 1 7 0 — n s 1 4 . 7 3 7 ND 6 . 8 6 7 — n s 1 1 . 6 2 0 — . 0 5 1 8 . 2 8 3 195 ns .02 .01 TG UD 1 . 4 2 9 143 n s PD 5 . 8 2 8 n s ND 9 . 1 8 9 n s 3 . 3 8 9 137 n s 3 . 0 8 9 1 5 . 7 4 3 .01 2 1 . 4 2 9 1 7 . 3 4 7 .01 2 5 . 4 4 5 280 n s .001 .001 NYI PPG UD PD ND 4 . 8 4 5 " 43 1 . 5 5 1 ! — 1 .978+ — ns 5 . 3 3 8 41 n s n s 1 .348< — n s n s 1 .469< — n s 9 . 2 3 3 84 ns 2 . 5 4 4 — ns 3 . 3 8 7 * — ns NPPG UD 3 . 3 6 8 96 ns 1 .878 82 n s 3 . 8 0 3 178 ns PD 1 0 . 9 4 2 — n s 3 . 9 9 8 — n s 1 4 . 1 8 5 .02 ND 1 3 . 3 7 0 - - . 0 5 5 . 5 4 4 — n s 1 7 . 2 9 9 .01 TG UD 4 . 4 3 7 139 n s PD 9 . 2 6 5 n s ND 1 2 . 5 4 9 . 0 5 4 . 6 4 0 123 n s 3 . 2 0 5 n s 6 . 3 4 7 n s 5 . 5 8 1 262 ns 1 0 . 7 3 0 ns 1 6 . 8 9 2 .01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: " = 2 , ! = 3 , + = 4 , < = 5 , * * = 6 113 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n l s T o t a l Games 2 X n l s NYR PPG UD PD ND 4 . 6 2 7 " 39 n s 1 .527 5 .474+ — n s 1 . 4 2 4 ' 6 .582+ — n s 2 . 1 4 2 ' 45 n s n s n s 2 . 7 7 6 4 . 1 9 9 6 . 7 9 8 84 n s n s n s NPPG UD 4 . 1 1 9 121 n s PD 8 .881 n s ND 1 0 . 8 3 1 n s 0 . 4 6 8 104 n s 2 . 3 0 5 6 . 5 2 1 n s 1 3 . 3 0 9 7 . 6 6 2 n s 1 6 . 9 7 9 225 ns . 0 5 .01 TG UD 2 . 1 0 0 160 n s PD 8 . 4 6 5 ns ND 1 1 . 5 4 9 . 0 5 0 . 3 0 5 149 n s 1 . 1 7 9 6 . 4 8 6 ns 1 4 . 0 9 6 9 . 4 8 2 n s 2 0 . 2 2 3 309 n s .02 .01 P h i PPG UD PD ND 4 . 2 5 1 54 n s 5 . 1 7 2 45 n s 5 . 8 9 1 3 . 6 8 2 " — n s 2 .159+ — n s 3 . 0 9 6 5 . 9 8 6 * — n s 3 .504+ — n s 6 . 6 8 6 99 n s n s ns NPPG UD 1 . 4 0 9 111 n s 5 . 7 1 7 PD 6 . 5 4 5 n s 1 5 . 6 7 5 ND 9 . 5 5 6 n s 1 5 . 8 3 2 96 n s .01 .01 1 . 9 1 4 1 8 . 6 6 6 2 1 . 4 1 4 207 n s .01 001 TG UD 1 .640 165 ns PD 7 . 1 3 2 n s ND 1 1 . 9 8 1 n s 4 . 9 9 2 141 n s 2 . 8 9 1 1 3 . 1 2 8 . 0 5 1 8 . 5 0 0 1 6 . 7 1 9 .01 2 7 . 0 2 9 306 n s .01 001 P i t PPG UD PD ND 3 . 3 3 3 3 .208" 4 . 6 3 4 ' 60 n s n s n s 0 . 8 5 2 1 . 7 1 9 2 . 9 5 2 59 n s n s n s 3 . 0 3 9 3 . 5 1 0 7 . 4 0 6 119 ns n s n s NPPG UD 4 . 0 1 1 125 n s PD 1 4 . 4 6 7 .02 ND 1 4 . 5 5 1 .02 1 .587 101 n s 1 .068 3 . 4 9 6 n s 1 4 . 7 4 6 5 . 9 9 3 — ns 1 7 . 6 3 8 226 n s .02 .01 TG UD 2 . 0 8 6 185 n s PD 1 2 . 5 7 6 . 0 5 ND 1 5 . 1 7 5 .01 2 . 2 4 5 160 n s 1 . 7 7 0 5 .148 n s 1 6 . 0 9 4 8 . 9 8 0 n s 2 3 . 6 7 2 345 ns .01 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : * = 1 , " = 2 , + = 4 114 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s Que PPG UD PD ND 5 . 6 5 1 " 44 n s 1 .621 2 .487+ - - n s 3 . 6 1 5 ! 4 .341+ — ns 4 . 6 1 6 ! 41 n s ns n s 4 . 6 8 0 4 . 6 3 0 7 . 8 4 9 85 ns ns ns NPPG UD 3 . 4 2 8 105 n s 4 . 2 9 8 PD 4 . 8 5 9 n s 8 . 6 5 8 ND 6 . 1 0 8 ns 9 . 4 9 0 77 n s n s n s 2 . 831 1 1 . 6 3 7 1 5 . 0 6 1 182 ns ,05 ,02 TG UD 3 . 8 6 3 149 n s PD 5 . 2 4 8 n s ND 9 . 9 2 2 n s 2 . 2 4 5 118 n s 3 . 5 2 0 9 . 0 8 9 n s 1 2 . 9 5 4 1 1 . 9 1 2 . 0 5 1 9 . 9 6 1 267 n s , 05 .01 S t L PPG UD PD ND 6 . 4 1 2 " 33 n s 3 . 0 0 8 * — n s 3 . 1 4 8 * * — n s 5 . 1 4 7 " 41 n s 1 0 . 2 1 0 0 .053+ — n s 1 . 3 9 3 ' 1 .357+ — n s 1 . 9 5 3 ' 70 n s n s ns NPPG UD 1 .041 113 n s 2 . 8 6 5 PD 8 . 2 7 7 n s 4 . 7 8 4 ND 1 1 . 5 6 0 . 0 5 5 . 0 7 5 ' 85 n s ns n s 1 . 8 8 9 1 2 . 7 2 4 1 6 . 1 1 8 198 ns ,05 ,01 TG UD 0 . 7 8 0 146 ns PD 6 . 4 8 3 n s ND 9 . 8 0 6 n s 0 . 9 6 0 126 n s 1 .074 4 . 1 4 7 n s 1 0 . 3 3 3 7 . 2 2 3 n s 1 7 . 0 0 0 268 ns ns ,01 T o r PPG UD PD ND 3 . 1 4 0 35 0 .520+ - -1 .192+ — n s 1.900>> 20 ns 2 . 9 5 8 ns 2.93.1<< — n s 1 .021 n s 2.409<< — n s 1 . 4 9 4 ' 55 n s ns n s NPPG UD 3 . 5 4 8 98 n s PD 1 3 . 2 3 2 — . 0 5 ND 1 4 . 0 6 1 ~ .02 6 . 2 6 5 105 n s 1 . 6 8 3 6 . 0 6 1 n s 1 1 . 0 8 0 6 . 6 8 4 ns 1 4 . 2 2 2 203 n s .05 ,02 TG UD 1 . 9 0 3 133 n s PD 9 . 3 6 6 n s ND 1 2 . 0 2 7 . 0 5 5 . 6 4 0 125 n s 2 . 4 2 8 5 .111 n s 8 . 9 9 7 6 . 1 7 9 ns 1 4 . 4 3 0 258 ns ns 02 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : " = 2 , ! = 3 , + = 4 , * * = 6 , << = 1 0 , >> = 11 115 T a b l e A7 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s F o r G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n l s T o t a l Games 2 X n l s V a n PPG UD PD ND 3 . 2 7 7 * 44 n s 1 .475+ — n s 1 .743+ — n s 1.8751 34 n s 1 .881 4 .841+ — n s 2 . 7 5 8 5 .024+ — n s 4 . 6 5 2 78 ns n s n s NPPG UD PD ND 2 . 4 1 9 7 . 5 6 6 7 . 9 5 2 77 n s n s n s 1 . 2 3 2 4 . 4 6 0 5 . 9 3 3 94 n s n s n s 2 . 4 3 5 1 0 . 2 8 9 1 3 . 6 7 5 171 n s n s ,02 TG UD 1 . 8 0 5 121 n s 0 . 5 5 4 128 n s 1 . 0 6 2 PD 4 . 2 6 2 n s 5 . 9 2 0 n s 9 . 5 3 6 ND 7 . 7 6 8 n s 8 . 6 1 4 n s 1 5 . 4 3 1 249 ns n s .01 Was PPG UD PD ND 0 . 2 6 6 2 . 9 2 6 4 . 1 1 5 55 n s ns n s 2 . 5 9 5 * 1.2641 -1 .976+ -40 n s n s n s 1 . 5 6 9 3 . 4 7 4 5 . 4 0 9 95 n s ns n s NPPG UD 1 .777 104 n s 4 . 4 9 7 104 n s 1 . 9 8 3 PD 8 . 2 8 0 n s 1 3 . 3 4 4 . 0 5 1 7 . 3 2 5 ND 1 0 . 1 6 5 n s 1 4 . 3 4 0 .02 1 9 . 0 9 4 208 ns ,01 ,01 TG UD 0 . 8 0 7 159 n s 3 . 9 7 6 144 n s 1 . 0 1 2 PD 9 . 6 6 7 n s 1 1 . 8 5 6 . 0 5 1 8 . 2 2 2 ND 1 2 . 3 4 7 . 0 5 1 3 . 8 7 8 .02 2 3 . 7 5 6 303 n s .01 001 Win PPG UD PD ND 2 . 1 2 4 1 .767 3 . 1 9 6 45 n s n s ns 5 . 0 7 7 ' 1 .087< -1 . 6 7 8 * * -34 n s n s n s 4 . 3 2 8 0 . 6 7 9 3 . 5 5 0 79 n s n s n s NPPG UD 5 . 7 8 2 104 n s 2 . 5 2 1 111 n s 3 . 0 5 5 PD 1 6 . 8 5 3 .01 7 . 8 7 2 n s 1 9 . 2 0 9 ND 1 7 . 0 4 6 .01 1 0 . 3 0 6 n s 2 2 . 5 8 7 215 n s .01 001 TG UD 3 . 4 3 3 149 n s 1 .741 145 n s 2 . 3 6 7 PD 1 5 . 1 2 9 .01 5 . 3 0 9 n s 1 7 . 8 7 8 ND 1 7 . 2 5 4 .01 9 . 6 0 9 n s 2 5 . 0 5 8 294 n s .01 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 ! = 3 , + = 4 , < = 5 , * * = 6 116 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s B o s PPG UD PD ND 3 .886+ 36 n s 1 . 2 2 1 * * — n s 1 . 9 7 5 * * — n s 3 .956+ 30 n s n s n s 5 . 1 3 5 2 . 7 8 9 5 . 8 1 7 66 n s n s n s NPPG UD PD ND 1 .878 9 . 4 4 7 9 . 8 6 3 97 n s n s n s 0 . 9 9 2 4 . 4 6 3 6 . 7 3 5 91 n s n s n s 5 . 7 8 6 1 1 . 3 9 8 1 5 . 2 2 3 188 n s .05 .01 TG UD 2 . 3 2 6 133 n s PD 6 . 8 9 0 n s ND 1 0 . 2 8 3 n s 1 .671 121 n s 2 . 0 5 9 5 . 3 7 4 n s 1 0 . 1 5 7 9 . 1 1 5 n s 1 7 . 1 9 0 254 n s n s 01 B u f PPG UD PD ND 2 . 3 4 0 46 n s 4 . 7 6 0 * — n s 5 . 2 5 9 * — n s € . 0 7 1 " 40 n s 3 . 0 1 2 1 .034+ - - n s 1.24.2 0 .674+ — n s 4 . 1 2 4 86 n s ns ns NPPG UD 1 . 2 4 7 93 n s 2 . 2 2 4 116 n s 2 . 6 7 1 PD 6 . 5 6 2 — n s 1 3 . 6 4 2 .02 1 8 . 5 2 2 ND 8 . 4 3 2 — n s 1 4 . 0 3 6 .02 2 2 . 1 3 8 209 ns .01 001 TG UD 2 . 3 8 1 139 n s PD 9 . 0 9 7 n s ND 1 3 . 6 9 7 .02 0 . 8 1 2 156 n s 2 . 2 8 2 8 . 8 3 6 n s 1 8 . 2 2 9 1 2 . 5 0 0 . 0 5 2 5 . 4 6 6 295 n s .01 001 C a i PPG UD PD ND 5 .401+ 34 n s 3 . 0 5 4 * * — n s 3 .1461 ~ n s 5 . 3 1 4 * 44 2 .044+ — n s n s n s 8 . 796 2 . 6 1 9 4 . 4 9 9 78 n s ns ns NPPG UD 0 . 8 2 1 68 n s 0 . 8 6 0 PD 5 . 7 1 4 — n s 3 . 6 6 9 ND 6 . 3 3 6 — n s 5 . 0 0 3 77 n s ns n s 0 . 4 1 2 7 . 552 1 0 . 1 8 6 145 n s n s n s TG UD 1 . 2 4 9 102 n s PD 2 . 9 6 9 - - - n s ' ND 5 . 8 2 7 n s 2 . 8 8 7 121 n s 3 . 3 0 7 2 . 0 5 2 n s 5 . 0 3 4 5 . 4 6 0 n s 1 1 . 1 7 8 223 ns n s .05 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: = 1, = 2 , = 3 , + = 4 , = 6 , = 8 117 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n and N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n T o t a l Games 2 l s X n l s C h i PPG UD PD ND 2 . 9 2 5 2 . 2 0 8 3 . 0 1 7 ' 51 n s n s n s 1 .416 3 . 4 3 9 5 . 1 5 0 71 n s n s n s 2 . 3 0 5 3 . 7 9 3 6 . 8 2 6 122 n s n s n s NPPG UD PD ND 5 . 3 2 5 102 n s 3 . 6 5 5 n s 4 . 3 3 0 n s 2 . 2 2 0 108 n s 4 . 4 9 8 9 . 2 8 4 n s 1 0 . 9 0 7 9 . 0 0 3 n s 1 3 . 4 8 5 210 ns n s .02 TG UD 7 . 1 4 0 153 n s PD 5 . 4 0 9 n s ND 7 . 1 3 2 n s 1 . 3 0 3 179 n s 4 . 2 3 0 1 1 . 1 7 4 . 0 5 1 3 . 0 4 9 1 3 . 9 0 5 .02 1 9 . 0 0 4 332 ns .05 .01 D e t PPG UD PD ND 1 .422< 32 n s 4 . 1 2 9 * 1 . 5 1 9 * * — n s 3 . 0 8 5 " 2 .236+ - - n s 2 . 8 7 0 " 46 n s n s n s 4 . 4 7 3 3 . 6 4 1 5 . 4 8 6 78 n s n s n s NPPG UD 5 . 3 4 3 108 n s PD 9 . 5 2 2 n s ND 1 2 . 0 5 4 . 0 5 2 . 3 3 3 129 n s 6 . 5 7 3 8 . 1 1 6 n s 1 5 . 8 2 1 1 0 . 5 2 8 n s 2 0 . 6 7 5 237 n s .01 .001 TG UD 4 . 4 2 1 140 n s PD 8 .411 n s ND 1 2 . 4 7 0 . 0 5 3 . 1 7 8 175 n s 6 . 3 7 3 7 . 5 3 3 n s 1 5 . 8 2 7 1 0 . 8 1 3 n s 2 2 . 7 5 2 315 n s .01 001 Edm PPG UD PD ND 2 . 7 7 8 * * 30 n s 0 . 9 9 5 " " — n s 3 . 0 0 4 * * — n s 3 . 5 8 5 47 n s 6 . 0 9 2 1 . 7 4 3 " — n s 3 . 0 0 3 3 .128+ — n s 5 . 7 3 5 77 ns n s n s NPPG UD PD ND 1 .491 119 n s 7 . 5 9 9 n s 9 . 5 5 2 n s 1 .628 107 n s 0 . 6 9 2 1 1 . 2 2 7 . 0 5 1 6 . 3 0 4 1 2 . 5 3 2 . 0 5 1 9 . 5 6 4 226 n s .01 .01 TG UD 1 . 0 6 9 149 n s PD 6 . 6 7 3 n s ND 9 . 6 6 4 ns 0 . 9 7 8 154 n s 0 . 5 5 0 9 . 5 3 5 n s 1 5 . 2 2 7 1 3 . 1 7 0 . 0 5 2 1 . 6 0 4 303 ns .01 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: " = 2 , < = 5 , * * = 6 118 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n T o t a l Games 2 I s X n I s H a r PPG UD PD ND 3 . 8 3 1 * 40 n s 6 . 4 4 8 * — n s 7 . 1 3 3 * — n s 30 n s 4 . 9 8 2 4 .029+ - - n s 9 . 6 1 8 3 .545< — n s 1 0 . 3 0 4 70 n s n s ns NPPG UD PD ND 2 . 2 8 2 102 n s 5 . 6 5 1 n s 7 . 8 2 1 n s 1 . 3 7 5 114 n s 2 . 3 8 9 8 . 4 6 4 n s 1 2 . 9 2 0 1 0 . 0 9 3 n s 1 6 . 1 1 4 216 n s . 05 .01 TG UD 3 . 6 1 8 142 n s PD 1 0 . 0 6 0 ns ND 1 2 . 1 2 4 . 0 5 1 . 7 7 4 144 ns 3 . 9 0 4 1 1 . 8 6 2 . 0 5 2 0 . 0 6 5 1 2 . 8 5 7 . 0 5 2 4 . 3 5 6 286 n s .01 001 LA PPG UD 4 . 5 6 6 ! 34 n s PD 2 .079+ — n s ND 2 .151+ — n s 0 . 3 8 1 45 n s 1 . 8 6 9 1 . 6 1 0 * — n s 2 . 0 8 1 2 . 1 9 6 " — n s 3 . 6 9 6 79 ns n s ns NPPG UD 4 . 8 1 3 134 n s PD 1 5 . 8 3 2 .01 ND 1 7 . 9 7 1 .01 8 . 1 6 5 121 n s 1 1 . 3 9 7 1 1 . 8 8 2 . 0 5 2 6 . 8 5 1 1 2 . 9 9 3 . 0 5 3 0 . 4 1 2 255 . 0 5 001 001 TG UD 2 . 5 7 9 168 n s PD 1 6 . 0 2 3 .01 ND 1 8 . 6 2 6 .01 6 . 7 8 8 166 n s 6 . 9 7 3 1 2 . 8 5 3 .01 2 7 . 6 9 6 1 7 . 4 9 2 .01 3 3 . 1 2 5 334 n s 001 001 M i n PPG UD PD ND 0 . 7 6 7 39 n s 0 . 3 4 2 3 . 1 7 4 * — n s 3 . 4 4 5 4 . 0 1 7 * — n s 3 . 2 9 8 43 n s n s n s 0 . 6 5 8 6 . 7 1 3 7 . 420 82 n s ns ns NPPG UD 3 . 9 8 1 95 n s 2 . 2 3 0 100 n s 4 . 4 2 3 PD 5 . 9 0 6 — n s 7 . 0 2 4 n s 1 0 . 9 5 0 ND 7 . 7 4 6 — n s 8 . 5 2 8 n s 1 4 . 7 7 9 195 n s n s .02 TG UD 3 . 0 6 5 134 n s PD 8 . 4 9 7 n s ND 1 1 . 0 4 3 n s 1 . 4 5 3 143 n s 3 . 5 1 6 9 . 7 8 4 n s 1 7 . 0 3 2 1 1 . 5 5 5 . 0 5 2 1 . 9 8 7 277 ns .01 ,001 NB, Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * =1, = 2 , = 3 , + = 4 , < = 5 , = 8 119 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n l s T o t a l Games 2 X n l s Mon PPG UD PD ND 2 . 6 1 4 " " 28 n s 2.511<< — n s 3.230++ — n s 4 .015+ 30 n s 2 . 2 9 5 0 . 4 3 3 * — n s 1 . 6 7 0 1 . 1 5 3 * — n s 2 . 4 7 5 58 n s ns n s NPPG UD PD ND 4 . 9 8 6 6 . 0 4 1 7 . 5 2 5 76 n s n s n s 0 . 5 1 0 4 . 0 3 4 4 . 9 4 9 84 n s n s n s 3 . 7 9 6 8 . 4 1 8 9 . 9 2 7 160 n s n s n s TG UD 3 . 9 2 2 104 n s 1 . 8 0 5 PD 6 . 3 9 6 n s 3 . 4 4 0 ND 7 . 3 8 9 n s 5 . 7 0 5 114 n s n s n s 4 . 2 4 8 8 . 0 8 7 1 2 . 4 5 0 218 n s n s .05 N J PPG UD PD ND 0 . 4 8 1 1 .958 2 . 7 8 8 56 n s n s n s 1 .866 3 . 1 5 9 3 . 9 1 0 57 n s n s n s 1 . 8 5 4 4 . 4 6 1 6 . 6 5 4 113 ns n s n s NPPG UD PD ND 3 . 4 8 5 8 . 5 6 6 1 0 . 4 2 6 91 n s n s n s 2 . 8 2 1 1 2 . 9 2 1 1 4 . 1 2 4 117 n s . 0 5 .02 4 . 0 0 1 1 9 . 5 4 7 2 2 . 9 0 8 208 n s .01 001 TG UD 2 . 8 5 9 147 ns 1 . 9 4 9 PD 1 0 . 2 6 0 n s 1 5 . 2 7 2 ND 1 2 . 4 2 8 . 0 5 1 7 . 1 3 9 174 n s .01 .01 3 . 8 3 2 2 2 . 8 4 9 2 8 . 8 2 7 321 ns 001 001 NY I PPG UD PD ND 0 . 3 0 0 2 . 4 1 4 3 . 3 1 1 * -49 n s n s n s 2 . 1 6 9 4 . 5 9 9 ' 6 . 3 4 4 ' 54 n s n s n s 0 . 9 4 3 6 . 4 2 5 7 . 776 103 n s n s n s NPPG UD 1 .250 100 n s PD 1 0 . 0 5 3 n s ND 1 0 . 4 1 4 n s 3 . 1 6 5 119 ns 2 . 6 7 9 4 . 5 6 7 n s 1 2 . 1 7 5 6 . 8 8 4 n s 1 6 . 6 6 4 219 n s ,05 .01 TG UD 0 . 7 7 3 149 n s 2 . 5 5 3 173 n s 1 . 8 6 2 PD 1 1 . 4 8 7 . 0 5 7 . 1 0 6 n s 1 7 . 1 5 1 ND 1 3 . 7 7 5 .02 1 0 . 7 0 5 n s 2 3 . 8 2 7 322 n s .01 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1 , " = 2 , + = 4 , " " = 7 , ++ = 9 , << = 10 120 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n I s Away Games 2 X n I s T o t a l Games 2 X n I s NYR PPG UD PD ND 12 .110< 37 n s 3 . 6 9 2 5 . 5 8 0 * * — n s 1 .601+ 4 .514+ — n s 3 . 1 9 8 ! 48 n s n s n s 9 . 7 9 2 1 . 4 2 3 3 . 3 0 4 85 n s n s n s NPPG UD PD ND 1 . 4 3 7 1 0 . 9 6 3 1 1 . 7 1 2 99 n s n s . 0 5 1 .457 4 . 4 9 5 6 . 3 2 6 112 n s n s n s 1 . 5 5 4 1 3 . 2 0 4 1 6 . 4 8 2 221 ns . 0 5 .01 TG UD 1 .127 146 n s 1 .594 PD 6 . 5 3 3 ns 4 . 1 8 3 ND 9 . 6 3 7 n s 7 . 4 6 1 160 n s n s n s 1 . 9 5 4 8 .946 1 6 . 4 5 1 306 n s n s .01 P h i PPG UD PD ND 4 .103+ 7 .421+ 7 .456+ 38 n s n s n s 2 . 7 7 6 4 . 4 4 6 ' 6 . 5 1 1 ' 42 n s n s n s 5 . 6 1 0 1 1 . 5 3 8 1 2 . 7 2 5 80 n s ,05 ,05 NPPG UD 1 . 7 4 4 92 n s 2 . 2 8 1 PD 2 . 1 7 3 — n s 5 . 4 8 1 ND 3 . 7 4 9 — n s 9 . 1 1 9 108 n s n s n s 2 . 4 8 0 5 . 4 2 5 1 1 . 2 1 2 200 ns n s .05 TG UD 1 . 4 0 5 130 ns 2 . 2 4 9 PD 6 . 0 7 3 n s 8 . 0 2 3 ND 9 . 6 9 7 ns 1 2 . 3 8 4 150 n s n s . 0 5 1 .658 1 2 . 1 9 7 2 0 . 1 2 8 280 n s ,05 ,01 P i t PPG UD PD ND 1 .728 3 . 3 5 5 5 . 4 5 9 55 n s n s n s 2 . 4 9 8 6 . 1 5 3 6 . 8 7 0 57 n s ns n s 2 . 0 9 4 7 . 7 0 2 9 . 8 5 7 112 ns n s n s NPPG UD 1 . 6 5 0 104 n s 0 . 8 7 0 PD 6 . 6 2 4 n s 8 . 0 0 2 ND 8 . 0 6 4 n s 1 0 . 6 2 4 132 n s n s n s 2 . 2 0 2 1 3 . 2 1 2 1 8 . 6 3 5 236 n s .05 ,01 TG UD 2 . 0 4 5 159 n s 1 .518 PD 8 . 6 9 8 n s 1 1 . 8 0 2 ND 1 2 . 7 8 1 . 0 5 1 4 . 8 8 9 189 n s ,05 02 3 . 1 6 2 1 9 . 7 3 0 2 6 . 7 6 7 348 n s .02 001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : = 3 , + = 4 , < = 5 , * * = 6 i _ i 121 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t G o a l T y p e D i s t ' n Home Games 2 X n l s Away Games 2 X n T o t a l Games 2 l s X n l s Que PPG UD PD ND i 33 n s 2 . 1 1 9 ! — n s 3 . 6 9 7 * * — n s 4 . 6 6 4 48 n s 6 . 9 7 0 4 . 3 0 8 ! ~ n s 5 . 9 5 3 5 .339+ — n s 8 . 5 4 8 81 n s n s n s NPPG UD PD ND 0 . 5 4 2 127 n s 6 . 2 8 3 n s 9 . 6 7 8 n s 0 . 9 8 2 133 n s 1 . 4 0 0 9 . 2 6 6 n s 1 5 . 4 4 9 1 1 . 3 5 8 . 0 5 3 0 . 9 4 6 260 n s .01 001 TG UD 1 . 6 8 9 160 n s PD 7 . 7 2 8 n s ND 1 1 . 5 5 2 . 0 5 1 . 9 7 0 181 n s 3 . 4 9 9 9 . 7 5 7 n s 1 6 . 9 7 5 1 4 . 5 6 2 .02 2 5 . 9 7 2 341 n s .01 001 S t L PPG UD PD ND 2 .660+ 30 n s 2 . 5 8 2 2 . 8 8 5 * * — n s 0 . 7 4 2 ' 3 . 5 3 8 * * — n s 1 . 5 4 6 ' 53 n s n s n s 3 . 2 4 6 1 . 3 2 6 3 . 9 2 0 83 n s ns n s NPPG UD PD ND 2 . 1 1 3 1 0 . 4 2 4 1 0 . 2 4 4 86 n s n s n s 115 1 .491 8 . 5 0 4 — 9 . 0 6 8 — n s n s n s 2 . 3 8 5 1 8 . 2 2 5 1 9 . 3 3 2 201 ns ,01 ,01 TG UD 0 . 9 8 7 116 n s PD 9 . 5 3 6 n s ND 1 1 . 2 2 2 . 0 5 4 . 4 5 4 174 n s 4 . 1 8 8 3 . 2 0 2 n s 8 . 9 3 3 6 . 5 9 6 n s 1 5 . 6 3 2 338 ns ns ,01 T o r PPG UD PD ND 5 . 1 2 2 46 n s 1 . 2 5 2 ! — n s 1 .765+ — n s 1 .272 52 n s 4 . 2 6 5 1 .022 — n s 0 . 8 2 0 2 . 0 0 7 * — n s 2 . 4 6 9 98 ns ns ns NPPG UD PD ND 1 . 4 2 7 118 n s 9 . 9 7 5 n s 1 0 . 9 0 4 n s 4 . 0 9 8 122 n s 2 . 4 1 7 3 . 2 2 7 n s 1 0 . 4 2 4 4 . 2 9 0 n s 1 4 . 6 7 0 240 ns n s ,02 TG UD 1 . 3 3 7 164 n s PD 6 . 9 3 8 n s ND 1 0 . 6 9 5 n s 4 . 4 5 4 174 n s 4 . 1 8 8 3 . 2 0 2 n s 8 . 9 3 3 6 . 5 9 6 n s 1 5 . 6 3 2 338 n s ns ,01 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5: * = 1 , " = 2 , ! = 3 , + = 4 , * * = 6 122 T a b l e A8 I n d i v i d u a l Team C h i - s q u a r e A n a l y s i s o f U n i f o r m , P o i s s o n a n d N o r m a l D i s t r i b u t i o n s f o r A c t u a l P l a y i n g T i m e (Ten M i n u t e D i v i s i o n s : d f = 5) ( c o n t i n u e d ) G o a l s A g a i n s t Home Games Away Games T o t a l Games G o a l T y p e 2 D i s t ' n X n l s 2 X n l s 2 X n l s V a n PPG UD 3 . 4 6 8 ! ! 29 n s 3 . 4 9 1 " 36 n s 3 . 2 7 9 65 n s PD 2 . 0 3 7 ! 1 — n s 2 . 8 8 2 * * — n s 1 . 516 — ns ND 2 . 4 3 5 ! I — n s 4 . 1 6 0 * * — n s 2 .870 — ns NPPG UD 4 . 5 3 5 79 n s 3 . 073 105 n s 5 . 0 4 5 184 n s PD 1 2 . 0 1 6 — . 0 5 9 .828 n s 19 . 934 .01 ND 1 2 . 5 2 7 — . 0 5 11 .391 05 22 .007 • . 001 TG UD 3 . 4 6 6 108 n s 2 . 440 141 n s 4 .107 249 ns PD 9 . 5 5 4 n s 5 .098 n s 18 .168 .01 ND 1 1 . 3 0 8 . 0 5 8 . 746 n s 22 .914 .001 Was PPG UD 2 . 3 7 3 ! ! 31 n s 8 .206+ 42 n s 6 . 2 0 9 73 n s PD 2 . 3 3 4 ! 1 — n s 2 .504 + — n s 0 . 8 6 3 -- n s ND 2 . 6 1 6 " " — n s 1 .204+ — n s 0 . 877 — n s NPPG UD 1 .238 84 n s 1 .950 98 n s 2 . 316 182 ns PD 3 . 0 4 4 — ns 5 . 769 — n s 8 . 367 n s ND 4 . 6 8 7 — n s 7 .544 -— n s 11 .051 n s TG UD 1 . 5 1 4 115 n s , 3 . 773 140 n s 4 .708 255 n s PD 3 . 1 6 6 n s 3 . 340 n s 5 .627 n s ND 5 . 6 8 5 n s 5 .751 n s 10 .638 ns Win PPG UD 5 .019+ 38 n s 1 . 643 44 n s 4 . 363 82 n s PD 2 . 4 3 3 * * — n s 4 . 4 6 3 -- n s 4 .934 -- n s ND 5 . 3 1 3 * * — n s 5 .861 — n s 8 . 5 5 9 — n s NPPG UD 4 . 0 8 8 128 n s 3 .504 143 n s 5 .327 271 ns PD 7 .468 n s 15 .504 01 21 . 516 .001 ND 9 . 2 9 7 n s 15 .602 01 23 .977 .001 TG UD 2 . 2 2 0 166 n s 2 . 896 187 ns 2 . 6 6 3 353 n s PD 5 . 5 7 7 n s 17 .132 01 21 .181 .001 ND 9 . 1 4 4 n s 19 . 309 — — — ^  01 27 . 2 0 7 .001 N B . Number o f c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 : 2 , + = 4 , * * = 6 , 7 , ! ! =8 123 T a b l e A9 T o t a l Team C h i - s q u a r e C o m p a r i s o n o f S h o t D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s TOTAL SHOTS S h o t s / G a m e E x p e c t e d N e g a t i v e / S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e B i n o m i a l 31 - 4 0 / 1 5 9 140 86 420 - O b s / E x p 41 - 5 0 / 2 103 93 140 153 210 0 1 0 8 / 1 0 9 51 - 6 0 / 3 296 302 140 180 105 1 2 9 6 / 3 0 2 61 - 7 0 / 4 323 316 140 159 53 2 3 2 3 / 3 0 0 71 - 8 0 / 5 95 107 140 113 26 3 9 5 / 95 81 - 9 0 / 6 18 12 140 67 13 4 1 8 / 18 91 - 1 0 0 / 7 1 C h i s q u a r e d f I s N o r m a l 3 .702 5 n . s . U n i f o r m 362 .290 5 001 P o i s s o n 192 .037 5 001 G e o m e t r i c 766 .770 5 001 N e g a t i v e B i n o m i a l 5 . 087 4 n . s . SHOTS FOR 12 - 1 6 / 1 22 33 209 101 834 0 27 17 - 2 1 / 2 131 131 209 215 417 1 144 22 - 2 6 / 3 370 319 209 304 209 2 338 27 - 3 1 / 4 472 466 209 323 104 3 458 32 - 3 6 / 5 359 411 209 275 52 4 391 37 - 4 1 / 6 203 218 209 195 26 5 216 42 - 4 6 / 7 92 70 209 118 13 6 76 47 - 5 1 / 8 19 13 209 63 7 7 16 52 - 5 6 / 9 7 57 - 6 1 / 1 0 0 C h i s q u a r e d f I s 62 - 6 6 / 1 1 1 N o r m a l 1 4 . 1 5 8 7 . 0 5 U n i f o r m 5 5 9 . 1 2 7 7 .001 P o i s s o n 1 4 2 . 0 3 5 7 .001 G e o m e t r i c 1 6 3 0 . 4 9 9 7 .001 N e g a t i v e '. B i n o m i a l 6 . 3 3 0 7 n . s . T a b l e A9 T o t a l Team C h i - s q u a r e C o m p a r i s o n o f S h o t D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s ( c o n t i n u e d ) SHOTS AGAINST S h o t s / G a m e E x p e c t e d E x p e c t e d E x p e c t e d / S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n c l c2 c 3 c l c2 c3 c l c2 c 3 c l c2 c3 1 2 - 1 6 / 1 23 23 23 32 34 34 209 210 210 102 102 102 1 7 - 2 1 / 2 129 129 129 132 132 132 209 210 210 216 215 214 2 2 - 2 6 / 3 374 374 374 320 317 317 209 210 210 305 305 305 2 7 - 3 1 / 4 468 468 468 468 463 462 209 210 210 324 325 325 3 2 - 3 6 / 5 367 367 367 411 412 412 209 210 210 275 277 277 3 7 - 4 1 / 6 198 198 198 217 222 224 209 210 210 195 197 197 4 2 - 4 6 / 7 92 92 92 69 73 74 209 210 210 118 120 120 4 7 - 5 1 / 8 18 25 27 18 15 15 209 210 210 63 64 64 5 2 - 5 6 / 9 7 5 7 - 6 1 / 1 0 0 6 2 - 6 6 / 1 1 2 S h o t s / G a m e E x p e c t e d E x p e c t e d / S c a l e O b s e r v e d G e o m e t r i c S c a l e N e g a t i v e B i n o m i a l c l c2 c3 c l c2 c3 c l c2 c3 1 2 - 1 6 / 1 23 23 23 835 838 839 0 27 27 28 1 7 - 2 1 / 2 129 129 129 417 419 420 1 144 145 145 2 2 - 2 6 / 3 374 374 374 209 210 210 2 339 338 338 2 7 - 3 1 / 4 468 468 468 104 105 105 3 460 456 455 3 2 - 3 6 / 5 367 367 367 52 52 52 4 392 391 390 3 7 - 4 1 / 6 198 198 198 26 26 26 5 215 219 220 4 2 - 4 6 / 7 92 92 92 13 13 13 6 75 80 81 48+ / 8 18 25 27 7 7 7 7 16 18 19 C h i s q u a r e d f I s C h i s q u a r e d f I s N o r m a l c l 1 3 . 1 2 1 7 n . s . U n i f o r m c l 5 6 3 . 6 6 2 7 .001 c2 1 5 . 5 2 6 7 . 05 c2 5 4 8 . 4 8 0 7 .001 c3 1 6 . 4 6 1 7 . 0 5 c 3 5 4 4 . 1 4 8 7 .001 P o i s s o n c l 1 4 5 . 0 1 2 7 .001 G e o m e t r i c c l 1 6 3 1 . 8 7 1 7 .001 c2 1 3 5 . 2 1 6 7 .001 c2 1 6 3 9 . 3 8 6 7 .001 c 3 1 3 2 . 6 0 4 7 .001 c3 1 6 4 2 . 8 0 1 7 .001 N e g a t i v e B i n o m i a l c l 6 . 3 0 2 7 n . s c2 7 . 0 2 3 7 n . s c3 7 . 3 7 4 7 n . s c l = S c a l e i s 1 t o 8, 9 t o 11 n o t i n c l u d e d ( t o t a l n - 9) c2 = S c a l e i s 1 t o 9 , 10 t o 11 n o t i n c l u d e d ( t o t a l n - 2) c 3 = S c a l e i s 1 t o 11 ( b a s e d on t o t a l n) 125 T a b l e A10 T o t a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s ( c o n t i n u e d ) TOTAL GOALS G o a l s / G a m e E x p e c t e d N e g a t i v e / S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e B i n o m i a l 0 - 2 / 1 22 24 105 61 421 0 21 3 -4 / 2 88 86 105 123 210 1 97 5 -6 / 3 205 185 105 164 105 2 197 7 -8 / 4 238 239 105 164 53 3 232 9 - 1 0 / 5 175 186 105 132 26 4 174 1 1 - 1 2 / 6 72 87 105 88 13 5 86 1 3 - 1 4 / 7 32 24 105 50 7 6 28 1 5 - 1 6 / 8 9 4 105 25 3 7 6 C h i s q u a r e d f l s N o r m a l 5 . 932 7 n . s . U n i f o r m 2 8 2 . 963 7 .001 P o i s s o n 6 0 . 736 7 .001 G e o m e t r i c 7 2 9 . 615 7 .001 N e g a t i v e B i n o m i a l 2 . 807 7 n . s . GOALS F O R / A G A I N S T S h o t s / G a m e E x p e c t e d N e g a t i v e / S c a l e O b s e r v e d N o r m a l U n i f o r m i P o i s s o n 1 G e o m e t r i c S c a l e B i n o m i a l 0 45 54 140 41 836 0 41 1 154 129 140 152 418 1 152 2 283 234 140 282 209 2 282 3 340 323 140 348 104 3 348 4 314 342 140 323 52 4 323 5 238 276 140 239 26 5 239 6 149 171 140 148 13 6 148 7 82 80 140 78 7 7 80 8 51 29 140 36 3 8 37 9 15 8 140 15 2 9 15 10 7 140 11 2 140 C h i s q u a r e d f I s N o r m a l 21 .642 9 .02 U n i f o r m • 685 .378 11 .001 P o i s s o n 3 .088 9 n . s . G e o m e t r i c 1556 .389 9 .001 N e g a t i v e B i n o m i a l 2 .668 9 n . s . 126 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l T o t a l Games B o s t o n B r u i n s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f I s 0 - 5 1 218 103 78 124 233 0 212 ND 70 . 5 7 9 5 .001 5 - 1 0 2 126 140 78 126 116 1 130 UD 182 .059 5 .001 1 0 - 1 5 3 59 107 78 85 58 2 66 PD 34 .594 5 .001 1 5 - 2 0 4 28 47 78 43 29 3 31 GD 5 .177 5 n . s . 2 0 - 2 5 5 17 11 78 18 15 4 14 NBD 6 . 2 0 3 5 n . s . 25+ 6 17 2 78 6 7 5 6 B u f f a l o S a b r e s 0 - 5 1 245 125 85 143 255 0 243 ND 65 .131 5 .001 5 - 1 0 2 147 168 85 138 127 1 145 UD 232 .024 5 .001 1 0 - 1 5 3 63 114 85 88 64 2 69 PD 31 . 496 5 .001 1 5 - 2 0 4 28 38 85 43 32 3 30 GD 2 . 819 5 n . s . 2 0 - 2 5 5 14 7 85 16 16 4 13 NBD 3 .267 5 n . s . 25+ 6 12 1 85 5 8 5 5 C a l g a r y F l a m e s 0 - 5 1 244 119 83 138 250 0 236 ND 69 .401 5 .001 5 - 1 0 2 130 160 83 135 125 1 141 UD 218 .083 5 .001 1 0 - 1 5 3 65 113 83 89 63 2 69 PD 36 . 9 0 5 5 .001 1 5 - 2 0 4 38 42 83 43 31 3 31 GD 7 .052 5 n . s . 2 0 - 2 5 5 7 8 83 17 16 4 14 NBD 8 . 279 5 n . s . 25+ 6 16 1 83 6 8 5 6 C h i c a g o B l a c k Hawks 0 - 5 1 299 142 92 161 276 0 292 ND 90 .622 5 .001 5 -10 2 132 180 92 149 138 1 143 UD 276 .967 5 .001 1 0 - 1 5 3 71 116 92 92 69 2 65 PD 51 .138 5 .001 1 5 - 2 0 4 21 38 92 42 35 3 29 GD 7 . 017 5 n . s . 2 0 - 2 5 5 13 6 92 16 17 4 13 NBD 6 . 5 9 9 5 n . s . 25+ 6 16 1 92 5 9 5 6 D e t r o i t R e d W i n g s 0 - 5 1 273 140 92 157 275 0 273 ND 73 . 1 3 3 5 .001 5 - 1 0 2 156 184 92 148 137 1 153 UD 269 .098 5 .001 1 0 - 1 5 3 69 119 92 93 69 2 71 PD 41 . 290 5 .001 1 5 - 2 0 4 27 38 92 44 34 3 31 GD 7 . 184 5 n . s . 2 0 - 2 5 5 8 6 92 17 17 4 13 NBD 7 .278 5 n . s . 25+ 6 16 0 92 5 9 5 5 127 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) T o t a l Games E d m o n t o n O i l e r s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f l s 0 - 5 1 292 146 92 162 276 0 288 ND 80 .621 5 .001 5 - 1 0 2 146 187 92 148 138 1 149 UD 279 .901 5 .001 1 0 - 1 5 3 60 114 92 90 69 2 67 PD 39 .837 5 .001 1 5 - 2 0 4 30 33 92 41 34 3 28 GD 2 .558 5 n . s . 2 0 - 2 5 5 12 5 92 15 17 4 12 NBD 2 . 7 5 9 5 n . s . 25+ 6 11 0 92 5 9 5 5 H a r t f o r d W h a l e r s 0 - 5 1 254 123 85 143 225 0 249 ND 76 .528 5 .001 5 - 1 0 2 137 163 85 138 127 1 140 UD 227 .724 5 .001 1 0 - 1 5 3 60 113 85 89 64 2 67 PD 37 . 839 5 .001 1 5 - 2 0 4 28 41 85 43 32 3 30 GD 1 . 949 5 n . s . 2 0 - 2 5 5 17 8 85 17 16 4 13 NBD 3 . 634 5 n . s . 25+ 6 13 1 85 5 8 5 6 L o s A n g e l e s K i n g s 0 - 5 1 335 184 105 189 316 0 333 ND 79 .408 4 .001 5 - 1 0 2 175 237 105 169 158 1 174 UD 341 .948 5 .001 1 0 - 1 5 3 70 121 105 101 79 2 75 PD 42 . 673 5 .001 1 5 - 2 0 4 31 24 105 45 39 3 30 GD 6 . 149 5 n . s . 2 0 - 2 5 5 10 2 105 16 20 4 11 NBD 2 .804 5 n . s . 25+ 6 10 105 5 10 5 4 M i n n e s o t a N o r t h S t a r s 0 - 5 1 221 108 76 123 228 0 209 ND 68 . 369 4 .001 5 - 1 0 2 113 149 76 123 114 1 129" UD 183 .971 5 .001 1 0 - 1 5 3 63 105 76 82 57 2 65 PD 30 .408 5 .001 1 5 - 2 0 4 28 38 76 41 29 3 30 GD 2 . 4 0 5 5 n . s . 2 0 - 2 5 5 22 7 76 16 14 4 13 NBD 4 . 390 5 n . s . 25+ 6 9 76 5 7 5 6 M o n t r e a l C a n a d i e n s 0 - 5 1 209 111 76 124 227 0 202 ND 50 .631 4 .001 5 - 1 0 2 125 160 76 123 114 1 137 UD 194 .074 5 .001 1 0 - 1 5 3 67 105 76 81 57 2 68 PD 21 .297 5 .001 1 5 - 2 0 4 34 32 76 40 28 3 29 GD 3 . 4 9 9 5 n . s . 2 0 - 2 5 5 10 4 76 14 12 4 13 NBD 3 . 174 5 n . s . 25+ 6 9 76 5 7 5 4 128 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) T o t a l Games New J e r s e y D e v i l s T i m e (min) Sc Obs ND UD PD GD Sc NBD 2 X d f I s 0 - 5 1 276 132 88 153 278 0 270 ND 85 . 013 5 .001 5 - 1 0 2 135 180 88 142 139 1 144 UD 261 .270 5 .001 1 0 - 1 5 3 64 127 88 87 70 2 65 PD 36 .611 5 .001 1 5 - 2 0 4 30 46 88 40 35 3 28 GD 8 .871 5 n . s . 2 0 - 2 5 5 10 9 88 15 17 4 12 NBD 2 . 861 5 n . s . 25+ 6 11 1 88 5 9 5 5 New Y o r k I s l a n d e r s 0 - 5 1 248 124 85 143 255 0 245 ND 70 .500 5 .001 5 - 1 0 2 145 166 85 138 128 1 144 UD 229 .248 5 .001 1 0 - 1 5 3 58 114 85 89 64 2 69 PD 34 .578 5 .001 1 5 - 2 0 4 33 40 85 43 32 3 31 GD 2 . 9 1 9 5 n . s . 2 0 - 2 5 5 13 7 85 17 16 4 13 NBD 4 . 584 5 n . s . 25+ 6 13 1 85 5 8 5 5 New Y o r k R a n g e r s 0 - 5 1 274 147 90 156 269 0 269 ND 65 .637 4 .001 5 - 1 0 2 143 197 90 145 134 1 152 UD 271 .174 5 .001 1 0 - 1 5 3 76 111 90 89 67 2 69 PD 33 .921 5 .001 1 5 - 2 0 4 22 26 90 41 34 3 29 GD 4 . 501 5 n . s . 2 0 - 2 5 5 12 3 90 15 17 4 11 NBD 4 .226 5 n . s . 25+ 6 10 90 5 8 5 4 P h i l a d e l p h i a F l y e r s 0 - 5 1 256 121 85 141 256 0 247 ND 78 . 130 5 .001 5 - 1 0 2 128 160 85 139 128 1 140 UD 218 . 8 8 5 5 .001 1 0 - 1 5 3 66 115 85 91 64 2 68 PD 38 . 949 5 .001 1 5 - 2 0 4 32 45 85 44 32 3 32 GD 2 .768 5 n . s . 2 0 - 2 5 5 14 9 85 17 16 4 14 NBD 5 . 249 5 n . s . 25+ 6 16 1 85 6 8 5 6 P i t t s b u r g h P e n g u i n s 0 - 5 1 330 174 102 183 308 0 323 ND 90 .697 4 .001 5 - 1 0 2 167 222 102 165 154 1 171 UD 326 . 899 5 .001 1 0 - 1 5 3 61 121 102 100 77 2 73 PD 46 . 7 6 5 5 .001 1 5 - 2 0 4 30 28 102 45 39 3 29 GD 4 .878 5 n . s . 2 0 - 2 5 5 21 3 102 16 19 4 11 NBD 7 . 2 5 5 4 n . s . 25+ 6 5 102 5 10 129 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) T o t a l Games Q u e b e c N o r d i q u e T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f l s 0 - 5 1 259 131 89 149 266 0 253 ND 66 .934 5 .001 5 - 1 0 2 146 177 89 144 133 1 152 UD 240 .951 5 .001 1 0 - 1 5 3 67 118 89 92 66 2 73 PD 31 . 099 5 .001 1 5 - 2 0 4 35 39 89 44 33 3 31 GD 1 .712 5 n . s . 2 0 - 2 5 5 13 6 89 17 17 4 13 NBD 2 .926 5 n . s . 25+ 6 11 1 89 5 8 5 5 S t . L o u i s B l u e s 0 - 5 1 237 111 80 131 240 0 227 ND 74 . 273 5 .001 5 - 1 0 2 121 149 80 130 120 1 132 UD 201 .688 5 .001 1 0 - 1 5 3 58 109 80 86 60 2 65 PD 37 . 896 5 .001 1 5 - 2 0 4 38 43 80 42 30 3 30 GD 4 . 614 5 n . s . 2 0 - 2 5 5 10 9 80 17 15 4 14 NBD 7 .208 5 n . s . 25+ 6 16 1 80 6 8 6 T o r o n t o M a p l e L e a f s 0 - 5 1 268 131 87 149 261 0 262 ND 75 .508 5 .001 5 - 1 0 2 135 170 87 141 130 1 141 UD 247 .006 5 .001 1 0 - 1 5 3 65 113 87 89 65 2 66 PD 39 .712 5 .001 1 5 - 2 0 4 27 38 87 42 33 3 29 GD 2 . 9 3 3 5 n . s . 2 0 - 2 5 5 12 6 87 16 16 4 13 NBD 4 . 557 5 n . s . 25+ 6 14 1 87 5 8 5 5 V a n c o u v e r C a n u c k s 0 - 5 1 177 86 71 181 212 0 168 ND 54 . 079 5 .001 5 - 1 0 2 114 123 71 96 106 1 125 UD 138 .912 5 .001 1 0 - 1 5 3 66 103 71 75 53 2 69 PD 14 . 550 5 .02 1 5 - 2 0 4 36 51 71 45 27 3 34 GD 11 .682 5 . 0 5 2 0 - 2 5 5 12 14 71 21 13 4 16 NBD 6 . 9 4 5 5 n . s . 25+ 6 19 2 71 8 7 5 7 W a s h i n g t o n C a p i t o l s 0 - 5 1 235 124 81 136 243 0 229 ND 63 .468 4 .001 5 - 1 0 2 136 170 81 131 121 1 141 UD 219 .539 5 .001 1 0 - 1 5 3 57 108 81 84 61 2 67 PD 27 . 4 1 2 5 .001 1 5 - 2 0 4 36 31 81 41 30 3 29 GD 1 . 782 5 n . s . 2 0 - 2 5 5 13 4 81 16 15 4 12 NBD 2 . 456 5 n . s . 25+ 6 8 81 5 8 5 5 130 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) T o t a l Games W i n n i p e g J e t s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f I s 0 - 5 1 290 158 96 167 288 0 286 ND 67 . 1 9 5 4 .001 5 - 1 0 2 159 212 96 155 144 1 165 UD 287 .461 5 .001 1 0 - 1 5 3 74 119 96 96 72 2 75 PD 34 . 2 8 5 5 .001 1 5 - 2 0 4 32 28 96 45 36 3 31 GD 3 . 2 9 5 5 n . s . 2 0 - 2 5 5 10 3 96 17 18 4 12 NBD 2 .911 5 n . s . 25+ 6 10 96 5 9 5 4 Home Games B o s t o n B r u i n s 0 - 5 1 110 51 39 62 118 0 106 ND 37 .732 5 .001 5 - 1 0 2 63 69 39 64 59 1 65 UD 88 .171 5 .001 1 0 - 1 5 3 29 55 39 44 30 2 34 PD 19 .487 5 .001 1 5 - 2 0 4 16 25 39 23 15 3 16 GD 3 . 080 5 n . s . 2 0 - 2 5 5 8 7 39 9 7 4 8 NBD 3 . 0 5 5 5 n . s . 25+ 6 10 1 39 3 4 5 4 B u f f a l o S a b r e s 0 - 5 1 139 72 45 78 134 0 133 ND 35 .954 4 .001 5 - 1 0 2 68 95 45 72 67 1 78 UD 129 . 840 5 .001 1 0 - 1 5 3 34 56 45 44 34 2 35 PD 15 .701 4 .001 1 5 - 2 0 4 16 15 45 21 17 3 14 GD 0 . 482 4 n . s . 2 0 - 2 5 5 6 2 45 8 8 4 5 NBD 3 .198 4 n . s . 25+ 6 5 45 C a l g a r y F l a m e s 0 - 5 1 130 68 44 74 132 0 128 ND 34 . 873 4 .001 5 - 1 0 2 71 92 44 71 66 1 73 UD 121 . 2 4 5 5 .001 1 0 - 1 5 3 33 58 44 46 33 2 35 PD 21 . 323 5 .001 1 5 - 2 0 4 17 17 44 22 16 3 16 GD 0 . 9 0 5 4 n . s . 2 0 - 2 5 5 3 12 44 9 8 4 7 NBD 4 .731 5 n . s . 25+ 6 9 44 3 5 3 C h i c a g o B l a c k Hawks • 0 - 5 1 162 78 46 84 139 0 155 ND 53 . 875 4 .001 5 - 1 0 2 60 96 46 74 70 1 71 UD 150 .242 5 .001 1 0 - 1 5 3 32 54 46 44 35 2 30 PD 31 . 4 5 9 5 .001 1 5 - 2 0 4 8 14 46 19 17 3 13 GD 7 . 746 5 n . s . 2 0 - 2 5 5 11 2 46 7 9 4 5 NBD 8 . 067 5 n . s . 25+ 6 5 46 131 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) Home Games D e t r o i t R e d W i n g s T i m e (min) Sc Obs ND UD PD GD Sc NBD 2 X d f I s 0 - 5 1 118 65 43 71 129 0 119 ND 30 . 319 4 .001 5 - 1 0 2 79 91 43 70 64 1 74 UD 116 .222 5 .001 1 0 - 1 5 3 33 58 43 46 32 2 36 PD 18 . 4 6 5 5 .001 1 5 - 2 0 4 14 17 43 22 16 3 16 GD 5 .437 5 n . s . 2 0 - 2 5 5 4 2 43 9 8 4 7 NBD 4 .242 5 n . s . 25+ 6 9 43 3 4 5 3 E d m o n t o n O i l e r s 0 - 5 1 154 80 47 84 141 0 147 ND 40 . 3 1 5 * 4 .001 5 - 1 0 2 68 103 47 75 71 1 80 UD 147 . 9 8 3 5 .001 1 0 - 1 5 3 35 55 47 45 35 2 34 PD 17 . 983 4 .001 1 5 - 2 0 4 16 12 47 20 18 3 13 GD 0 .640 4 n . s . 2 0 - 2 5 5 5 1 47 _7 9 4 5 NBD 2 .587 4 n . s . 25+ 6 4 47 H a r t f o r d W h a l e r s 0 - 5 1 131 65 44 74 131 0 126 ND 42 .441 4 .001 5 - 1 0 2 70 88 44 71 65 1 75 UD 115 . 746 5 .001 1 0 - 1 5 3 28 58 44 45 33 2 35 PD 19 .632 4 .001 1 5 - 2 0 4 16 19 44 21 16 3 15 GD 1 .441 5 n . s . 2 0 - 2 5 5 9 3 44 _8 8 4 _6 NBD 5 . 5 9 5 4 n . s . 25+ 6 7 44 L o s A n g e l e s K i n g s 0 - 5 1 195 111 57 108 172 0 193 ND 44 .550+ 4 .001 5 - 1 0 2 95 135 57 90 86 1 96 UD 210 . 6 8 9 5 .001 1 0 - 1 5 3 32 56 57 51 43 2 37 PD 23 .982 4 .001 1 5 - 2 0 4 14 8 57 21 22 3 13 GD 11 . 566 5 . 0 5 2 0 - 2 5 5 2 0 57 7 11 4 4 NBD 1 . 747 4 n . s . 25+ 6 6 57 5 M i n n e s o t a N o r t h S t a r s 0 - 5 1 113 60 39 66 118 0 108 ND 27 .492 4 .001 5 - 1 0 2 63 83 39 64 59 1 73 UD 106 .091 5 .001 1 0 - 1 5 3 36 53 39 41 30 2 34 PD 11 .568 4 .02 1 5 - 2 0 4 13 15 39 20 15 3 14 GD 1 . 712 4 n . s . 2 0 - 2 5 5 8 2 39 8 7 4 5 NBD 3 . 184 4 n . s . 25+ 6 3 39 132 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) Home Games M o n t r e a l C a n a d i e n s T i m e 2 (min) S c Obs ND UD PD GD Sc NBD X d f I s 0 - 5 1 101 54 38 62 114 0 97 ND 26 . 3 4 9 4 .001 5 - 1 0 2 67 78 38 61 57 1 71 UD 92 . 9 8 3 5 .001 1 0 - 1 5 3 29 54 38 41 28 2 36 PD 9 .310 4 n . s . 1 5 - 2 0 4 19 18 38 20 14 3 15 GD 3 .148 4 n . s . 2 0 - 2 5 5 7 3 38 _8 7 4 _6 NBD 2 . 883 4 n . s . 25+ 6 4 38 New J e r s e y D e v i l s 0 - 5 1 132 67 42 73 126" 0 125 ND 36 .146 4 .001 5 - 1 0 2 61 88 42 68 63 1 72 UD 120 .701 5 .001 1 0 - 1 5 3 34 53 42 42 32 2 33 PD 16 .778 4 .01 1 5 - 2 0 4 13 15 42 19 16 3 13 GD 1 . 244 4 n . s . 2 0 - 2 5 5 8 2 42 J 7 8 4 5 NBD 3 .966 4 n . s . 25+ 6 4 42 New Y o r k I s l a n d e r s 0 - 5 1 114 60 42 68 125 0 109 ND 30 .558 4 .001 5 - 1 0 2 72 86 42 68 63 1 78 UD 103 .904 5 .001 1 0 - 1 5 3 31 58 42 45 31 2 38 PD 11 . 7 2 5 4 .02 1 5 - 2 0 4 21 19 42 22 16 3 16 GD 2 . 483 4 n . s . 2 0 - 2 5 5 7 3 42 _9 8 4 _6 NBD 3 . 720 4 n . s . 25+ 6 5 42 New Y o r k R a n g e r s 0 - 5 1 133 76 45 79 134 0 132 ND 28 .408 4 .001 5 - 1 0 2 81 102 45 72 67 1 82 UD 144 . 859 5 .001 1 0 - 1 5 3 35 53 45 43 33 2 35 PD 15 . 726 4 .001 1 5 - 2 0 4 8 11 45 19 17 3 13 GD 4 . 729 4 n . s . 2 0 - 2 5 5 5 1 45 7 8 4 4 NBD 3 . 770 4 n . s . 25+ 6 5 45 P h i l a d e l p h i a F l y e r s 0 - 5 1 135 68 43 75 130 0 131 ND 44 .298 5 .001 5 - 1 0 2 70 88 43 70 65 1 72 UD 122 .407 5 .001 1 0 - 1 5 3 26 55 43 44 33 2 33 PD 22 .227 4 .001 1 5 - 2 0 4 13 17 43 20 16 3 14 GD 3 . 9 6 3 4 n . s . 2 0 - 2 5 5 7 2 43 8 8 4 6 NBD 5 .471 4 n . s . 25+ 6 9 43 133 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) Home Games P i t t s b u r g h P e n g u i n s T i m e 2 (min) Sc Obs ND UD PD GD S c NBD X d f l s 0 - 5 1 171 85 51 91 152 0 164 ND 51 .822 4 .001 5 -10 2 72 106 51 81 76 1 81 UD 159 .826 5 .001 1 0 - 1 5 3 32 60 51 48 38 2 35 PD 27 .112 4 .001 1 5 - 2 0 4 14 15 51 21 19 3 14 GD 3 . 363 4 n . s . 2 0 - 2 5 5 11 2 51 8 10 4 6 NBD 4 .641 4 n . s . 25+ 6 4 51 Q u e b e c N o r d i q u e 0 - 5 1 129 70 5 - 1 0 2 77 98 1 0 - 1 5 3 35 59 1 5 - 2 0 4 20 16 2 0 - 2 5 5 5 2 25+ 6 4 45 77 135 0 124 45 73 68 1 84 45 46 34 2 39 45 22 17 3 15 45 8 8 4 5 45 ND 2 9 . 896 4 .001 UD 1 2 6 . 117 5 .001 PD 1 1 . 065 4 . 0 5 GD 0 . 891 4 n . s . NBD 2 . 460 4 n . s . S t . L o u i s B l u e s 0 - 5 1 109 51 38 60 113 5 - 1 0 2 51 72 38 61 56 1 0 - 1 5 3 29 53 38 41 28 1 5 - 2 0 4 22 21 38 21 14 2 0 - 2 5 5 4 4 38 8 7 25+ 6 9 38 0 113 ND 3 5 . 9 7 6 4 .001 1 56 UD 8 4 . 8 7 0 5 .001 2 34 PD 1 6 . 1 4 6 4 .01 3 15 GD 4 . 3 2 5 4 n . s . 4 7 NBD 6 . 4 2 3 4 n . s . T o r o n t o M a p l e L e a f s 0 - 5 1 126 64 5 - 1 0 2 68 88 1 0 - 1 5 3 33 58 1 5 - 2 0 4 17 19 2 0 - 2 5 5 4 3 25+ 6 10 43 72 129 0 120 43 70 65 1 77 43 45 32 2 37 43 22 16 3 15 43 8 8 4 6 43 ND 3 5 . 613 4 .001 UD 1 1 1 . 885 5 .001 PD 1 5 . 456 4 .001 GD 1 . 659 4 n . s . NBD 4 . 241 4 n . s . V a n c o u v e r C a n u c k s 0 - 5 1 73 39 5 - 1 0 2 55 60 1 0 - 1 5 3 34 50 1 5 - 2 0 4 14 22 2 0 - 2 5 5 7 5 25+ 6 10 32 47 97 0 68 32 52 48 1 62 32 38 24 2 36 32 21 12 3 17 32 9 6 4 7 43 ND 21 .171 4 .001 UD 56 .741 5 .001 PD 7 . 9 6 5 4 n . s . GD 10 . 911 4 . 0 5 NBD 5 . 087 4 n . s . 134 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n F rom T i m e o f F i r s t G o a l ( c ' d ) Home Games W a s h i n g t o n C a p i t a l s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f l s 0 - 5 1 107 59 39 66 118 0 104 ND 26 . 092 4 .001 5 - 1 0 2 73 85 39 64 59 1 75 UD 105 .467 5 .001 1 0 - 1 5 3 30 54 39 41 30 2 35 PD 9 .601 4 . 0 5 1 5 - 2 0 4 16 15 39 20 15 3 14 GD 2 . 4 7 9 4 n . s . 2 0 - 2 5 5 6 2 39 8 7 4 5 NBD 2 . 2 3 5 4 n . s . 25+ 6 4 39 W i n n i p e g J e t s 0 - 5 1 139 73 5 - 1 0 2 74 99 1 0 - 1 5 3 39 61 1 5 - 2 0 4 15 17 2 0 - 2 5 5 7 2 25+ 6 6 B o s t o n B r u i n s 47 80 140 0 133 47 76 70 1 84 47 48 35 2 39 47 22 18 3 16 47 _8 9 4 6 47 Away Games ND 35 .817 4 .001 UD 130 .132 5 .001 PD 15 . 374 4 .01 GD 1 . 2 1 5 4 n . s . NBD 3 . 3 6 9 4 n . s . 0 - 5 1 108 55 38 63 115 5 - 1 0 2 63 76 38 62 57 1 0 - 1 5 3 30 53 38 41 29 1 5 - 2 0 4 12 18 38 20 14 2 0 - 2 5 5 9 3 38 8 7 25+ 6 7 38 B u f f a l o S a b r e s 0 103 ND 3 3 . 6 9 3 4 .001 1 68 UD 9 3 . 4 5 8 5 .001 2 33 PD 1 5 . 4 3 1 4 .01 3 15 GD 4 . 1 0 5 4 n . s . 4 6 NBD 5 . 2 9 6 4 n . s . 0 - 5 1 106 59 5 - 1 0 2 79 84 1 0 - 1 5 3 29 56 1 5 - 2 0 4 12 18 2 0 - 2 5 5 8 3 25+ 6 7 40 66 121 0 106 40 65 60 1 75 40 43 30 2 37 40 21 15 3 15 40 8 8 4 6 40 ND 30 .424 4 .001 UD 103 .950 5 .001 PD 14 .827 4 .01 GD 5 . 967 4 n . s . NBD 5 . 256 4 n . s . C a l g a r y F l a m e s 0 - 5 1 114 57 5 - 1 0 2 59 80 1 0 - 1 5 3 32 55 1 5 - 2 0 4 21 18 2 0 - 2 5 5 4 3 25+ 6 7 40 65 119 0 106 40 64 59 1 72 40 42 30 2 35 40 21 15 3 15 40 8 7 4 6 ND 31 . 866 4 .001 UD 98 .622 5 .001 PD 12 .398 4 .02 GD 1 . 956 4 n . s . NBD 4 .167 4 n . s . 135 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) Away Games C h i c a g o B l a c k Hawks T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f I s 0 - 5 1 137 72 46 79 137 0 131 ND 34 .966 4 .001 5 - 1 0 2 72 97 46 74 69 1 82 UD 131 .833 5 .001 1 0 - 1 5 3 39 59 46 46 34 2 37 PD 15 . 783 4 .01 1 5 - 2 0 4 13 16 46 22 17 3 15 GD 1 . 5 4 9 4 n . s . 2 0 - 2 5 5 2 2 46 _8 9 4 _6_ NBD 3 .542 4 n . s . 25+ 6 11 46 D e t r o i t Red W i n g s 0 - 5 1 155 83 49 87 146 0 150 ND 38 .644 4 .001 5 - 1 0 2 77 107 49 78 73 1 84 UD 153 . 8 9 5 5 .001 1 0 - 1 5 3 36 58 49 47 37 2 36 PD 18 . 2 1 5 4 .01 1 5 - 2 0 4 13 13 49 21 18 3 14 GD 1 . 2 5 5 4 n . s . 2 0 - 2 5 5 4 1 49 _8 9 4 5 NBD 2 .658 4 n . s . 25+ 6 7 49 E d m o n t o n O i l e r s 0 - 5 1 138 72 45 79 135 0 136 ND 41 . 795 4 .001 5 - 1 0 2 78 95 45 72 67 1 76 UD 133 . 6 5 5 5 .001 1 0 - 1 5 3 25 56 45 44 34 2 34 PD 20 . 083 4 .001 1 5 - 2 0 4 14 15 45 20 17 3 14 GD 4 .047 4 n . s . 2 0 - 2 5 5 7 2 45 _8 8 4 _5 NBD 5 . 647 4 n . s . 25+ 6 7 45 H a r t f o r d W h a l e r s 0 - 5 1 123 63 41 71 124 0 118 ND 35 .477 4 .001 5 - 1 0 2 67 86 41 67 62 1 73 UD 112 . 5 1 9 5 .001 1 0 - 1 5 3 32 54 41 42 31 2 34 PD 15 .537 4 .01 1 5 - 2 0 4 12 16 41 20 16 3 14 GD 2 . 322 4 n . s . 2 0 - 2 5 5 8 2 41 8 8 4 6 NBD 3 .757 4 n . s . 25+ 6 6 41 L o s A n g e l e s K i n g s 0 - 5 1 140 75 48 82 144 0 135 ND 34 . 0 9 9 4 .001 5 - 1 0 2 80 103 48 77 72 1 87 UD 134 . 770 5 .001 1 0 - 1 5 3 38 62 48 49 36 2 40 PD 13 . 848 4 .01 1 5 - 2 0 4 17 17 48 23 18 3 16 GD 0 . 876 4 n . s . 2 0 - 2 5 5 8 2 48 9 9 4 6 NBD 2 . 450 4 n . s . 25+ 6 4 48 136 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F r o m T i m e o f F i r s t G o a l ( c ' d ) Away Games  M i n n e s o t a N o r t h S t a r s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f I s 0 - 5 1 108 49 37 58 110 0 98 ND 40 . 879 4 .001 5 - 1 0 2 50 67 37 60 55 1 63 UD 80 .292 5 .001 1 0 - 1 5 3 27 51 37 40 28 2 32 PD 22 .104 4 .001 1 5 - 2 0 4 15 22 37 21 14 3 15 GD 6 . 487 4 n . s . 2 0 - 2 5 5 14 5 37 _8 7 4 _1_ NBD 8 .608 4 - n . s . 25+ 6 6 37 M o n t r e a l C a n a d i e n s 0 - 5 1 108 58 38 64 114 0 101 ND 23 . 837* 4 .001 5 - 1 0 2 58 82 38 61 57 1 72 UD 102 .911 5 .001 1 0 - 1 5 3 38 51 38 39 28 2 34 PD 8 . 667 4 n . s . 1 5 - 2 0 4 15 14 38 19 14 3 13 GD 1 .678 4 n . s . 2 0 - 2 5 5 3 1 38 _7 7 4 5 NBD 2 .791 4 n . s . 25+ 6 5 38 New J e r s e y D e v i l s 0 - 5 1 144 77 46 81 137 0 140 ND 37 . 0 8 3 * 4 .001 5 - 1 0 2 74 100 46 73 69 1 80 UD 142 .824 5 .001 1 0 - 1 5 3 30 55 46 44 34 2 34 PD 16 . 133 4 .01 1 5 - 2 0 4 17 13 46 20 17 3 13 GD 0 .481 4 n . s . 2 0 - 2 5 5 2 1 46 _7 9 4 5 NBD 2 . 2 0 9 4 n . s . 25+ 6 7 46 New Y o r k I s l a n d e r s 0 - 5 1 134 70 43 76 130 0 131 ND 38 .920 4 .001 5 - 1 0 2 73 91 43 70 65 1 74 UD 127 .626 5 .001 1 0 - 1 5 3 27 54 43 43 33 2 33 PD 20 .178 4 .01 1 5 - 2 0 4 12 15 43 20 16 3 14 GD 3 .208 4 n . s . 2 0 - 2 5 5 6 2 43 7 8 4 5 NBD 5 .041 4 n . s . 25+ 6 8 43 New Y o r k R a n g e r s 0 - 5 1 141 71 45 78 135 0 132 ND 38 .830 4 .001 5 - 1 0 2 62 95 45 73 68 1 79 UD 128 . 4 9 3 5 .001 1 0 - 1 5 3 41 58 45 45 34 2 36 PD 17 . 4 5 9 4 .01 1 5 - 2 0 4 14 16 45 21 17 3 15 GD 2 .031 4 n . s . 2 0 - 2 5 5 7 2 45 8 8 4 6 NBD 4 .698 4 n . s . 25+ 6 5 45 137 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l ( c ' d ) Away Games P h i l a d e l p h i a F l y e r s T i m e 2 (min) Sc Obs ND UD PD GD Sc NBD X d f l s 0 - 5 1 121 59 42 68 126 0 111 ND 3 4 . 2 0 7 4 .001 5 - 1 0 2 58 84 42 68 63 1 76 UD 9 9 . 5 6 9 5 .001 1 0 - 1 5 3 40 59 42 45 32 2 38 PD 1 4 . 2 4 7 4 .01 1 5 - 2 0 4 19 21 42 23 16 3 17 GD 2 . 9 9 2 4 n . s . 2 0 - 2 5 5 7 4 42 9 8 4 7 NBD 5 . 327 4 n . s . 25+ 6 7 42 P i t t s b u r g h P e n g u i n s 0 - 5 1 159 87 52 92 156 5 - 1 0 2 95 114 52 84 78 1 0 - 1 5 3 29 63 52 50 39 1 5 - 2 0 4 16 14 52 23 20 2 0 - 2 5 5 10 1 52 8 10 25+ 6 3 52 Q u e b e c N o r d i q u e 0 158 ND 4 4 . 079 4 .001 1 91 UD 1 6 4 . 535 5 .001 2 39 PD 2 1 . 474 4 .001 3 15 GD 3 . 875 4 n . s . 4 6 NBD 4 . 157 4 n . s . 0 - 5 1 130 66 44 74 131 5 - 1 0 2 69 89 44 71 65 1 0 - 1 5 3 32 58 44 45 33 1 5 - 2 0 4 15 18 44 21 16 2 0 - 2 5 5 8 3 44 8 8 25+ 6 7 44 0 124 ND 3 7 . 8 5 3 4 .001 1 76 UD 1 1 5 . 9 3 9 5 .001 2 36 PD 1 7 . 1 8 3 4 .01 3 15 GD 2 . 1 7 7 4 n . s . 4 6 NBD 4 . 5 4 1 4 n . s . S t . L o u i s B l u e s 0 - 5 1 128 67 5 - 1 0 2 70 89 1 0 - 1 5 3 29 55 1 5 - 2 0 4 16 16 2 0 - 2 5 5 5 2 25+ 6 7 43 73 128 0 124 43 69 64 1 75 43 43 32 2 34 43 20 16 3 14 43 8 8 4 5 43 ND 35 .248 4 .001 UD 119 .771 5 .001 PD 15 .378 4 .01 GD 1 . 1 1 9 4 n . s . NBD 3 .631 4 n . s . T o r o n t o M a p l e L e a f s 0 - 5 1 142 74 44 78 132 5 - 1 0 2 67 95 44 70 66 1 0 - 1 5 3 32 52 44 42 33 1 5 - 2 0 4 10 12 44 19 16 2 0 - 2 5 5 8 1 44 7 8 25+ 6 4 44 0 138 ND 3 8 . 9 3 9 4 .001 1 74 UD 1 3 7 . 9 5 7 5 .001 2 32 PD 1 9 . 7 2 2 4 .001 3 13 GD 2 . 4 5 0 4 n . s . 4 5 NBD 3 . 6 7 6 4 n . s . 138 T a b l e A l l I n d i v i d u a l Team C h i - s q u a r e C o m p a r i s o n o f G o a l D i s t r i b u t i o n s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s From T i m e o f F i r s t G o a l ( c ' d ) Away Games V a n c o u v e r C a n u c k s T i m e (min) Sc Obs ND UD PD GD Sc NBD 2 X d f l s 0 - 5 1 104 52 39 61 116 0 95 ND 30 .321 4 .001 5 -10 2 59 75 39 62 58 1 71 UD 84 .062 5 .001 1 0 - 1 5 3 32 56 39 42 29 2 37 PD 11 .351 4 . 0 5 1 5 - 2 0 4 22 22 39 22 14 3 17 GD 4 . 815 4 n . s . 2 0 - 2 5 5 5 4 39 _9 7 4 _1_ NBD 4 . 817 4 n . s . 25+ 6 9 39 W a s h i n g t o n C a p i t o l s 0 - 5 1 128 64 42 71 125 0 121 ND 39 .140 4 .001 5 - 1 0 2 63 86 42 67 62 1 72 UD 115 .151 5 .001 1 0 - 1 5 3 27 54 42 42 31 2 33 PD 16 . 640 4 .01 1 5 - 2 0 4 20 16 42 20 16 3 14 GD 1 .138 4 n . s . 2 0 - 2 5 5 7 _2 42 _8 8 4 6 NBD 3 .908 4 n . s . 25+ 6 4 42 W i n n i p e g J e t s 0 - 5 1 151 85 49 88 148 0 147 ND 32 .149+ 4 .001 5 - 1 0 2 85 113 49 79 74 1 90 UD 158 .439 5 .001 1 0 - 1 5 3 35 57 49 47 37 2 38 PD 14 .176 4 .01 1 5 - 2 0 4 17 11 49 21 18 3 14 GD 0 .986 4 n . s . 2 0 - 2 5 5 3 1 49 8 9 4 4 NBD 1 . 422 4 n . s . 25+ 6 4 49 + Two c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 T a b l e A12 Means a n d V a r i a n c e s f o r I n d i v i d u a l Teams Home, Away a n d T o t a l Games f o r F r e q u e n c y D i s t r i b u t i o n s F rom T i m e o f F i r s t G o a l Home Away T o t a l 2 _ 2 _ 2 Team X S X S X S B o s 2 . 0 6 1 . 8 5 2 . 0 0 1 . 6 6 2 . 0 3 1 . 7 5 B u f 1 .87 1 .37 2 . 0 0 1 .53 1 . 9 3 1 .44 C a i 1 . 93 1 .53 2 . 0 0 1 .60 1 . 9 6 1 .56 C h i 1 .78 1 .41 1 .91 1 .54 1 . 8 5 1 .48 D e t 1 .96 1 .51 1 .82 1 .31 1 . 8 9 1 .41 Edm 1 .80 1 . 2 5 1 .87 1 . 4 5 1 . 8 3 1 . 3 5 H a r 1 .94 1 .60 1 .92 1 .49 1 . 9 3 1 .54 LA 1 .69 1 . 0 6 1 .90 1 .33 1 . 7 9 1 .19 M i n 1 .94 1 .36 2 . 0 7 1 .90 2 . 0 0 1 .62 Mon 2 . 0 1 1 . 4 7 1 . 9 5 1 .38 1 .98 1 .43 N J 1 .87 1 .39 1 .83 1 .33 1 . 8 5 1 .36 NYI 2 . 0 0 1 . 4 9 1 .87 1 . 4 9 1 . 9 4 1 . 4 9 NYR 1 .82 1 . 1 9 1 .89 1 .40 1 .78 0 . 9 9 P h i 1 .90 1 .61 2 . 0 2 1 .64 1 .92 1 .57 P i t 1 .80 1 . 3 5 1 .82 1 .22 1 . 7 9 1 .23 Que 1 .91 1 . 3 0 1 .94 1 .56 1 . 9 3 1 .43 S t L 2 . 0 5 1 .82 1 .91 1 .48 1 .98 1 .66 T o r 1 .97 1 . 6 5 1 .81 1 .30 1 . 8 9 1 .47 V a n 2 . 2 6 1 .94 2 . 1 0 1 .79 2 . 1 7 1 .86 Was 1 . 9 5 1 . 3 6 1 .90 1 . 4 5 1 . 9 3 1 . 4 0 Win 1 .91 1 .41 1 .81 1 .14 1 . 8 6 1 .27 140 T a b l e A13 T o t a l Team C h i - s q u a r e C o m p a r i s o n o f W i n , L o s s a n d T o t a l S t r e a k F r e q u e n c i e s w i t h N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s C o n v e n t i o n a l M e t h o d W i n n i n g S t r e a k s S t r e a k E x p e c t e d L e n g t h O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e N e g a t i v e B i o n o m i a l 1 185 75 58 97 201 0 179 2 102 99 58 108 101 1 102 3 46 88 58 79 50 2 56 4 26 54 58 44 25 3 30 5 16 22 58 19 13 4 16 6 11 6 58 7 6 5 9 7 16 1 58 2 3 6 5 C h i s q u a r e d f I s N o r m a l 81 . 3 2 5 6 .001 U n i f o r m 233 .100 6 .001 P o i s s o n 50 .402 6 .001 G e o m e t r i c 11 . 5 1 9 6 n . s N e g a t i v e B i n o m i a l 7 .296 6 n . s L o s i n g S t r e a k s 1 2 3 4 5 239 93 57 29 18 116 151 91 25 3 89 89 89 89 89 127 117 72 33 12 218 109 55 27 14 0 1 2 3 4 223 122 54 23 9 C h i s q u a r e d f I s N o r m a l 7 2 . 452 4 .001 U n i f o r m 153 .244 4 .001 P o i s s o n 33 .457 4 .001 G e o m e t r i c 2 . 6 4 3 4 n . s N e g a t i v e B i n o m i a l 8 .211 4 n . s 141 T a b l e A13 T o t a l Team C h i s q u a r e C o m p a r i s o n o f W i n , L o s s and T o t a l S t r e a k F r e q u e n c i e s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c and N e g a t i v e B i n o m i a l D i s t r i b u t i o n s ( c o n t i n u e d ) T o t a l S t r e a k s S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e N e g a t i v e B i n o m i a l 1 424 180 120 223 420 0 408 2 195 234 120 227 210 1 214 3 103 187 120 154 105 2 109 4 55 92 120 79 53 3 55 5 30 28 120 32 26 4 28 6 13 5 120 11 13 5 14 7 20 1 120 3 7 6 7 C h i s q u a r e d f I s N o r m a l 149 .201 6 .001 U n i f o r m 424 . 690 6 .001 P o i s s o n 84 .657 6 .001 G e o m e t r i c 7 .154 6 n . s N e g a t i v e B i n o m i a l 7 .711 6 n . s U n c o n v e n t i o n a l M e t h o d W i n n i n g S t r e a k s E x p e c t e d S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e N e g a t i v e B i o n o m i a l 1 185 87 64 104 193 0 178 2 102 118 64 104 97 1 108 3 46 89 64 70 48 2 54 4 26 37 64 35 24 3 26 5 16 9 64 14 12 4 12 6. 11 1 64 5 6. 5 5 7 7 8 3 C h i s q u a r e d f I s 9 4 N o r m a l 5 9 . 8 2 7 5 .001 10 0 U n i f o r m 1 5 2 . 7 3 7 5 .001 11 0 P o i s s o n 2 7 . 4 9 3 5 .001 12 1 G e o m e t r i c 2 . 4 1 3 5 n . s . 13 1 N e g a t i v e B i n o m i a l 3 . 7 5 6 5 n . s . 142 T a b l e A13 T o t a l Team C h i s q u a r e C o m p a r i s o n o f W i n , L o s s a n d T o t a l S t r e a k F r e q u e n c i e s w i t h T h e o r e t i c a l N o r m a l , U n i f o r m , P o i s s o n , G e o m e t r i c a n d N e g a t i v e B i n o m i a l D i s t r i b u t i o n s ( c o n t i n u e d ) L o s i n g S t r e a k s 1 239 119 86 128 216 0 224 2 93 154 86 116 108 1 121 3 57 87 86 70 54 2 53 4 29 22 86 32 27 3 21 5 14 2 86 11 14 4 8 6 1 7 1 C h i s q u a r e d f l s 8 1 N o r m a l 6 8 . 9 1 4 4 n . s . 9 0 U n i f o r m 1 5 8 . 2 7 1 4 .001 10 0 P o i s s o n 3 1 . 0 9 3 4 .001 11 1 G e o m e t r i c 2 . 2 3 6 4 n . s . N e g a t i v e : B i n o m i a l 7 . 1 8 2 4 n . s . T o t a l S t r e a k s S c a l e O b s e r v e d N o r m a l U n i f o r m P o i s s o n G e o m e t r i c S c a l e N e g a t i v e B i n o m i a l 1 424 192 118 228 414 0 406 2 195 252 118 224 207 1 220 3 103 184 118 147 104 2 108 4 55 76 118 72 52 3 51 5 30 17 118 28 26 4 24 6 13 2 118 9 13 5 11 7 8 0 118 3 6 6. 5 8 4 9 5 C h i s q u a r e d f l s 10 0 N o r m a l 1 3 4 . 0 4 0 * 6 .001 11 0 U n i f o r m 4 4 8 . 1 7 8 6 .001 12 2 P o i s s o n 6 5 . 5 0 0 6 .001 13 1 G e o m e t r i c 1 . 1 1 6 6 n . s . N e g a t i v e B i n o m i a l 3 . 6 8 6 6 n . s . 2 c e l l s w i t h e x p e c t e d v a l u e s l e s s t h a n 5 . C h i s q u a r e f o r N o r m a l D i s t r i b u t i o n w i t h d f = 5 was 5 9 . 8 2 7 ( s i g n , a t . 0 0 1 ) . T a b l e A14 C h i - s q u a r e C o m p a r i s o n o f B a c k - t o - B a c k a n d N o n - B a c k - t o - B a c k Game P o i n t R e c o r d s f o r 1 9 8 5 - 8 6 N a t i o n a l H o c k e y L e a g u e S e a s o n 1 s t 2nd O t h e r 1 s t v 2nd 1 s t v OG 2nd v OG 1 v 2 v 0 N i g h t N i g h t Games 2 2 2 2 PG PL PG PL PG PL X I s X I s X I s X I s B o s 16 22 23 15 47 34 2 . 5 8 1 n s 2 . 0 0 9 n s 0 . 2 2 4 n s 2 . 9 3 9 n s B u f 10 20 12 18 58 42 0 . 2 8 7 ns 5 . 6 2 8 .02 3 . 0 0 9 n s 7 . 0 9 3 . 0 5 C a i 13 17 19 11 57 43 2 . 4 1 1 n s 1 . 7 3 4 n s 0 . 3 8 1 n s 2 . 6 3 5 n s C h i 19 23 28 14 39 37 3 . 9 1 3 . 0 5 0 . 4 0 0 n s 2 . 5 9 8 ns 4 . 2 2 4 n s D e t 12 24 7 29 21 67 1 . 7 8 7 n s 1 . 1 7 3 n s 0 . 2 8 5 n s 1 .987 n s Edm 19 11 25 5 75 25 3 . 0 6 8 n s 1 . 5 6 9 n s 0 . 9 0 3 n s 3 . 2 0 3 n s H a r 22 12 17 17 45 47 1 . 5 0 3 n s 2 . 4 8 7 n s 0 . 0 1 2 n s 2 . 5 9 1 n s L A 14 20 12 20 28 66 0 . 0 9 3 n s 1 . 4 6 9 n s 0 . 6 5 5 n s 1 . 7 0 0 n s M i n 14 12 10 16 61 47 1 .238 ns 0 . 0 5 9 n s 2 . 7 3 2 n s 2 . 7 3 9 ns Mon 19 13 16 16 52 44 0 . 5 6 7 n s 0 . 2 6 4 n s 0 . 1 6 7 n s 0 . 5 7 1 n s N J 12 30 16 26 31 45 0 . 8 5 7 n s 1 . 7 4 3 n s 0 . 0 8 2 n s 1 .771 n s NYI 21 17 15 23 54 30 1 . 9 0 0 ns 0 . 8 9 9 n s 6 . 5 5 6 02 6 . 5 6 5 .05 NYR 20 22 16 26 42 34 0 . 7 7 8 n s 0 . 6 3 4 n s 3 . 1 9 0 n s 3 . 2 2 0 n s P h i 29 11 29 11 52 32 0 . 0 0 0 n s 2 . 4 3 5 n s 2 . 4 3 5 n s 3 . 8 5 7 n s P i t 21 23 15 29 40 32 1 . 6 9 2 n s 0 . 6 7 1 n s 5 . 0 4 7 05 5 . 0 4 7 . 0 5 Que 21 11 14 18 57 39 3 . 0 9 0 n s 0 . 3 9 4 n s 2 . 3 7 2 n s 3 . 4 7 8 n s S t L 23 17 17 23 43 37 1 . 8 0 0 n s 0 . 1 5 2 n s 1 . 3 5 0 n s 2 . 0 2 8 n s T o r 12 32 19 25 26 46 2 . 4 4 0 n s 0 . 9 6 9 n s 0 . 5 7 5 ns 2 . 4 4 1 ns V a n 14 22 11 25 34 44 0 . 5 5 1 n s 0 . 2 2 3 n s 1 . 7 5 1 n s 1 .758 n s Was 28 16 26 18 53 19 0 . 1 9 2 n s 1 .290 n s 2 . 6 5 1 n s 2 . 8 8 7 n s W i n 11 23 10 24 38 54 0 . 0 6 9 n s 0 . 8 3 7 ns 1 . 4 8 9 n s 1 .888 ns PG = P o i n t s G a i n e d , PL = P o i n t s L o s s , OG = O t h e r Games 1 s t v 2 n d , 1 s t v O G , 2nd v OG : : d f = 1 1 v 2 v OG : : d f = 2 T a b l e A15 S i g n i f i c a n c e L e v e l s f o r Runs T e s t s on I n d i v i d u a l Teams C o n d i t i o n A C o n d i t i o n B W,T = W; L = L Home ( n , 4 0 ) Away ( n , 4 0 ) T o t a l ( n , 8 0 ) I f home, W = W; L , T = L I f away , W,T = W; L = L Home T o t a l ( n , 4 0 ) ( n , 8 0 ) B o s B u f C a i C h i D e t Edm H a r LA M i n Mon N J NY I NYR P h i P i t Que S t L T o r V a n Was W i n .6688 . 6456 .4548 .1801 . 3232 . 7893 .1360 . 6 4 5 9 . 7974 . 8974 . 7957 . 3650 . 8827 .9318 . 5702 1 . 0 0 0 0 . 7 0 0 9 . 2 0 0 3 . 1 9 9 5 . 1 4 5 5 .4766 * S i g n i f i c a n t i n . 9437 . 8569 . 5 9 5 3 . 5702 .5218 . 1743 . 0 0 9 9 * . 0 2 3 8 * . 4766 . 3 9 9 3 . 6114 . 4 3 0 5 . 1206 . 9872 . 5218 .6720 . 6886 . 4376 . 9467 . 6114 . 9467 n o n - r a n d o m . 8028 .2194 . 0774 . 9 9 5 5 . 8359 . 0692 .6358 .6311 . 3392 . 7 2 1 3 . 8 2 6 3 . 1 1 5 9 .1452 . 8 9 1 5 . 8572 . 9816 . 5 4 8 5 . 2594 . 1166 . 8028 . 3552 d i r e c t i o n . 6 8 4 5 . 9 8 1 5 . 0 2 1 7 * . 2 0 6 5 . 8691 . 9437 . 8403 . 2203 . 3 9 9 3 . 7557 .8691 . 2662 . 3232 . 0 4 6 5 * . 1 9 9 5 .3498 . 2 2 0 3 .1448 . 2 6 3 9 .6258 . 7 8 9 3 . 6147 . 9820 .9942 .6218 .6528 . 1 0 2 3 . 3683 .5044 .8864 . 2460 .9816 .1867 .2801 . 8 2 6 3 . 9276 . 7187 .9820 . 4612 . 4 4 0 5 . 7 0 7 5 .4540 145 T a b l e A16 I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r Runs T e s t W i t h E a c h P e r i o d T r e a t e d a s a Game (W,T = W i n , L = L o s s ) P I P2 P3 P1+P2 P1+P2 P2+P3 B o s Home Away-T o t a l 9815x 9437x 0 0 0 3 * . 1801 .0851 . 2376 .7991 . 4 3 0 5 . 5377 ,7893 7974 3399 . 5702 . 7 9 7 4 .0602 .3498 . 0 4 0 9 * . 5482 Bu'f Home Away T o t a l 7974 1936 2594 9815x 6720 8310 7974 4625 0588 8691 3367 0937 3503 7360 3693 1964 3436 8028 C a i Home Away T o t a l 0 1 4 9 * 5493 0623 3606 9839x ,2149 0 3 9 4 * 3808 7461 1848 5493 3662 3606 4305 5377 8974 ,1964 0 4 7 6 * C h i Home Away T o t a l 0851 5313 9951x 0905 5478 1867 8569 9437 2801 3993 2048 3267 1801 4766 7161 0927 5254 2809 D e t Home Away T o t a l 3993 5254 2084 3232 2046 0 0 5 8 * 2853 0 3 4 7 * 1788 2065 5603 6311 5254 0549 2594 8827 5104 5855 Edm Home Away T o t a l ,5702 0 0 9 9 * 3693 ,0668 ,5603 ,7576 1609 3993 6921 5702 2003 6055 ,3367 ,2046 7161 9437 5254 8819 H a r Home Away T o t a l ,1460 ,0404* 9795x ,7592 6456 ,3614 ,2331 4766 3269 6978 1801 1893 3503 0927 8172 .3503 ,5104 ,7075 LA Home Away T o t a l ,5493 6720 2256 7592 ,1033 ,9216 5254 9872x 3267 1385 6978 0 4 9 1 * 7777 6978 5736 1801 0200* 0246* M i n Home Away T o t a l 8691 8403 2924 3467 ,5313 7161 4414 7603 0 1 2 2 * 2446 1360 1155 4766 5313 9594x 0 4 9 3 * ,7488 .1301 Mon Home Away T o t a l 7991 6666 0 1 4 7 * 7009 3993 4449 6590 0 3 1 4 * 7490 5422 2639 2288 7991 5702 2284 3808 3993 ,7609 N J Home Away T o t a l ,7893 , 8569 ,7761 0827 7957 , 8001 9318 2048 ,1746 0 4 9 3 * 2003 ,7103 2582 9872x 2502 6886 2190 6683 * = s i g n , n o n r a n d o m d i r e c t i o n ; x = s i g n , i n a l t e r n a t i n g d i r e c t i o n 146 T a b l e A16 I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r Runs T e s t W i t h E a c h P e r i o d T r e a t e d as a Game (W,T = W i n , L = L o s s ) ( c o n t i n u e d ) P I P2 P3 P1+P2 P1+P2 P2+P3 NYI Home Away-T o t a l . 7360 .6688 .5044 . 7 0 0 9 . 0 5 4 9 . 6396 . 6 3 4 5 .3367 . 8608 .6886 . 0 4 5 0 x . 6 6 8 3 .2048 . 3006 . 2 6 2 9 4625 5256 1835 NYR Home .9467 . 8 4 0 3 .7777 Away .7777 . 5 9 5 3 . 6720 T o t a l . 0 7 1 9 . 5 8 9 7 . 5 5 2 5 9467 9484 8001 .7974 . 1609 . 8080 .0691 .7957 .1452 P h i Home .9839 . 3 9 9 3 . 4073 Away . 9484 . 4 3 0 5 .0958 T o t a l . 2210 . 9 5 0 6 x . 5194 6712 3467 3166 . 0 1 4 6 * .5218 . 3 3 9 9 .4376 .1936 .5751 P i t Home Away T o t a l 1385 2582 8080 .3498 . 6114 . 1752 .2331 . 0927 .6218 0827 4219 4540 4305 3006 3269 . 8403 . 5313 .8001 Que Home Away T o t a l 2853 5603 5044 5493 8569 8041 . 6 3 4 5 . 4 6 2 5 . 8263 1095 5478 2199 . 7957 . 5313 . 6566 .2190 .4414 .8608 S t L Home .4904 . 8 4 0 3 . 1 0 3 3 Away . 3006 . 9467 . 1 7 4 3 T o t a l . 9044 . 1 4 5 7 . 0 5 7 3 3993 5603 8001 .5478 . 3 2 7 5 . 1592 .7777 . 1743 . 1460 T o r Home .7603 . 8827 l . O O O O x . 5 3 1 3 .5218 Away . 0 1 3 5 * l . O O O O x .2662 .0608 . 2446 T o t a l . 0 5 6 5 . 6147 . 7533 .6566 . 6147 ,4766 ,5104 ,6485 V a n Home .7592 . 3232 . 1936 Away . 9839 . 4 9 0 4 . 3650 T o t a l . 7359 . 9 7 9 5 x . 8864 9389 ,7957 ,2799 . 0 4 9 3 * l . O O O O x . 3174 . 9 8 3 9 x . 1095 .3460 Was Home .8974 . 1 9 9 5 1 . 0 0 0 0 Away .5702 .0958 . 4032 T o t a l . 9 9 3 8 x . 0672 .5751 6666 ,1385 0766 1 . 0 0 0 0 . 5953 . 3746 . 9389 .9437 .1299 Win Home . 7 8 9 3 . 2 8 5 3 . 9437 Away . 7603 . 9 8 7 2 x .2048 T o t a l . 6147 . 4074 .7761 9872x 8569 ,0679 . 3467 . 1 9 9 5 l . O O O O x .5218 l . O O O O x .8220 * S i g n i f i c a n t i n n o n r a n d o m d i r e c t i o n x S i g n i f i c a n t i n a l t e r n a t i n g W / L / W / L d i r e c t i o n T a b l e A17 S i g n i f i c a n c e L e v e l s f o r Runs T e s t s On Number o f S h o t s By P e r i o d f o r I n d i v i d u a l Teams Team (Team w i t h P e r i o d 1 M o s t S h o t s , P e r i o d 2 X P e r i o d P e r i o d 3 = Win) T o t a l S h o t s B o s . 9079 . 1938 .3344 . 9216 B u f . 2563 . 1547 . 8043 . 4 8 6 3 C a i . 4 7 4 5 . 1 9 6 5 . 9 8 1 5 x . 1532 C h i . 2697 .6971 . 9 9 4 9 x . 2829 D e t • 9 6 8 8 x . 4272 . 5164 .4351 Edm .1146 . 7860 . 6 3 7 5 . 4697 H a r . 8787 . 9087 .4674 .2144 LA . 3 2 4 5 . 7104 . 2 2 6 5 . 3460 M i n . 6260 . 7 2 4 9 .4517 . 0 1 2 4 * Mon . 6856 . 9 5 3 3 .6348 . 2660 N J .4221 . 1 7 9 5 . 6260 .1408 NY I .9368 . 3014 .4697 . 0708 NYR .5770 . 6280 .3437 . 1 7 7 3 P h i . 2346 . 0558 .6968 . 0 3 6 0 * P i t . 4468 . 2 4 7 5 .4008 . 7808 Que .4116 . 2576 .3779 . 8136 S t L . 8174 . 5992 . 0 4 3 5 * . 6438 T o r .2371 . 0 9 7 3 .1626 . 0 0 3 5 * V a n .0958 . 6442 .3487 . 0 5 9 5 Was .4860 . 1 0 4 2 .1900 . 8905 Win .3288 . 6 9 2 3 .1994 . 1460 * S i g n i f i c a n t i n n o n r a n d o m d i r e c t i o n x S i g n i f i c a n t i n a l t e r n a t i n g W / L / W / L d i r e c t i o n 148 T a b l e A18 I n d i v i d u a l Team Runs T e s t R e s u l t s on T o t a l G o a l s F o r Home, Away a n d T o t a l Games" B o s B u f C a l C h i D e t Edm H a r LA M i n Mon N J Home . 1 6 4 7 .9174 . 5236 . 3726 .0802 . 9 3 9 5 . 0 0 1 6 * . 2884 .3386 . 8645 . 9230 Away .4701 .4714 .1069 . 0 2 9 5 * ,7914 .6093 . 0 1 3 3 * 1339 2750 2786 8967 T o t a l . 1348 . 7423 . 7847 . 2 0 7 5 . 3 2 0 5 .9070 . 2364 .6331 . 0 8 2 3 .7838 . 9012 NYI NYR P h i P i t Que S t L T o r V a n Was Win Home . 9166 . 7846 . 5 0 4 5 . 6654 . 9 3 0 3 . 7 9 0 9 .7001 . 4486 .5532 . 2 0 4 9 Away .7361 . 3136 . 8301 . 0 5 4 9 . 4757 . 7 7 8 3 .0927 . 4 7 8 5 . 4 8 0 9 . 0 3 5 1 * T o t a l . 6491 . 5553 .4328 .4054 .9110 .5791 .1601 .1931 . 3643 .6416 G o a l s f o r (1) and g o a l s a g a i n s t (0) t r a c k e d f r o m p e r i o d t o p e r i o d , game t o game * D e n o t e s . 0 5 l e v e l o f s i g n i f i c a n c e o r l e s s i n n o n r a n d o m d i r e c t i o n . 149 T a b l e A19 I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s A b o v e / B e l o w Mean V a r i a t i o n s (P1G1. GF Home Games GA SF SA GF f o r Runs T e s t s on . . P 2 G 1 . . . P 3 G 4 0 ) Away Games GA SF SA .7250 .9262 .7300 . 8059 B o s . 9 2 6 2 B u f . 3 4 4 5 C a l . 0 3 5 0 * .8617 C h i . 7267 . 5 2 5 5 D e t . 9 5 8 8 x . 4 5 7 5 Edm . 2 8 4 9 H a r . 0 1 3 7 * LA . 2 0 8 3 M i n . 2 2 0 0 Mon . 2 6 6 3 N J .8211 NYI . 1621 NYR . 4 4 9 9 P h i . 3 0 2 5 P i t . 4322 Que . 4 4 5 3 S t L . 8134 T o r . 4 5 7 5 V a n . 9 5 1 9 Was . 3187 Win . 1426 . 8759 . 4556 . 4 9 1 9 . 8926 .4556 . 0 3 2 8 * . 9 1 4 3 . 6 5 4 5 . 3797 .0586 .1468 .9391 . 0 2 5 9 * . 6 5 9 9 9780x 9975X . 9 5 1 9 x . 4 7 0 5 . 3797 . 6764 . 0 3 5 7 * . 1344 .4931 . 0 7 3 3 . 0 4 2 8 * . 6192 . 9 8 8 7 x . 3 5 9 5 .6568 . 5 0 8 4 . 1 1 6 3 . 6 5 5 5 . 2152 . 2764 . 4 6 5 2 * D e n o t e s p r o b a b i l i t y o f . 0 5 x D e n o t e s p r o b a b i l i t y o f . 9 5 .1891 .6335 .2825 .6928 .1041 .0992 .1984 .3175 .5824 ,6335 ,6007 , 0174* ,4150 ,3175 , 0 1 5 9 * ,3976 0 0 0 0 X ,1457 ,5476 7182 7267 o r l e s s , o r more , .0672 ,5366 , 0147* ,5999 ,8199 9357 , 8905 ,1156 3587 6809 ,4914 6205 2577 3368 4499 9037 6240 6700 2641 9685 8706 ,4523 , 3752 ,8074 .7339 , 8470 , 8520 9973x ,7321 9451 ,4008 , 8520 9975x 4080 4914 7955 7228 8751 2663 1541 1245 2716 . 8259 .1390 .0992 .5290 .9410 .1494 .2076 . 9 7 8 0 x . 0 8 8 5 .3578 .6928 .8926 . 0 0 1 9 * . 4 7 0 5 . 4 9 1 9 .4937 . 7570 . 0929 . 7 5 3 3 .5841 . 4705 .3595 ,2564 2152 .2244 ,4008 .9477 ,0000 6148 6005 ,2701 6005 5802 8926 2716 6928 9410 9902 1014 9608 1390 9030 150 T a b l e A20 I n d i v i d u a l Team S i g n i f i c a n c e L e v e l s f o r Runs T e s t s A b o v e / B e l o w Mean V a r i a t i o n s ( P 1 G 1 . . . P 2 G 1 . . . P 3 G 8 0 ) T o t a l Games G o a l s F o r G o a l s A g a i n s t G o a l s F o r G o a l s Agad B o s .1704 . 3135 . 4487 .6907 B u f . 8200 . 2 7 1 5 .6106 . 3675 C a i . 0 1 1 2 * . 5787 . 2327 . 0 4 3 2 * C h i . 9 5 3 4 x . 3752 .7992 . 8792 D e t . 0777 . 4 6 1 5 • 9 5 1 7 x . 8754 Edm .3008 . 0 3 9 6 * .8954 .1824 H a r .1878 . 8083 . 1113 . 9107 LA . 0 3 6 9 * . 5834 .6106 .6196 M i n . 0653 .4821 . 0 7 5 5 . 2482 Mon . 6 8 2 3 . 5 7 1 5 . 5673 . 5063 N J . 7 3 4 5 . 5294 . 0 5 0 9 .4486 NYI . 7134 . 6 7 9 4 . 7609 . 8350 NYR .0580 . 4 4 7 9 . 0 1 2 1 * .5643 P h i . 1693 . 6 2 6 5 .4210 .3987 P i t . 7801 . 7924 . 6800 . 4599 Que . 5635 . 0 1 6 2 * . 1246 . 5 2 3 9 S t L . 8295 . 0 6 6 9 . 0 3 9 1 * . 1133 T o r . 9 5 9 0 x . 3 1 7 5 .6462 . 0604 V a n . 7979 .3951 . 0692 .7050 Was . 1910 . 2300 . 6383 . 8 2 4 5 Win .1744 . 3507 .3014 .8756 * D e n o t e s . 0 5 l e v e l o f s i g n i f i c a n c e o r l e s s ( n o n r a n d o m d i r e c t i o n ) x D e n o t e s . 9 5 l e v e l o f s i g n i f i c a n c e o r g r e a t e r ( a l t e r n a t i n g d i r e c t i o n ) 151 T a b l e A21 R u n s T e s t L e v e l s o f S i g n i f i c a n c e f o r I n d i v i d u a l Teams a n d S e t - t o - S e t C o r r e l a t i o n s o n F o u r - G a m e S e t s T r e a t e d a s Games ( T i e s = Wins ) Team Runs p 4G S e t r Teams Runs p 4G S e t B o s . 8250 . 026 NYI . 3 5 8 3 - . 2 6 7 B u f . 1 1 8 9 - . 3 5 7 NYR . 2880 - . 1 3 1 C a i . 7390 - . 0 5 6 P h i . 1158 .321 C h i . 6096 - . 1 6 9 P i t . 3766 - . 2 1 8 D e t . 1 8 4 9 . 186 Que . 6096 - . 1 6 9 Edm .7668 . 050 S t L . 3 4 9 3 .186 H a r . 5020 - . 2 0 9 T o r l . O O O O x - . 0 5 6 LA .7402 - . 1 4 9 V a n . 3297 - . 2 9 5 M i n . 3297 - . 2 9 5 Was . 3 4 9 3 .050 Mon .7390 - . 0 5 6 Win . 3 2 9 7 - . 2 6 7 N J .6478 - . 1 3 1 N b . X i s i n a l t e r n a t i n g ( n o n s t r e a k ) d i r e c t i o n o f s i g n i f i c a n c e . T a b l e A22 E m p i r i c a l P r o b a b i l i t y o f R e p e a t i n g a W i n , L o s s o r T i e C o n d i t i o n e d o n t h e Outcome o f P r e v i o u s Game(s) P P P P P P P P P W:1W W:2W W:3W L : 1 L L : 2 L L : 3 L T : 1 T T : 2 T T : 3 T w / n w / n w / n 1 / n 1 / n 1 / n t / n t / n t / n B o s .444 1 6 / 3 6 . 400 6 / 1 5 . 333 2 / 6 . 3 4 5 1 0 / 2 9 .300 3 / 1 0 . 333 1 /3 . 1 4 3 2 / 1 4 .000 0 / 2 0 / 0 B u f . 4 5 9 1 7 / 3 7 . 500 8 / 1 6 .500 4 / 8 . 514 1 8 / 3 5 .500 9 / 1 8 .444 4 / 9 . 286 2 / 7 .000 0 / 2 0 / 0 C a i . 6 7 9 3 6 / 5 3 .686 2 4 / 3 5 .652 1 5 / 2 3 . 0 5 9 1 /17 . 000 0 /1 . 000 0 / 0 . 000 0 / 9 0 / 0 0 / 0 C h i . 2 6 9 7 / 2 6 . 286 2 / 7 .500 1 /2 . 512 2 1 / 4 1 .714 1 5 / 2 1 .667 1 0 / 1 5 .167 2 / 1 2 . 000 0 / 2 0 / 0 D e t . 353 1 2 / 3 4 . 500 6 / 1 2 .667 4 / 6 . 394 1 3 / 3 3 . 333 4 / 1 2 .500 2 / 4 . 0 8 3 1 /12 .000 0 /1 0 / 0 Edm . 3 9 5 1 5 / 3 8 . 400 6 / 1 5 .667 4 / 6 . 3 0 3 1 0 / 3 3 . 400 4 / 1 0 .250 1 /4 . 1 2 5 1 /8 .000 0 /1 0 / 0 H a r . 459 1 7 / 3 7 .294 5 / 1 7 .200 1 / 5 . 432 1 6 / 3 7 . 500 8 / 1 6 . 3 7 5 3 / 8 . 000 0 / 5 0 / 0 0 / 0 LA . 537 2 2 / 4 1 .476 1 0 / 2 1 . 333 3 / 9 . 355 1 1 / 3 1 .364 4 / 1 1 .000 0 / 4 . 1 4 3 1 /7 .000 0 /1 0 / 0 M i n . 296 8 / 2 7 .500 4 / 8 . 500 2 / 4 . 389 1 4 / 3 6 . 429 6 / 1 4 .500 3 / 6 .188 3 / 1 6 . 333 1 / 3 .000 0 /1 Mon .604 3 2 / 5 3 . 6 2 5 2 0 / 3 2 . 650 1 3 / 2 0 .167 3 / 1 8 . 000 0 / 3 . 000 0 / 0 . 1 2 5 1 /8 . 000 0 /1 0 / 0 N J .308 8 / 2 6 . 1 2 5 1 /8 1 . 0 0 0 1/1 . 512 2 1 / 4 1 .524 1 1 / 2 1 . 4 5 5 5 /11 . 250 3 / 1 2 . 000 0 / 3 0 / 0 NYI .444 1 2 / 2 7 . 333 4 / 1 2 .250 1 /4 .660 3 1 / 4 7 . 6 4 5 2 0 / 3 1 .600 1 2 / 2 0 . 0 0 0 0 / 5 0 / 0 0 / 0 NYR .541 2 0 / 3 7 .600 1 2 / 2 0 . 5 8 3 7 / 1 2 . 5 2 9 1 8 / 3 4 . 529 9 / 1 7 .500 4 / 8 . 000 0 / 8 0 / 0 0 / 0 P h i . 472 1 7 / 3 6 . 529 9 / 1 7 . 5 5 5 5 / 9 . 457 1 6 / 3 5 . 375 6 / 1 6 .167 1 /6 . 1 2 5 1 /8 .000 0 /1 0 / 0 153 T a b l e A22 E m p i r i c a l P r o b a b i l i t y o f R e p e a t i n g a W i n , L o s s o r T i e C o n d i t i o n e d on t h e Outcome o f P r e v i o u s Game(s ) ( c o n t i n u e d ) P P P P P P P P P W:1W W:2W W:3W L : 1 L L : 2 L L : 3 L T : 1 T T : 2 T T : 3 T w / n w / n w / n 1 / n 1 / n 1 / n t / n t / n t / n P i t . 462 1 8 / 3 9 .412 7 / 1 7 . 571 4 / 7 . 424 1 4 / 3 3 . 429 6 / 1 4 .167 1 /6 . 1 4 3 1 /7 1 . 0 0 0 1/1 0 / 0 Que . 2 5 9 7 / 2 7 . 143 1 /7 .000 0 /1 .578 2 6 / 4 5 .600 1 5 / 2 5 .571 8 / 1 4 . 1 4 3 1 /7 .000 0 /1 0 / 0 S t L . 344 1 1 / 3 2 .400 4 / 1 0 . 333 1 / 3 . 400 1 4 / 3 5 .500 7 / 1 4 . 571 4 / 7 . 0 8 3 1 /12 . 000 0 /1 0 / 0 T o r . 286 8 / 2 8 . 1 2 5 1 /8 .000 0 /1 .511 2 3 / 4 5 . 500 1 1 / 2 2 . 5 4 5 6 / 1 1 . 000 0 / 6 0 / 0 0 / 0 V a n . 4 5 5 1 5 / 3 3 .400 6 / 1 5 .667 4 / 6 . 5 7 9 2 2 / 3 8 .476 1 0 / 2 1 . 333 3 / 9 . 1 2 5 1 /8 .000 0 /1 0 / 0 Was . 4 3 9 1 8 / 4 1 .500 9 / 1 8 .778 7 / 9 .321 9 / 2 8 . 333 3 / 9 .000 0 / 3 . 100 1 / 1 0 . 000 0 /1 0 / 0 Win . 240 6 / 2 5 . 333 2 / 6 . 500 1 /2 .476 2 0 / 4 2 .450 9 / 2 0 .556 5 / 9 . 0 8 3 1 /12 .000 0 /1 0 / 0 T o t a l . 439 . 4 6 5 . 556 .452 .491 . 4 6 5 .119 .087 .000 n 322 147 80 331 160 73 23 2 0 733 316 144 733 326 157 193 23 1 T a b l e A23 I n d i v i d u a l Team I n t e r p e r i o d Outcome a n d Game Outcome C o r r e l a t i o n s (W = 2 , T = 1, L = 0 ) PI v P2 PI v P3 P2 v P3 PI v GO P2 v GO P3 v GO B o s B u f C a i C h i D e t Edm H a r LA M i n Mon N J NYI NYR P h i P i t Que S t L T o r V a n Was W i n .079 .090 .078 .086 .085 .150 .001 .037 .117 .232 .129 .096 022 .177 ,047 ,057 ,018 ,121 024 019 015 .393 - . 0 8 8 .092 .104 - . 1 1 0 .098 .157 .049 .168 . 0 9 9 - . 0 1 4 - . 1 1 1 .092 .032 . 206 . 0 8 3 .017 .141 .189 .004 .043 .057 .037 .088 .051 - . 0 3 0 . 1 0 5 . 1 4 3 . 2 5 9 . 0 4 3 .008 . 0 4 9 .024 . 0 2 5 - . 1 4 9 . 323 . 0 3 9 . 0 3 3 . 1 2 5 .012 - . 0 5 3 . 0 8 3 ,624 ,352 .437 ,308 ,315 , 579 .441 .363 ,382 ,441 304 ,215 380 353 433 458 517 543 564 401 534 .370 .517 .587 .435 ,419 .469 ,454 ,536 .383 ,306 ,518 .486 ,425 ,459 668 514 362 420 403 518 428 .628 .421 .328 .672 .424 .398 .642 .648 .618 .510 . 4 7 5 .401 .641 .360 .577 .468 .536 .579 .550 . 4 0 5 .434 N o t e : E a c h p e r i o d i s t r e a t e d a s a s e p a r a t e game, T a b l e A24 P e r i o d t o P e r i o d C o r r e l a t i o n s o f G o a l s F o r , G o a l s A g a i n s t , S h o t s F o r and S h o t s A g a i n s t F o r I n d i v i d u a l Teams F o r P l v P 2 P l v P 3 P2vP3 P l v P 2 A g a i n s t P l v P 3 P2vP3 B o s G o a l s S h o t s . 046 .049 233 142 .204 .144 .174 . 0 6 9 021 084 031 195 B u f G o a l s S h o t s . 0 8 3 .164 006 176 124 013 014 023 083 052 005 246 C a l G o a l s S h o t s .168 .068 257 141 . 0 5 5 . 2 0 5 . 156 .070 015 043 063 206 C h i G o a l s S h o t s .172 .202 006 060 .061 . 1 3 5 062 073 239 068 039 125 D e t G o a l s S h o t s . 109 .052 343 143 .088 .212 .056 . 1 1 9 050 050 052 244 Edm G o a l s S h o t s .222 .028 050 097 . 0 5 9 . 040 .204 .057 122 001 072 069 H a r G o a l s S h o t s .006 .241 067 114 . 1 7 9 .126 .002 .121 236 061 129 061 LA G o a l s S h o t s 150 030 107 097 049 010 144 098 058 024 052 123 M i n G o a l s S h o t s .154 .148 096 038 037 ,283 199 146 052 083 200 135 Mon G o a l s S h o t s .031 .006 ,176 078 .071 .170 . 0 4 9 .262 134 260 096 009 N J G o a l s S h o t s . 046 .066 ,153 020 . 0 9 3 .071 .056 .091 002 195 131 142 NYI G o a l s S h o t s .120 .061 142 024 .001 .008 .029 . 1 2 9 017 042 275 051 NYR G o a l s S h o t s 109 324 069 121 . 146 .011 .064 . 0 9 3 002 117 146 139 P h i G o a l s S h o t s 121 170 063 008 .226 . 0 9 9 .032 . 0 9 3 025 129 033 138 T a b l e A24 P e r i o d t o P e r i o d C o r r e l a t i o n s o f G o a l s F o r , G o a l s A g a i n s t , S h o t s F o r and S h o t s A g a i n s t f o r I n d i v i d u a l Teams ( c o n t i n u e d ) F o r A g a i n s t P i t G o a l s . 097 . 0 7 5 . 0 2 5 - . 0 6 9 .072 .242 S h o t s . 0 5 9 .101 . 0 7 5 .286 .061 .107 Que G o a l s - . 0 0 8 - . 0 0 4 .027 .072 .099 - . 0 9 1 S h o t s .170 .124 .191 . 2 4 5 - . 1 1 4 .152 S t L G o a l s - . 0 4 2 . 1 0 9 . 0 4 9 - . 0 5 4 - . 0 0 4 . 0 3 5 S h o t s . 2 9 3 .172 - . 1 2 3 . 119 - . 0 2 9 . 1 4 5 T o r G o a l s - . 0 0 8 .231 .094 .148 .087 .098 S h o t s - . 0 5 0 .094 .037 .154 - . 0 4 9 - . 0 3 5 V a n G o a l s . 102 .143 - . 0 1 3 - . 2 0 2 .154 - . 0 1 2 S h o t s .164 - . 0 5 5 .028 - . 0 1 3 . 006 . 1 6 5 Was G o a l s - . 0 4 2 . 0 3 5 - . 0 0 1 . 103 .014 - . 0 6 8 S h o t s - . 0 5 8 - . 0 5 1 .032 . 100 .077 . 230 Win G o a l s .046 . 216 - . 0 3 5 . 136 - . 0 3 4 - . 0 1 4 S h o t s . 1 7 9 .032 . 1 0 5 .017 - . 0 4 5 - . 1 1 5 T a b l e A25 I n d i v i d u a l Team P e r i o d - t o - P e r i o d G o a l a n d S h o t C o r r e l a t i o n s T o t a l G o a l s B e t w e e n P e r i o d s T o t a l S h o t s B e t w e e n P e r i o d s Team PI v P2 P I v P3 P2 v P3 P I v P2 PI v P3 P2 v P3 B o s B u f C a l C h i D e t Edm H a r LA M i n Mon NJD NYI NYR P h i P i t Que S t L T o r V a n Was Win .042 .000 . 036 - . 1 7 7 . 063 .268 .007 .049 - . 0 8 6 .121 . 030 . 215 .138 - . 0 6 5 . 0 4 9 - . 0 7 4 - . 0 1 2 .024 . 016 .068 . 1 8 9 064 .100 .161 .212 .120 . 0 6 5 .062 .024 . 070 .068 .057 .102 .037 .012 .029 .106 .141 .166 .121 .103 .261 . 1 1 3 - . 1 0 8 .034 - . 1 0 9 - . 1 0 6 - . 0 5 2 . 189 . 0 1 5 .111 .138 . 116 - . 1 9 3 - . 0 9 6 - . 0 1 1 - . 0 7 3 . 0 1 9 .019 . 040 . 060 - . 0 0 6 - . 1 7 4 066 .044 .107 . 1 9 5 . 1 3 3 .017 . 2 2 3 .106 . 100 .007 .093 .011 .118 .057 .124 .003 . 2 6 9 .203 . 1 5 3 . 026 . 070 149 . 110 .034 .112 . 0 2 4 .127 .160 .122 .062 .032 . 0 7 5 . 0 6 5 .032 .232 .077 . 0 4 3 .062 .278 . 1 3 9 . 020 .201 . 036 .090 .297 .061 .151 . 086 .143 .092 . 180 .030 .098 .019 .049 .162 .101 .041 .041 .038 .254 .242 .123 T a b l e A26 I n d i v i d u a l Team C o r r e l a t i o n s o f G o a l D i f f e r e n c e s B e t w e e n P e r i o d s a n d w i t h T o t a l G o a l D i f f e r e n c e P I v P2 PI v P3 P2 v P3 PI v TGD P2 v TGD P3 v TGD B o s - . 1 5 0 .337 .122 .633 .494 . 7 7 5 B u f - . 1 0 1 - . 0 1 9 . 1 4 5 .408 .658 . 6 6 9 C a i .068 .187 .118 .573 . 670 .682 C h i - . 0 6 3 - . 0 1 0 .092 .477 . 6 0 9 .653 D e t - . 0 0 2 - . 1 8 3 - . 0 4 6 .451 .617 .520 Edm .117 .112 - . 0 5 2 . 626 .691 .496 H a r . 002 . 0 9 5 .136 . 506 . 6 1 5 .729 LA - . 0 8 5 .083 .071 .481 . 5 0 9 .759 M i n - . 2 6 7 . 079 .061 .443 .504 .707 Mon - . 1 4 8 . 210 - . 0 6 3 .607 .423 .699 N J - . 1 3 8 - . 0 8 1 . 1 1 3 .412 . 6 2 5 .630 NYI - . 0 5 6 .024 - . 0 7 5 . 555 .578 .532 NYR - . 0 1 5 .087 .008 .568 .501 .698 P h i - . 1 0 4 . 0 6 5 - . 1 5 6 .516 .538 .552 P i t - . 0 3 1 .090 .302 .536 .674 .720 Que .117 .191 - . 0 5 1 .662 .561 .648 S t L - . 1 3 6 - . 0 5 1 . 0 2 3 .508 .474 .646 TML . 1 6 9 .156 . 1 9 9 .629 .718 .661 V a n - . 1 3 3 .164 - . 1 1 8 . 529 . 506 .578 Was - . 0 1 2 - . 0 4 8 - . 0 6 0 . 543 .613 .502 Win .040 - . 0 9 3 .166 .539 . 6 7 9 .578 T o t a l - . 0 3 6 .061 .051 . 5 3 5 . 5 9 3 .646 159 T a b l e A27 I n d i v i d u a l Team C o r r e l a t i o n s o f S h o t D i f f e r e n c e B e t w e e n P e r i o d s a n d T o t a l S h o t D i f f e r e n c e PI v P2 PI v P3 P2 v P3 PI v TSD P2 v TSD P3 v TSD B o s .083 .002 . 2 6 5 .527 .723 .672 B u f . 084 . 0 5 3 . 1 0 5 .551 .667 . 6 4 3 C a l . 3 2 9 . 036 . 3 5 9 .614 .811 .682 C h i . 040 - . 0 0 9 .097 .542 .587 .669 D e t - . 0 6 9 - . 1 1 6 .278 .429 . 696 .659 Edm .044 . 1 4 5 - . 0 6 4 .681 .518 .602 H a r . 1 5 5 .039 - . 0 4 9 .671 .647 .491 LA - . 0 4 0 . 1 0 9 - . 1 8 2 . 6 0 3 .532 . 526 M i n .182 - . 0 5 8 .040 .621 .662 .541 Mon - . 1 3 0 .167 .114 .481 .631 .691 N J - . 1 5 6 . 0 4 9 .104 .484 . 5 3 5 .709 NYI .110 .092 - . 0 3 4 .648 .588 .590 NYR . 0 8 9 .037 .142 .522 .709 .640 P h i . 126 - . 0 9 4 . 0 7 5 .529 . 6 2 3 .631 P i t . 372 . 1 4 5 .161 .723 .666 .687 Que .287 . 000 . 226 . 630 . 776 .600 S t L . 156 .059 - . 0 4 6 . 696 . 5 9 5 .534 T o r .107 . 020 . 120 . 550 . 676 .640 V a n - . 0 1 8 - . 2 0 6 - . 0 1 4 . 4 9 9 .612 .477 Was .074 - . 0 0 2 . 296 .582 .699 .650 Win .248 - . 1 5 0 - . 0 7 7 . 643 . 6 7 5 .426 T o t a l .101 . 0 1 5 .100 .584 .652 .616 N B . T o t a l s h o t d i f f e r e n c e c o r r e l a t i o n w i t h Game Outcome = .142 160 T a b l e A28 C o r r e l a t i o n o f W i n s , G o a l s and S h o t s B e t w e e n P e r i o d s f o r A l l Teams C o m b i n e d P I v P2 PI v P3 P2 v P3 W i n / T i e s / L o s s e s - . 0 3 4 .082 .060 G o a l s F o r . 009 - . 0 0 2 . 075 G o a l s A g a i n s t .014 . 0 3 5 .073 S h o t s F o r .118 .034 .124 S h o t s A g a i n s t . 077 .054 .106 T o t a l G o a l s B o t h Teams .057 .052 .013 T o t a l S h o t s B o t h Teams .098 .087 .141 T a b l e A29 I n d i v i d u a l Team L a g g e d C o r r e l a t i o n s o f W i n s , T i e s , L o s s e s f o r P e r i o d s 1 , 2 and 3 B e t w e e n Games n and n + l P1G1 v P 1 G 2 . . n P2G1 v P 2 G 2 . . n P3G1 v P 3 G 2 . . n G l v G 2 . . n B o s B u f C a i C h i D e t Edm H a r LA M i n Mon N J NYI NYR P h i P i t Que S t L T o r V a n Was Win - . 2 2 8 - . 1 3 3 .102 - . 0 3 4 - . 0 3 5 - . 0 9 9 - . 0 2 0 .142 .017 - . 3 3 1 - . 1 2 2 - . 0 9 6 .034 - . 1 0 2 . 000 - . 0 4 0 .034 - . 2 6 7 . 140 - . 0 6 5 - . 0 6 4 - . 1 2 4 - . 1 4 3 .181 - . 1 8 3 - . 2 4 8 - . 1 0 1 - . 1 2 2 - . 0 6 2 .047 - . 0 9 2 .027 . 0 8 3 . 050 - . 0 2 8 - . 1 0 6 - . 0 9 8 . 1 8 5 .022 - . 0 9 5 .091 - . 1 5 6 .061 248 .024 .099 .102 .087 .083 .121 .293 .030 .168 .112 . 1 6 9 .108 . 0 2 5 . 0 1 9 .228 . 0 4 5 .098 .059 .087 - . 0 2 5 .041 - . 0 9 1 - . 0 7 2 - . 0 9 0 - . 2 0 0 - . 0 4 1 - . 0 1 8 - . 1 1 3 - . 1 6 4 - . 0 3 8 .161 .167 .028 - . 0 3 4 - . 0 6 8 - . 0 9 1 - . 1 2 5 .121 - . 1 2 2 - . 1 3 2 162 T a b l e A30 I n d i v i d u a l Team C o r r e l a t i o n s o f G o a l s F o r , G o a l s A g a i n s t and T o t a l G o a l s B e t w e e n P e r i o d s f o r Game l . . n v s . Game 2 . . . n G o a l s F o r G o a l s A g a i n s t T o t a l G o a l s P I G l P2 G l P3 G l P I G l P2 G l P3 G l P I P2 P3 B o s - . 0 7 0 - . 2 7 8 . 0 3 5 - . 0 4 6 . 1 1 5 - . 0 0 2 . 0 9 3 - . 0 8 6 - . 0 0 9 B u f - . 0 1 9 - . 0 2 5 . 1 4 5 - . 1 5 0 .018 . 190 - . 0 1 4 .121 .033 C a l - . 0 9 1 .037 .086 .298 - . 0 2 4 . 0 8 5 . 056 - . 0 3 0 .094 C h i - . 1 4 4 - . 1 3 2 . 1 0 5 . 0 9 3 - . 0 3 1 - . 0 3 0 - . 0 1 3 .021 - . 0 5 4 D e t . 037 .117 - . 0 4 6 - . 1 7 0 - . 0 5 4 - . 0 2 7 .017 . 206 - . 0 5 5 Edm .046 - . 1 2 3 - . 0 5 0 - . 0 9 8 .078 - . 0 0 1 - . 0 0 7 .130 .027 H a r - . 0 4 5 .004 .124 - . 1 8 7 - . 3 0 7 - . 0 1 6 - . 1 7 3 - . 2 0 5 - . 0 8 6 LA . 0 2 5 - . 1 2 1 .050 - . 0 2 7 .090 - . 1 3 4 - . 1 2 3 - . 0 5 1 - . 0 1 1 M i n - . 0 8 9 .046 - . 2 3 2 .042 - . 1 2 5 - . 0 6 1 - . 1 4 7 - . 0 6 2 - . 0 9 3 Mon - . 1 7 1 - . 0 5 8 - . 0 3 3 - . 0 9 9 .044 - . 0 2 4 - . 0 0 6 .057 .030 N J - . 1 6 8 .067 .217 - . 1 7 9 - . 1 2 6 - . 1 6 5 - . 2 4 5 - . 0 3 1 . 003 NYI . 012 - . 0 6 6 - . 0 5 5 - . 0 4 8 . 049 - . 0 7 2 . 0 2 3 - . 0 7 3 - . 0 6 8 NYR - . 0 5 2 .326 . 1 1 5 - . 0 3 0 - . 0 3 9 - . 2 0 0 - . 0 4 1 .118 - . 3 2 5 P h i - . 1 8 9 - . 1 8 6 - . 0 6 1 - . 0 5 1 - . 0 6 2 - . 0 8 8 - . 1 3 2 - . 2 1 4 - . 0 2 6 P i t - . 0 3 3 - . 0 6 6 .027 - . 0 9 7 .058 - . 1 2 1 - . 1 4 9 . 0 3 9 - . 1 0 0 Que - . 0 4 7 - . 1 4 4 - . 1 5 1 .084 - . 1 1 2 - . 0 6 1 . 0 7 3 - . 2 4 7 - . 2 2 6 S t L . 0 3 5 .030 - . 0 8 3 .007 .044 - . 0 4 2 - . 0 9 5 - . 0 0 4 .020 T o r - . 0 3 9 - . 1 0 8 .091 - . 0 7 4 - . 1 6 0 - . 0 3 6 . 176 - . 2 5 4 - . 0 1 7 V a n - . 1 0 8 - . 1 6 2 - . 2 3 6 . 2 0 3 .073 .096 . 0 3 5 - . 0 7 3 - . 0 8 3 Was - . 1 8 7 - . 0 1 6 .082 . 106 - . 1 7 0 . 0 2 5 - . 0 2 1 - . 1 5 2 .178 Win - . 0 6 8 .032 - . 0 0 8 - . 0 9 9 - . 1 6 8 - . 0 9 4 .018 - . 0 4 5 - . 0 4 6 163 T a b l e A31 I n d i v i d u a l Team C o r r e l a t i o n s o f S h o t s F o r , S h o t s A g a i n s t and T o t a l S h o t s B e t w e e n P e r i o d s f o r Game l . . n v s . Game 2 . . . n S h o t s F o r S h o t s A g a i n s t T o t a l S h o t s P I G l P2 G l P3 G l P I G l P2 G l P3 G l P I G l P2 G l P3 G l v v v v v v v v v P I G2 P2 G2 P3 G2 P I G2 P2 G2 P3 G2 P I G2 P2 G2 P2 G3 B o s B u f C a i C h i D e t Edm H a r LA M i n Mon N J NYI NYR P h i P i t Que S t L T o r V a n Was Win . 080 . 1 0 3 . 016 .008 . 030 .144 . 1 1 5 .187 .051 . 0 4 5 .227 .031 .094 .174 . 0 2 5 . 1 4 9 .112 .098 . 096 . 006 .142 .080 .001 .198 .109 ,110 ,078 ,081 .029 .127 ,053 ,044 ,050 106 035 ,108 054 ,034 ,186 087 019 069 - . 0 9 4 - . 0 4 9 - . 0 8 0 - . 1 0 4 - . 1 1 2 - . 1 3 5 - . 1 3 4 .121 .026 .068 . 156 - . 0 8 8 . 000 - . 0 6 1 - . 0 3 9 - . 0 6 7 - . 0 6 6 . 129 .001 .072 .062 ,135 ,167 ,071 ,028 066 ,002 ,043 ,003 .116 041 ,023 136 ,051 115 153 016 073 072 235 002 067 .157 .061 .090 .084 ,053 .029 ,166 ,096 ,093 ,020 ,002 .075 ,177 ,100 ,139 ,047 053 ,106 ,015 ,140 .009 .154 .077 .047 - . 1 1 0 - . 0 1 7 - . 0 8 7 - . 0 5 1 - . 0 3 9 .047 - . 0 8 6 .082 . 0 1 9 - . 1 5 3 .052 - . 0 2 0 - . 0 3 3 - . 0 0 7 .118 - . 1 4 1 - . 1 5 5 - . 0 2 1 .186 ,163 .026 ,075 077 .027 ,128 .081 ,077 ,095 ,049 ,020 ,077 ,002 015 009 141 052 077 137 ,101 ,070 ,183 202 021 024 018 019 165 090 012 058 090 073 080 006 135 073 094 057 235 098 - . 1 1 6 - . 0 1 8 . 099 - . 0 9 9 - . 0 2 1 - . 1 7 6 - . 1 7 5 . 0 6 5 - . 0 2 4 .081 . 155 - . 0 1 7 - . 1 2 3 - . 0 0 9 - . 0 6 7 - . 0 4 7 .016 .180 - . 0 3 6 - . 0 4 7 .018 T a b l e A32 Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r G o a l s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) Home GF GA Away GF GA T o t a l GF GA B o s raw b i n - . 0 2 7 - . 0 1 2 160 029 .128 .154 O i l 053 . 103 .086 .070 .064 B u f raw b i n - . 0 1 9 .078 .024 . 0 1 2 026 053 035 ,091 .008 . 010 081 073 C a i raw b i n .037 .186 191 009 . 1 6 5 .219 085 036 074 161 079 ,039 C h i raw b i n - . 1 0 1 - . 0 4 1 .017 . 0 5 3 026 044 . 1 2 5 .022 .052 . 0 0 9 .020 .054 D e t raw b i n . 0 4 9 - . 0 1 4 . 1 0 9 .057 . 0 2 3 . 0 1 5 . 1 2 5 .027 . 0 1 5 .021 .090 . 0 5 3 Edm raw b i n - . 0 5 3 .092 .152 . 190 . 0 5 5 .017 085 008 .018 .062 .063 .130 H a r raw b i n 098 ,214 007 021 .031 .017 .007 . 0 0 5 .038 .080 .016 .010 LA raw b i n - . 0 0 6 .107 .024 .028 046 135 089 046 .032 . 1 3 0 071 031 M i n raw b i n - . 1 1 1 - . 1 1 6 . 0 0 3 .018 .078 .071 .050 .004 . 0 9 3 .121 ,010 051 Mon raw b i n 097 093 117 072 .122 .042 .002 . 090 .001 . 0 2 4 .027 .032 N J raw b i n - . 0 6 8 - . 0 3 2 092 202 047 059 .023 .008 . 006 .017 .003 . 0 4 5 NYI raw b i n . 113 .124 .001 .067 .096 .048 196 006 .008 .022 .092 . 0 2 9 NYR raw b i n .062 .063 082 190 ,183 098 .066 . 0 8 5 .142 . 120 049 052 P h i raw b i n - . 1 9 8 - . 1 0 4 054 005 .114 .092 .018 . 0 5 9 .134 . 0 9 4 .014 .029 Raw = raw s c o r e B i n = raw s c o r e s c o n v e r t e d t o 0 ,1 s c o r i n g s y s t e m (no g o a l s i n p e r i o d = 0 ; 1 o r more g o a l s i n a p e r i o d T a b l e A32 Raw and B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r G o a l s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) ( c o n t i n u e d ) Home GF GA Away GF GA T o t a l GF GA P i t raw - . 0 8 0 . 0 7 9 . 1 2 9 - . 0 1 2 .030 . 0 2 5 b i n - . 0 8 2 - . 0 4 6 . 063 . 013 . 0 1 5 .011 Que raw - . 0 3 0 - . 0 2 8 . 0 4 5 - . 0 0 2 . 029 . 0 2 5 b i n . 059 . 0 7 2 . 006 . 0 2 3 .031 .151 S t L raw - . 0 4 2 - . 0 2 3 .040 - . 0 6 5 .008 - . 0 3 1 b i n .017 . 1 6 5 - . 0 5 9 - . 0 2 0 - . 0 2 0 - . 1 2 0 T o r raw - . 0 5 5 . 2 1 3 .087 .064 .017 .090 b i n . 057 .127 . 036 .093 .002 .060 V a n raw - . 0 9 1 - . 0 0 3 .161 - . 1 0 8 .008 - . 0 6 8 b i n - . 0 1 8 - . 0 1 3 . 090 - . 1 3 3 . 010 - . 0 6 2 Was raw - . 0 3 4 .164 .010 - . 1 1 3 . 020 .001 b i n . 079 . 1 9 9 - . 0 0 8 - . 1 6 0 .077 - . 0 8 4 Win raw - . 0 4 0 . 0 2 3 .094 .070 - . 0 0 4 .061 b i n - . 1 3 7 .034 - . 0 2 0 .092 - . 0 9 0 .057 Raw = raw s c o r e B i n = raw s c o r e s c o n v e r t e d t o 0 ,1 s c o r i n g s y s t e m (no g o a l s i n p e r i o d = 0 ; 1 o r more g o a l s i n a p e r i o d T a b l e A33 Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) Home Away T o t a l SF SA SF SA SF S A _ B o s raw - . 0 5 8 - . 1 2 5 .041 .160 .027 . 045 b i n .006 - . 1 2 9 . 0 0 9 .076 . 0 4 5 B u f raw - . 0 3 8 - . 0 0 3 . 076 .180 .031 b i n - . 0 7 8 . 0 3 5 - . 1 4 4 .098 . 030 C a i raw .086 . 120 .142 .038 . 120 b i n . 053 . 0 8 9 . 1 4 3 . 1 0 5 .074 C h i raw .162 - . 0 5 4 .067 .094 . 0 8 5 b i n .004 - . 0 4 4 .051 . 1 0 3 .012 D e t raw .084 .144 . 0 3 9 - . 0 2 4 .050 b i n - . 0 7 8 .141 - . 0 0 2 - . 0 8 2 . 000 Edm raw - . 0 6 5 . 0 5 5 .101 - . 0 5 0 - . 0 0 6 b i n .001 . 1 4 3 . 1 2 5 - . 0 1 5 - . 0 1 4 H a r raw . 2 1 5 - . 0 3 7 . 0 8 3 - . 0 2 0 . 1 7 5 b i n - . 0 1 2 - . 1 2 6 .107 - . 0 0 8 .098 LA raw - . 0 4 3 - . 0 4 8 - . 1 0 7 - . 0 2 2 . 0 0 5 b i n - . 0 0 6 - . 1 0 0 - . 0 1 2 .038 . 030 M i n raw .127 - . 0 2 9 . 113 - . 0 4 3 . 1 4 3 b i n .058 .042 . 1 4 7 .040 .111 Mon raw .057 - . 1 6 2 .008 - . 1 2 8 .047 b i n - . 0 4 6 . 0 3 5 - . 0 9 3 - . 1 1 0 - . 0 4 2 N J raw .066 - . 1 4 2 - . 0 1 5 .016 .004 b i n - . 1 4 7 - . 0 5 9 - . 0 4 6 .040 - . 1 3 1 NYI raw .097 .028 - . 0 9 4 - . 1 2 3 . 026 b i n . 157 .210 .004 - . 0 5 9 . 0 1 5 NYR raw .027 - . 1 4 0 .176 .077 .162 b i n . 0 3 9 - . 0 8 2 .277 .004 . 1 5 9 P h i raw .167 .014 .093 .037 .138 b i n . 076 . 0 8 5 .058 .092 . 0 4 9 P i t raw - . 0 0 7 .197 - . 0 3 7 .126 . 0 5 2 b i n . 032 .211 . 056 - . 0 4 6 . 0 2 3 T a b l e A33 Raw a n d B i n a r y I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s ( P 1 G 1 . . P 2 G 1 . . P 3 G n ) ( c o n t i n u e d ) Home Away T o t a l SF SA SF SA SF SA Que raw .010 .181 .028 .063 .131 .083 b i n . 053 . 0 6 9 .054 - . 0 0 2 . 0 9 5 .037 S t L raw . 0 8 5 .037 .083 .077 .101 .120 b i n .138 - . 0 0 8 . 020 - . 0 0 9 . 1 2 9 .098 T o r raw - . 0 4 2 . 0 6 5 - . 0 4 2 . 013 .027 . 0 6 5 b i n - . 0 5 1 . 1 2 5 - . 1 6 3 .142 - . 0 3 4 .117 V a n raw - . 0 3 0 - . 0 7 1 .064 . 0 3 5 .063 .046 b i n . 1 0 5 - . 0 6 4 - . 0 4 1 - . 0 1 2 . 1 1 3 .020 Was raw .147 . 130 - . 0 6 6 .027 .037 .086 b i n .092 .024 .044 - . 1 4 4 . 026 .010 Win raw .204 - . 0 0 7 . 030 - . 1 4 4 . 1 3 5 - . 0 3 8 b i n . 059 - . 0 4 1 .057 . 0 0 5 . 0 6 3 - . 0 1 4 raw = raw s c o r e s b i n = raw s c o r e s c o n v e r t e d t o 0 /1 s c o r i n g s y s t e m : no g o a l s i n p e r i o d = 0 1 o r more g o a l s i n p e r i o d = 1 168 T a b l e A34 I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s a n d G o a l s F o r and A g a i n s t GF v SF GA v SA GF v GA SF v SA GA v SF B o s Home .351 .427 .024 - . 2 4 8 - . 1 1 1 Away .341 .207 - . 0 0 2 - . 0 7 1 .116 T o t a l . 322 .317 .009 - . 1 7 3 - . 0 0 3 B u f Home . 3 7 5 .391 - . 0 4 6 - . 2 3 3 .082 Away .339 .312 - . 1 9 8 - . 0 6 8 - . 0 6 0 T o t a l .371 .351 - . 1 2 9 - . 1 6 5 . 003 C a l Home .372 .324 - . 0 0 2 - . 2 1 9 - . 1 3 9 Away .390 . 226 .140 - . 2 4 8 .194 T o t a l . 3 8 9 . 270 .048 - . 2 2 7 .017 C h i Home .388 .294 . 0 2 3 - . 2 6 8 - . 0 2 5 Away .240 .242 .028 - . 2 4 6 .020 T o t a l .344 .282 .013 - . 2 8 0 - . 0 2 2 D e t Home .207 . 4 3 5 - . 1 0 6 - . 1 1 0 - . 0 0 1 Away .476 . 2 9 5 - . 0 0 3 - . 1 6 4 - . 0 7 8 T o t a l . 349 . 3 7 0 - . 0 4 9 - . 1 5 5 - . 0 5 2 Edm Home .382 .278 .051 - . 2 2 2 - . 1 2 6 Away . 3 4 9 .263 .030 - . 2 7 4 - . 1 3 0 T o t a l .371 . 270 . 0 3 9 - . 2 5 4 - . 1 2 9 H a r Home .392 . 4 6 5 - . 1 5 1 - . 2 3 6 - . 2 6 5 Away .397 .240 .130 - . 1 1 5 . 069 T o t a l . 3 9 5 . 3 4 9 - . 0 2 8 - . 1 7 9 - . 1 2 3 LA Home . 4 0 3 .292 .011 - . 1 5 6 - . 2 0 6 Away .359 .347 - . 0 8 1 - . 2 0 9 .016 T o t a l .411 . 3 1 9 - . 0 2 9 - . 1 8 2 - . 1 0 4 M i n Home .336 .199 - . 0 3 2 - . 1 8 3 .082 Away . 2 7 3 .112 .052 - . 2 6 3 .100 T o t a l .312 .152 . 0 0 5 - . 2 2 7 .087 Mon Home .394 . 336 .030 - . 1 9 4 .008 Away .272 .262 - . 0 2 5 - . 1 9 8 .038 T o t a l . 333 . 296 - . 1 0 0 - . 2 1 7 .018 N J Home .209 . 3 6 3 .109 - . 3 1 8 .088 Away .374 .277 .060 - . 0 7 2 .082 T o t a l .288 . 330 .081 - . 2 0 8 .077 169 T a b l e A34 I n d i v i d u a l Team C o r r e l a t i o n C o e f f i c i e n t s f o r S h o t s a n d G o a l s F o r and A g a i n s t ( c o n t i n u e d ) GF v SF GA v SA GF v GA SF v SA GA v SF NYI Home .249 Away .424 T o t a l . 336 NYR Home .283 Away .330 T o t a l . 304 P h i Home .199 Away .320 T o t a l . 2 6 5 P i t Home .157 Away .329 T o t a l .247 Que Home .374 Away .300 T o t a l .362 S t L Home . 2 6 3 Away .341 T o t a l . 309 T o r Home .417 Away . 2 7 4 T o t a l .341 V a n Home .186 Away . 3 3 3 T o t a l .257 Was Home . 2 3 5 Away .355 T o t a l . 290 Win Home .423 Away .414 T o t a l .418 .168 .092 .244 - . 0 0 9 . 2 0 9 .031 .482 - . 0 1 0 .341 .134 .410 .052 .277 - . 0 2 6 .498 - . 0 1 6 . 406 - . 0 2 8 . 216 .018 .411 - . 1 7 3 .334 - . 0 8 7 . 366 - . 0 8 1 . 270 - . 1 4 9 . 3 1 9 - . 1 2 0 .300 - . 1 4 5 . 470 .012 .408 - . 0 7 8 .350 - . 0 5 7 .337 .132 . 3 4 5 .048 .384 .168 .263 . 0 6 5 . 336 . 116 . 316 .086 .270 . 0 3 5 .304 .052 .112 .064 .351 - . 0 4 5 .254 .002 - . 2 1 5 - . 0 2 4 - . 3 6 2 - . 0 8 7 - . 3 0 0 - . 0 6 7 - . 1 4 0 - . 0 6 0 - . 0 8 7 .144 - . 1 2 4 . 0 2 5 - . 0 9 6 .040 - . 2 3 1 - . 0 1 6 - . 1 9 1 .001 - . 2 1 6 .020 - . 2 2 5 - . 0 3 2 - . 2 4 3 - . 0 3 2 - . 0 2 6 .036 - . 1 3 9 - . 0 6 4 - . 1 4 8 - . 0 3 8 - . 1 7 2 - . 0 4 4 - . 1 3 8 .001 - . 1 7 0 - . 0 4 7 - . 2 0 6 - . 1 3 0 - . 2 6 0 - . 0 3 5 - . 2 3 8 - . 0 8 3 - . 0 7 2 . 013 - . 1 5 7 .016 - . 1 4 5 - . 0 0 4 - . 2 8 8 - . 0 1 6 - . 0 6 4 - . 0 6 3 - . 1 9 6 - . 0 5 0 - . 2 4 8 - . 0 1 5 - . 1 2 8 .028 - . 1 9 0 .007 o 170 T a b l e A35 I n d i v i d u a l Team T e s t s o f S t a t i o n a r i t y C h i - s q u a r e A n a l y s i s o f Mean V a r i a t i o n s o n F o u r - G a m e S e t s Games ( T i e s = W i n s ) G o a l s F o r G o a l s A g a i n s t S h o t s F o r S h o t s A g a i n s t X l s e f X l s X l s X l s 2 X l s B o s 6 . 8 3 6 . 0 5 U L 1 . 3 4 8 1 . 0 0 6 ns n s 0 . 9 5 4 0 . 9 5 4 n s n s 0 . 1 5 4 0 . 1 5 4 n s n s 0 . 4 7 7 n s 0 . 6 1 9 ns B u f 1 .381 n s U L 0 . 1 4 4 0 . 1 4 4 n s n s 0 . 1 4 4 0 . 1 4 4 n s n s 0 . 4 7 7 0 . 6 1 9 n s n s 0 . 4 7 7 n s 0 . 6 1 9 n s C a l 2 . 3 4 1 n s U L 0 . 6 1 9 0 . 1 5 8 n s ns 1 . 1 4 5 0 . 6 5 0 n s ns 0 . 5 8 3 1 . 0 1 0 n s ns 0 . 4 7 7 ns 0 . 6 1 9 ns C h i 1 . 6 3 0 n s U L 0 . 0 0 0 0 . 1 5 4 n s n s 0 . 6 5 0 1 .418 n s ns 4 . 1 3 3 * n s 4 . 6 6 9 * n s 0 . 1 5 4 n s 0 . 1 5 4 ns D e t 2 . 9 3 0 n s U L 0 . 0 0 0 n s 0 . 1 5 4 ns 0 . 6 1 9 0 . 4 7 7 ns ns 0 . 1 5 8 0 . 6 1 9 ns n s 2 . 6 0 0 n s 2 . 2 0 6 n s Edm 4 . 5 4 8 n s U L 0 . 0 0 0 0 .154 n s n s 4 . 2 4 8 * n s 3 . 7 4 4 * n s 0 . 6 1 9 0 . 4 7 7 n s n s 0 . 4 7 7 n s 0 . 6 1 9 n s H a r 1 4 . 8 4 9 .001 U L 0 . 1 5 4 0 . 1 5 4 n s ns 0 . 6 1 9 0 . 1 5 8 ns ns 0 . 6 1 9 0 . 4 7 7 n s n s 0 . 1 5 4 n s 0 . 1 5 4 ns LA 4 . 0 4 8 n s U L 0 . 4 7 7 0 . 1 5 8 n s n s 0 . 1 5 8 0 . 6 1 9 n s n s 0 . 1 5 8 0 . 6 1 9 n s n s 1 . 1 4 5 ns 1 .142 ns M i n 6 . 2 4 5 .05 U L 1 . 1 4 5 1 . 4 1 8 n s ns 0 . 1 5 4 0 . 1 5 4 n s n s 0 . 1 4 4 0 . 4 4 0 n s n s 0 . 4 4 0 ns 0 . 1 4 4 ns Mon 0 . 7 0 9 n s U L 0 .000 0 . 1 5 4 n s n s 1 . 1 4 5 1 .418 n s ns 1 . 3 4 8 1 . 8 9 5 n s n s 0 . 0 0 0 n s 0 . 1 5 4 n s N J 1 7 . 4 6 3 .001 U L 0 . 144 0 . 1 4 4 n s n s 0 . 650 1 . 1 4 5 n s n s 0 . 477 0 . 1 5 8 n s n s 2 . 1 3 6 n s 2 . 2 6 7 n s NYI 2 . 9 6 9 n s U L 1 .171 1 . 1 7 1 n s n s 0 . 9 5 4 0 . 9 5 4 n s n s 0 . 1 4 4 0 . 1 4 4 ns n s 0 . 1 4 4 ns 0 . 1 4 4 n s NYR 0 . 2 8 8 n s U L 0 . 0 0 0 0 . 1 5 4 n s ns 0 . 0 0 0 0 . 1 5 4 ns ns 0 . 4 4 0 0 . 1 4 4 n s n s 0 . 1 5 8 ns 0 . 4 7 7 n s P h i 2 . 4 0 1 n s U L 1 . 1 4 5 0 . 6 5 0 n s ns 0 . 5 8 3 0 . 4 1 8 n s n s 0 . 0 0 0 0 . 1 5 4 n s n s 0 . 4 7 7 n s 0 . 6 1 9 ns 171 T a b l e A35 I n d i v i d u a l Team T e s t s o f S t a t i o n a r i t y C h i - s q u a r e A n a l y s i s o f Mean V a r i a t i o n s o n F o u r - G a m e ( c o n t i n u e d ) S e t s Games ( T i e s = W i n s ) G o a l s F o r G o a l s A g a i n s t S h o t s F o r S h o t s A g a i n s t X I s e f X I s X I s X I s 2 X I s P i t 6 . 0 1 5 ,05 U L 1 . 1 1 5 1 . 4 1 8 n s ns 2 . 2 6 7 2 . 1 3 6 ns ns 2 . 2 0 6 1 . 5 2 9 ns ns 0 . 4 4 0 ns 0 . 1 4 4 n s Que 5 . 0 4 4 ns U L 0 . 1 5 8 0 . 4 7 7 n s n s 0 . 1 5 8 0 . 4 7 7 n s n s 0 . 1 5 4 0 . 1 5 4 ns ns 0 . 4 4 0 n s 0 . 1 4 4 ns S t L 6 . 6 8 5 05 U L 0 . 1 5 4 0 . 1 5 4 n s n s 0 . 0 0 0 0 . 1 5 4 n s ns 0 . 6 1 9 0 . 4 7 7 ns ns 1 .418 ns 1 . 1 4 5 ns T o r 0 . 2 8 2 n s U L 0 . 1 5 4 n s 0 . 1 5 4 n s 0 . 4 1 8 1 . 0 1 0 n s n s 3 . 8 4 4 * n s 3 . 5 9 3 * n s 0 . 0 0 0 n s 0 . 1 5 4 ns V a n 2 . 8 3 6 n s U L 0 . 1 4 4 ns 0 . 1 4 4 ns 0 . 1 5 8 0 . 4 7 7 n s n s 0 . 1 5 4 0 . 1 5 4 ns n s 1 .798 n s 1 .622 n s Was 1 . 6 7 9 n s U L 0 .477 0 . 6 1 9 n s n s 0 .477 0 . 6 1 9 n s n s 0 . 1 5 8 0 . 6 1 9 n s n s 0 . 1 5 8 n s 0 . 4 7 7 n s Win 9 . 1 6 9 02 U L 2 . 2 0 6 2 . 6 0 0 ns n s 0 . 5 8 3 0 . 4 1 8 ns n s 1 . 0 0 6 1 . 3 4 8 n s ns 0 . 1 5 8 ns 0 . 4 7 7 n s d f = 2 f o r a l l c a s e s 2 c e l l s w i t h e x p e c t e d c o u n t s l e s s t h a n 5 . 0 172 T a b l e A36 I n d i v i d u a l Team N o r m a l D i s t r i b u t i o n C h e c k s and S e t - t o - S e t C o r r e l a t i o n s f o r f o u r game s e t s F o r A g a i n s t T o t a l n d c * r n d c r n d c r_ B o s g o a l s .986 .173 .959 . 2 5 3 .966 .184 s h o t s .990 .192 . 9 9 5 .123 . 9 8 5 .148 B u f g o a l s . 9 9 5 - . 1 9 2 . 9 9 5 - . 1 9 2 .992 - . 1 1 7 s h o t s . 969 . 291 .973 . 056 .977 .458 C a i g o a l s . 990 - . 3 2 4 . 979 .078 .977 - . 2 8 2 s h o t s .946 - . 1 1 5 . 9 8 5 .297 .961 .489 C h i g o a l s . 989 - . 4 5 2 . 9 8 5 . 146 . 9 8 9 - . 1 4 1 s h o t s .887 .224 . 9 9 5 - . 1 6 7 .984 .178 D e t g o a l s . 9 9 9 . 186 . 9 9 5 - . 3 2 3 .987 - . 3 1 5 s h o t s . 9 6 5 - . 0 2 8 .987 - . 0 8 4 .971 .329 Edm g o a l s .958 . 3 3 5 . 9 7 9 - . 2 5 4 . 9 9 5 .077 s h o t s .992 - . 1 2 4 .981 - . 0 7 3 .992 - . 0 0 2 H a r g o a l s .981 .004 . 9 8 9 - . 2 1 8 .957 .051 s h o t s .967 . 4 5 6 .990 - . 0 6 7 .978 - . 1 4 4 LA g o a l s . 9 8 5 - . 2 1 3 .962 .227 .991 - . 0 7 8 s h o t s .976 - . 0 3 7 .978 - . 2 0 7 .992 . 455 M i n g o a l s . 980 - . 0 3 5 . 9 4 5 - . 0 3 9 .982 - . 1 8 3 s h o t s . 987 - . 2 0 6 . 9 6 5 - . 1 4 9 .980 - . 0 7 5 Mon g o a l s . 992 - . 0 9 7 .974 - . 2 2 6 . 969 - . 3 1 5 s h o t s .971 . 0 8 3 .979 - . 1 9 4 .984 - . 0 7 3 N J g o a l s .994 .010 . 9 5 3 - . 0 8 0 .961 .039 s h o t s .972 . 0 0 3 .977 - . 2 0 0 .981 .017 NYI g o a l s . 990 - . 0 6 6 .984 - . 1 0 0 .992 - . 0 1 4 s h o t s .982 . 1 9 9 .948 - . 6 3 0 .981 .066 NYR g o a l s . 987 .137 .990 .306 . 9 9 2 .256 s h o t s .988 - . 1 3 3 .988 .114 .983 .190 P h i g o a l s .994 - . 0 9 7 .980 .302 . 9 9 3 .093 s h o t s .991 - . 1 7 1 . 986 - . 2 7 7 .987 .218 P i t g o a l s . 986 - . 3 7 6 . 9 8 5 - . 2 5 7 . 980 - . 5 3 0 s h o t s .972 - . 2 1 4 .968 .171 . 970 - . 0 4 5 173 T a b l e A36 I n d i v i d u a l Team N o r m a l D i s t r i b u t i o n C h e c k s a n d S e t - t o - S e t C o r r e l a t i o n s f o r f o u r game s e t s ( c o n t i n u e d ) Que F o r n d c * A g a i n s t n d c T o t a l n d c g o a l s s h o t s .990 .980 645 024 978 975 007 188 988 964 . 106 . 1 8 3 S t L g o a l s s h o t s ,991 984 - . 5 7 9 - . 0 7 0 . 9 8 5 .976 .151 - . 0 7 2 . 980 .948 - . 2 3 5 . 055 T o r g o a l s s h o t s .996 .961 199 203 989 979 329 554 987 991 .010 .283 V a n g o a l s s h o t s . 985 . 9 7 9 266 042 987 984 - . 0 5 5 .161 975 975 - . 0 3 1 .232 Was g o a l s s h o t s 985 959 031 241 978 976 - . 2 4 1 .170 990 986 .310 .090 Win g o a l s s h o t s . 969 .966 . 0 6 5 .111 994 - . 0 2 6 986 - . 2 2 8 960 979 . 050 - . 3 3 9 * n d c = n o r m a l d i s t r i b u t i o n c h e c k 174 T a b l e A37 A r t i f i c i a l and R e a l S e a s o n R e c o r d s S e a s o n W L T P t s % S e a s o n W L T P t s % B o s t o n B r u i n s B u f f a l o S a b r e s 8 8 - 8 9 37 29 14 88 . 550 8 8 - 8 9 38 35 7 83 . 519 P 1 35 23 22 92 . 5 7 5 P 1 23 34 23 69 .431 P 2 25 24 31 81 .506 P 2 37 23 20 94 .588 P 3 34 18 28 96 .600 P 3 30 31 19 79 . 494 P 1+2 30 31 19 79 . 494 P 1+2 34 33 13 81 .506 P 1 + 3 40 28 12 92 . 5 7 5 P 1+3 30 34 16 76 . 4 7 5 P 2 + 3 35 30 15 85 . 5 3 1 ' P 2+3 37 29 14 88 .550 C a l g a r y F l a m e s C h i c a g o B l a c k Hawks 8 8 - 8 9 54 17 9 117 .731 8 8 - 8 9 27 41 12 66 . 413 P 1 34 23 23 91 . 5 6 9 P 1 30 29 21 81 . 506 P 2 43 19 18 104 .650 P 2 26 30 24 76 . 4 7 5 P 3 45 13 22 112 .700 P 3 23 43 14 60 . 3 7 5 P 1+2 47 20 13 107 . 6 6 9 P 1+2 30 33 17 77 .481 P 1+3 47 19 14 108 . 6 7 5 P 1+3 30 36 14 74 . 463 P 2+3 53 16 11 117 .731 P 2 + 3 25 48 7 57 .356 D e t r o i t R e d W i n g s E d m o n t o n O i l e r s 8 8 - 8 9 34 34 12 80 .500 8 8 - 8 9 38 34 8 84 . 5 2 5 P 1 34 26 20 88 . 500 P 1 21 34 25 67 . 419 P 2 31 30 19 81 . 506 P 2 37 27 16 90 . 563 P 3 27 39 14 68 . 4 2 5 P 3 31 28 21 83 . 519 P 1+2 33 31 16 82 . 5 1 3 P 1+2 33 33 14 80 .500 P 1+3 34 34 12 80 . 500 P 1+3 32 36 12 76 . 4 7 5 P 2+3 31 36 13 75 . 4 6 9 P 2+3 39 28 13 91 . 569 H a r t f o r d W h a l e r s L o s A n g e l e s K i n g s 8 8 - 8 9 37 38 5 79 . 494 8 8 - 8 9 42 31 7 91 . 5 6 9 P 1 28 26 26 82 . 5 1 3 P 1 28 29 23 79 .494 P 2 27 24 29 83 . 519 P 2 32 25 23 87 . 543 P 3 31 29 20 82 . 5 1 3 P 3 34 33 13 81 .506 P 1 + 2 35 32 13 83 . 5 1 9 P 1+2 36 32 12 84 .524 P 1+3 35 31 14 84 . 5 2 5 P 1 + 3 33 32 15 81 .506 P 2+3 37 32 11 85 .531 P 2+3 38 32 10 86 .538 175 T a b l e A37 A r t i f i c i a l a n d R e a l S e a s o n R e c o r d s ( c o n t i n u e d ) S e a s o n W L T P t s % S e a s o n W L T P t s % M i n n e s o t a N o r t h S t a r s M o n t r e a l C a n a d i e n s 8 8 - 8 9 27 37 16 70 .438 8 8 - 8 9 53 18 9 115 . 7 1 9 P 1 28 26 26 82 . 513 P 1 43 17 20 106 .663 P 2 27 36 17 71 .444 P 2 29 23 28 86 .538 P 3 32 34 14 78 .488 P 3 40 19 21 101 .631 P 1+2 29 40 11 69 .431 P 1+2 47 16 17 111 .694 P 1 + 3 32 37 11 75 . 4 6 9 P 1+3 51 20 9 111 .694 P 2+3 30 35 15 75 . 4 6 9 P 2+3 41 19 20 102 .638 New J e r s e y ] D e v i l s New Y o r k I s l a n d e r s 8 8 - 8 9 27 41 12 66 . 4 1 3 8 8 - 8 9 28 47 5 61 .381 P 1 25 33 22 72 . 450 P 1 24 34 22 70 .438 P 2 22 34 24 68 . 4 2 5 P 2 27 31 22 76 . 4 7 5 P 3 26 34 20 72 . 450 P 3 25 37 18 68 . 425 P 1+2 24 35 21 69 .431 P 1+2 29 38 13 71 .444 P 1 + 3 26 38 16 68 . 4 2 5 P 1+3 21 41 18 60 . 3 7 5 P 2+3 26 38 16 68 . 4 2 5 P 2+3 27 42 11 65 .406 New Y o r k R a n g e r s P h i l a d e l p h i a F l y e r s 8 8 - 8 9 37 35 8 82 . 513 8 8 - 8 9 36 36 8 80 . 500 P 1 26 30 24 76 . 4 7 5 P 1 30 29 21 81 .506 P 2 30 26 24 84 . 5 2 5 P 2 34 23 23 91 . 569 P 3 38 28 14 90 . 563 P 3 34 24 22 90 . 563 P 1+2 28 34 18 74 . 4 6 3 P 1+2 35 26 19 89 .556 P 1+3 36 31 13 85 .531 P 1+3 36 31 13 85 .531 P 2 + 3 35 35 10 80 . 500 P 2+3 38 27 15 91 . 569 P i t t s b u r g h P e n g u i n s Q u e b e c N o r d i g u e 8 8 - 8 9 40 33 7 87 . 5 4 3 8 8 - 8 9 27 46 7 61 .381 P 1 31 31 18 80 . 500 P 1 25 34 21 71 .444 P 2 28 39 13 69 .431 P 2 23 38 19 65 .406 P 3 36 29 15 87 . 5 4 3 P 3 21 41 18 60 . 375 P 1+2 34 33 13 81 . 506 P 1+2 25 44 11 61 .381 P 1+3 39 29 12 90 . 563 P 1+3 24 39 17 65 .406 P 2+3 37 34 9 83 . 5 1 9 P 2+3 23 43 14 60 . 3 7 5 176 T a b l e A37 A r t i f i c i a l a n d R e a l S e a s o n R e c o r d s ( c o n t i n u e d ) S e a s o n W L T P t s % S e a s o n W L T P t s % S t . L o u i s i B l u e s T o r o n t o M a p l e L e a f s 8 8 - 8 9 33 35 12 78 .488 - 88 - 8 9 28 46 6 62 .388 P 1 30 30 20 80 . 500 P 1 22 36 22 66 . 413 P 2 25 29 26 76 . 4 7 5 P 2 26 37 17 69 .431 P 3 28 31 21 77 .481 P 3 23 35 22 68 . 4 2 5 P 1+2 32 34 14 78 .488 P 1+2 30 41 9 69 .431 P 1+3 31 37 12 74 . 4 6 3 P 1+3 22 43 15 59 . 369 P 2+3 30 36 14 74 . 4 6 3 P 2+3 26 41 13 65 .406 V a n c o u v e r • C a n u c k s W a s h i n g t o n C a p i t a l s 8 8 - 8 9 33 39 8 74 . 463 88 - 8 9 41 29 10 92 . 575 P 1 30 23 27 87 .544 P 1 35 19 26 96 .600 P 2 29 26 25 83 . 519 P 2 28 28 24 80 . 500 P 3 27 35 18 72 . 450 P 3 31 27 22 84 . 5 2 5 P 1+2 29 30 21 79 . 494 P 1+2 36 24 20 92 . 5 7 5 P 1+3 31 35 14 76 . 4 7 5 P 1 + 3 39 23 18 96 .600 P 2+3 28 31 21 77 .481 P 2+3 35 26 19 89 . 556 W i n n i p e g J e t s 8 8 - 8 9 26 42 12 64 .400 P 1 21 37 22 64 .400 P 2 21 35 24 66 . 4 1 3 P 3 25 33 22 72 . 450 P 1+2 26 42 12 64 . 400 P 1 + 3 25 40 15 65 .406 P 2+3 27 40 13 67 . 419 T a b l e A38 C o r r e l a t i o n s f o r O u t c o m e s B a s e d on A r t i f i c i a l S e a s o n s a s C o m p a r e d t o R e a l S e a s o n O u t c o m e s O u t c o m e s P 1 P 2 P 3 P l+2 P l + 3 P2+3 W i n s .642 . 6 7 5 .871 .921 . 887 .917 L o s s e s .731 .648 .816 . 863 . 9 2 3 .877 T i e s - . 0 3 4 .164 .066 .260 - . 3 7 7 .191 P o i n t s . 717 .724 . 8 8 3 . 926 .943 .930 S t a n d i n g s * . 569 . 6 9 3 . 849 . 846 . 916 . 891 * S p e a r m a n r a n k c o r r e l a t i o n s . R e m a i n d e r a r e P e a r s o n p r o d u c t -moment c o r r e l a i t o n s . 178 T a b l e A39 U p s e t s C o m p a r i s o n s : 3 7 - 3 8 , 4 6 - 4 7 , 8 8 - 8 9 N a t i o n a l H o c k e y L e a g u e S e a s o n s P r e - G a m e Rank S e a s o n Rank 2 2 U NU %U X I s U NU %U X 1. : s . 100% 8 8 - 8 9 NHL 312 409 4 3 . 3 274 460 37 . 3 v . 75% 8 8 - 8 9 NHL 228 325 4 1 . 2 2 .016 n . s . 200 353 36 .2 0 . 183 n . s . v . 50% 8 8 - 8 9 NHL 163 255 3 9 . 0 7 . 4 1 5 .01 142 272 34 .6 3 .024 n . s . v . 1 0 0 % 4 6 - 4 7 NHL 61 90 4 0 . 4 0 . 422 n . s . 51 103 35 . 9 0 .974 n . s . v . 1 0 0 % 3 7 - 3 8 NHL 53 96 3 5 . 6 3 .008 n . s . 47 104 31 .1 2 . 085 n . s . 50% 8 8 - 8 9 NHL 163 255 3 9 . 0 144 272 34 .6 v . 50% 4 6 - 4 7 NHL 30 48 3 8 . 5 0 .008 n . s . 28 50 35 .9 0 .048 n . s . v . 50% 3 7 - 3 8 NHL 27 61 3 0 . 7 2 . 1 4 3 n . s . 24 51 32 .0 0 .193 n . s . 100% 4 6 - 4 7 NHL 61 90 4 0 . 4 51 103 33 .1 v . 50% 4 6 - 4 7 NHL 30 48 38 . 5 0 .080 n . s . 28 50 35 .9 0 .178 n . s . v . 1 0 0 % 3 7 - 3 8 NHL 53 96 3 5 . 6 0 .742 n . s . 47 104 31 .1 0 . 139 n . s . 100% 3 7 - 3 8 NHL 53 96 3 5 . 6 47 104 31 .1 v . 50% 3 7 - 3 8 NHL 27 61 3 0 . 7 0 .591 n . s . 24 51 32 .0 0 .018 n . s . T o p 8 8 8 - 8 9 NHL 46 55 4 5 . 5 40 67 37 .4 v . 1 0 0 % 8 8 - 8 9 NHL 312 409 4 3 . 3 0 .186 n . s . 274 460 37 . 3 0 .000 n . s . v . 75% 8 8 - 8 9 NHL 228 325 4 1 . 2 0 . 653 n . s . 200 353 36 .2 0 .057 n . s . v . 50% 8 8 - 8 9 NHL 163 255 3 9 . 0 1 .451 n . s . 142 272 34 .6 0 .356 n . s . v . 1 0 0 % 3 7 - 3 8 NHL 53 96 3 5 . 6 2 . 504 n . s . 47 104 31 .1 1 . 097 n . s . v . 50% 3 7 - 3 8 NHL 27 61 3 0 . 7 4 . 382 . 0 5 24 51 32 .0 0 .560 n . s . T o p 6 8 8 - 8 9 NHL 22 28 4 4 . 0 18 35 34 .0 v . 1 0 0 % 8 8 - 8 9 NHL 312 409 4 3 . 3 0 . 010 n . s . 274 460 37 .3 0 .240 n . s . v . 75% 8 8 - 8 9 NHL 228 325 4 1 . 2 0 . 1 4 5 n . s . 200 353 36 .2 0 .101 n . s . 50% 8 8 - 8 9 NHL 163 255 3 9 . 0 144 272 34 .6 v . 1 0 0 % 4 6 - 4 7 NHL 61 90 4 0 . 4 0 .201 n . s . 51 103 33 .1 0 . 013 n . s . v . 50% 4 6 - 4 7 NHL 30 48 38 . 5 0 . 387 n . s . 28 50 35 .9 0 .052 n . s . 50% 4 6 - 4 7 NHL 30 48 3 8 . 5 28 50 35 . 9 v . 50% 3 7 - 3 8 NHL 27 61 3 0 . 7 1 .110 n . s . 24 51 3 2 . 0 0 . 2 5 9 n . s . 179 T a b l e A40 L e a g u e C h i - S q u a r e C o m p a r i s o n s o f U p s e t s B a s e d on P r e - G a m e v s . S e a s o n Rank P o s i t i o n P r e - G a m e U ' s N U ' s % U ' s S e a s o n U ' s N U ' s % U ' s X l . s . NHL 100% 8 8 - 8 9 312 75% 8 8 - 8 9 228 50% 8 8 - 8 9 163 100% 4 6 - 4 7 61 50% 4 5 - 4 7 30 100% 3 7 - 3 8 53 50% 3 7 - 3 8 27 T o p 8 8 8 - 8 9 46 T o p 6 8 8 - 8 9 22 1 D i v S o c c e r 8 7 - 8 8 100% 3 p t 50% 3 p t 100% 2 p t 50% 2 p t NBA 104 46 79 46 409 325 255 90 48 96 61 55 28 171 136 124 85 4 3 . 3 4 1 . 2 3 9 . 0 4 0 . 4 3 8 . 5 3 5 . 6 3 0 . 7 4 5 . 5 4 4 . 0 37 . 8 2 5 . 2 3 8 . 9 3 5 . 1 274 460 200 353 144 272 51 103 28 50 47 104 24 40 18 51 67 35 81 198 52 105 72 194 31 105 3 7 . 3 36 .2 3 4 . 6 3 3 . 1 3 5 . 9 3 1 . 1 3 2 . 0 3 7 . 4 3 4 . 0 2 9 . 0 3 3 . 1 2 7 . 1 2 2 . 8 5 . 3 4 2 2 . 9 8 8 1 . 7 1 9 1 . 7 3 9 0 . 1 1 0 0 . 6 6 7 0 . 0 3 3 1 .427 1 .091 4 . 8 0 6 2 . 5 2 5 7 . 4 0 4 4 . 9 3 5 .05 n . s . n . s , n . s , n . s , n . s , n . s . n . s . n . s , . 05 n . s , .01 . 05 50% 8 8 - 8 9 177 351 3 3 . 5 165 363 3 1 . 3 0 . 6 2 3 n . s 180 T a b l e A41 C r o s s S p o r t C h i - S q u a r e C o m p a r i s o n s Pregame S e a s o n 2 2 Us NUs % Us X l . s . Us NUs % Us X l . s . 100% 8 8 - 8 9 NHL 312 409 4 3 . 3 274 460 3 7 . 3 100% S o c 2 p t 79 124 3 8 . 9 1 . 2 3 2 n s 76 194 2 8 . 1 7 . 3 2 8 .01 100% S o c 3 p t 104 171 3 7 . 8 2 . 4 3 6 n s 81 198 2 9 . 0 6 . 1 1 2 .02 50% NBA 177 351 3 3 . 5 1 2 . 1 6 4 .001 165 363 3 1 . 3 5 . 0 0 4 . 0 5 75% 8 8 - 8 9 NHL 228 325 4 1 . 2 200 353 3 6 . 2 50% NBA 177 351 3 3 . 5 6 . 8 4 8 .02 165 363 3 1 . 3 2 . 9 1 9 ns 100% S o c 2 p t 79 124 3 8 . 9 0 . 3 3 0 ns 76 194 2 8 . 1 5 . 2 3 3 . 0 5 100% S o c 3 p t 104 171 3 7 . 8 0 . 8 9 0 n s 81 198 2 9 . 0 4 . 2 2 0 .05 50% S o c 2 p t 46 85 3 5 . 1 1 . 6 4 9 n s 31 136 2 2 . 8 1 8 . 2 4 1 .001 50% S o c 3 p t 46 136 3 3 . 8 1 4 . 9 0 8 .001 52 105 3 3 . 1 0 . 4 9 5 n s 50% 8 8 - 8 9 NHL 163 255 3 9 . 0 144 272 3 4 . 6 50% S o c 2 p t 46 85 3 5 . 1 0 . 6 3 7 ns 31 136 2 2 . 8 6 . 6 1 5 .01 50% S o c 3 p t 46 136 2 5 . 2 1 0 . 5 1 5 .01 52 105 3 3 . 1 0 . 0 7 0 ns 50% NBA 177 351 3 3 . 5 3 . 0 3 5 ns 165 363 3 1 . 3 0 . 0 9 2 n s 100% 4 6 - 4 7 NHL 61 90 4 0 . 4 51 103 3 3 . 1 100% S o c 2 p t 79 124 3 8 . 9 0 . 0 7 9 n s 76 194 2 8 . 1 1 .154 n s 100% S o c 3 p t 104 171 3 7 . 8 0 . 2 7 3 n s 81 198 2 9 . 0 0 . 7 8 1 ns 50% NBA 177 351 3 3 . 5 2 . 4 3 8 ns 165 363 3 1 . 3 0 . 1 9 2 n s 50% 4 6 - 4 7 NHL 30 48 3 8 . 5 28 50 3 5 . 9 50% S o c 2 p t 46 85 3 5 . 1 0 . 2 3 7 ns 31 105 2 2 . 8 4 . 2 6 2 . 0 5 50% S o c 3 p t 46 136 2 5 . 2 0 . 4 6 6 ns 52 105 3 3 . 1 0 . 1 7 9 n s 50% NBA 177 351 3 3 . 5 0 . 7 3 7 n s 165 363 3 1 . 3 0 . 6 7 6 n s 181' T a b l e A41 C r o s s S p o r t C h i S q u a r e C o m p a r i s o n s ( c o n t i n u e d ) 2 2 Us NUs % Us X l . s . Us NUs % Us X l . s . 100% 3 7 - 3 8 NHL 53 96 3 5 . 6 47 104 3 1 . 1 100% S o c 2 p t 79 124 3 8 . 9 0 . 4 1 0 ns 76 194 2 8 . 1 0 . 4 1 5 n s 100% S o c 3 p t 104 171 3 7 . 8 0 . 2 0 9 n s 81 198 2 9 . 0 0 . 2 0 5 n s 50% NBA 177 351 3 3 . 5 0 . 2 1 7 n s 165 363 3 1 . 3 0 . 0 0 1 ns 50% 3 7 - 3 8 NHL 27 61 3 0 . 7 24 51 3 2 . 0 50% S o c 2 p t 46 85 3 5 . 1 0 . 4 6 5 ns 31 105 2 2 . 8 2 . 1 2 6 n s 50% S o c 3 p t 46 136 2 5 . 2 0 . 8 7 9 n s 52 105 3 3 . 1 0 . 0 2 9 n s 50% NBA 177 351 3 3 . 5 0 . 2 7 5 ns 165 363 3 1 . 3 0 . 3 1 2 n s 50% NBA 177 351 3 3 . 5 165 363 3 1 . 3 100% S o c 2 p t 79 124 3 8 . 9 1 . 8 7 4 n s 76 194 2 8 . 1 0 . 8 1 5 ns 100% S o c 3 p t 104 171 3 7 . 8 1 . 4 6 7 ns 81 198 2 9 . 0 0 . 4 2 4 n s 50% S o c 2 p t 46 85 3 5 . 1 0 . 1 1 9 n s 31 105 2 2 . 8 3 . 7 1 7 ns 50% S o c 3 p t 46 136 2 5 . 2 4 . 2 7 4 .05 52 105 3 3 . 1 0 . 1 9 6 n s 182 T a b l e A42 I n d i v i d u a l Team Means a n d S t a n d a r d D e v i a t i o n s F o r , A g a i n s t a n d T o t a l G o a l s a n d F o r , A g a i n s t a n d T o t a l S h o t s P e r Game F o r A g a i n s t T o t a l X s d X s d X s d B o s G o a l s 3 . 6 7 5 2 . 1 2 7 3 . 1 3 8 1 . 5 8 9 6 . 8 1 3 2 . 5 8 6 S h o t s 3 1 . 3 7 5 7 . 3 5 9 2 6 . 7 6 3 6 . 2 2 6 5 8 . 1 3 8 8 . 9 1 4 B u f G o a l s 3 . 6 3 8 1 . 8 4 4 3 . 7 0 0 2 . 0 4 0 7 . 3 3 8 2 . 4 7 0 S h o t s 2 7 . 9 0 0 6 . 7 4 2 2 9 . 7 5 0 6 . 8 8 3 5 7 . 6 5 0 8 . 5 7 7 C a l G o a l s 4 . 4 2 5 2 . 2 9 3 2 . 8 3 8 1 .522 7 . 2 6 2 2 . 8 3 6 S h o t s 3 4 . 6 3 8 7 . 6 5 6 2 5 . 7 3 7 6 . 0 1 6 6 0 . 3 8 0 8 . 9 9 0 C h i G o a l s 3 . 6 8 7 1 . 8 1 1 4 . 2 1 3 2 . 0 1 7 7 . 9 0 0 2 . 6 1 7 S h o t s 3 2 . 7 1 3 7 . 5 9 4 3 1 . 5 3 8 7 . 7 8 7 6 4 . 2 5 0 9 . 3 3 0 D e t G o a l s 3 . 9 1 2 1 . 7 8 0 3 . 9 6 3 1 .958 7 . 8 7 5 2 . 5 8 2 S h o t s 2 8 . 8 7 5 6 . 8 7 9 3 1 . 1 8 7 7 . 4 9 2 6 0 . 0 0 0 9 . 4 1 0 Edm G o a l s 4 . 0 6 2 2 . 1 7 2 3 . 8 0 0 1 . 9 7 7 7 . 8 6 3 2 . 9 8 4 S h o t s 2 7 . 8 2 5 6 . 5 5 0 3 0 . 4 1 3 6 . 2 9 0 5 8 . 2 3 7 7 . 6 3 1 H a r G o a l s 3 . 7 2 5 2 . 0 6 8 3 . 1 6 2 2 . 1 2 6 7 . 3 3 8 2 . 9 0 8 S h o t s 3 1 . 0 5 0 7 . 5 2 5 2 8 . 9 2 5 6 . 0 3 5 5 9 . 9 7 0 9 . 3 2 0 LA G o a l s 4 . 7 0 0 2 . 1 7 8 4 . 2 1 3 1 . 9 9 2 8 . 9 1 3 2 . 8 3 4 S h o t s 3 1 . 7 5 0 7 . 0 4 1 3 3 . 4 7 5 6 . 0 4 2 6 5 . 2 2 5 8 . 4 7 5 M i n G o a l s 3 . 2 1 3 1 . 7 9 1 3 . 4 8 7 1 . 8 6 9 6 . 7 0 0 2 . 6 6 4 S h o t s 3 1 . 4 3 7 7 . 5 2 9 3 2 . 4 5 0 6 . 4 2 2 6 3 . 8 8 8 8 . 6 8 9 Mon G o a l s 3 . 9 5 0 1 .848 2 . 7 1 3 1 .624 6 . 6 6 2 2 . 6 1 9 S h o t s 3 0 . 8 5 0 5 . 7 1 5 2 6 . 6 6 3 5 . 5 7 1 5 7 . 5 1 3 6 . 9 1 0 N J G o a l s 3 . 5 2 5 1 . 8 4 9 4 . 0 6 2 2 . 0 2 1 7 . 5 8 8 2 . 9 2 4 S h o t s 3 1 . 2 0 0 6 . 2 3 5 2 9 . 5 4 4 6 . 5 4 4 6 0 . 9 3 7 7 . 9 5 9 NYI G o a l s 3 . 3 2 5 1 . 8 4 7 4 . 0 5 0 1 .698 7 . 3 7 5 2 . 5 7 2 S h o t s 3 0 . 2 7 5 6 . 4 7 5 3 0 . 8 1 3 6 . 4 3 2 6 1 . 0 8 8 7 . 2 6 4 NYR G o a l s 3 . 8 7 5 1 . 9 1 2 3 . 8 3 8 2 . 0 4 0 7 . 7 1 3 2 . 8 2 9 S h o t s 3 3 . 9 5 0 7 . 0 9 2 3 1 . 2 0 0 7 . 0 9 9 6 5 . 1 5 0 9 . 5 5 0 P h i G o a l s 3 . 8 5 0 1 . 7 5 1 3 . 5 6 3 1 . 8 7 5 7 . 5 1 2 2 . 5 8 4 S h o t s 2 9 . 9 3 7 6 . 5 3 0 3 0 . 5 3 8 6 . 8 6 4 6 0 . 4 7 5 8 . 8 0 0 P i t G o a l s 4 . 3 3 8 2 . 1 0 5 4 . 3 6 3 2 . 2 2 3 8 . 7 0 0 2 . 7 1 1 S h o t s 2 8 . 5 1 3 6 . 5 5 5 3 5 . 8 2 5 7 . 4 1 0 6 4 . 3 3 8 8 . 5 7 6 183 T a b l e A42 I n d i v i d u a l F o r , A g a i n s t a n d Team Means and S t a n d a r d D e v i a t i o n s T o t a l G o a l s and F o r , A g a i n s t a n d T o t a l ( c o n t i n u e d ) S h o t s F o r A g a i n s t T o t a l X s d X s d X s d Que G o a l s S h o t s 3 . 3 6 2 2 8 . 4 5 0 1 7 884 289 4 . 2 3 8 3 2 . 5 0 0 1 . 9 5 7 6 . 7 0 7 7 . 6 0 0 6 0 . 9 5 0 2 8 309 172 S t L G o a l s S h o t s 3 . 4 3 7 2 8 . 8 6 2 1. 6 785 092 3 . 5 6 3 2 8 . 2 7 5 1 . 9 6 7 6 . 547 7 . 0 0 0 5 7 . 1 3 8 2 8 619 422 T o r G o a l s S h o t s 3 . 2 3 7 2 7 . 0 8 9 2. 5, 020 938 4 . 2 7 5 3 4 . 1 2 7 2 7. 256 235 7 . 5 1 2 6 1 . 2 1 5 2 8 968 882 V a n G o a l s S h o t s 3 . 1 1 2 3 0 . 5 1 3 1 7 779 135 3 . 1 6 2 2 8 . 2 1 2 642 ,103 6 . 2 7 5 5 8 . 7 2 0 2 9 600 980 Was G o a l s S h o t s 3 . 812 3 1 . 1 7 5 1 6 910 207 3 . 2 2 7 2 7 . 5 5 0 1 .817 7 . 0 9 9 7 . 050 5 8 . 7 2 5 2 8 815 392 Win G o a l s S h o t s 3 . 7 5 0 2 9 . 5 1 9 2 6, 072 867 4 . 4 3 8 3 2 . 2 1 5 2 .116 6 . 1 6 2 8 . 1 8 8 6 1 . 7 3 4 3 8 , 011 847 184 REFERENCES A b e l s o n , R. P . ( 1 9 8 9 ) . A b a y e s b a l l p r o b l e m . C h a n c e , 2, 4 1 - 4 3 . A n a s t a s i , A . ( 1 9 5 4 ) . P s y c h o l o g i c a l t e s t i n g . New Y o r k : M a c m i l l a n . B o x , G . E . P . & J e n k i n s , G . M. ( 1 9 7 6 ) . R e v . e d . T i m e s e r i e s  a n a l y s i s f o r e c a s t i n g a n d c o n t r o l . San F r a n c i s c o : H o l d e n - D a y . C i n l a r , E . ( 1 9 7 5 ) . I n t r o d u c t i o n t o s t o c h a s t i c p r o c e s s e s . E n g l e w o o d C l i f f s , N . J . : P r e n t i c e - H a l l . C o o k , E . ( 1 9 7 7 ) . An a n a l y s i s o f b a s e b a l l a s a game o f c h a n c e by t h e M o n t e C a r l o m e t h o d . I n S . P . L a d a n y & R. E . M a c h o l ( E d . ) , S t u d i e s i n management s c i e n c e and s y s t e m s : v o l 5 o p t i m a l  s t r a t e g i e s i n s p o r t s ( p p . 5 0 - 5 4 ) . A m s t e r d a m : N o r t h - H o l l a n d . C o o k , E . ( 1 9 6 4 ) . P e r c e n t a g e b a s e b a l l . C a m b r i d g e , M a s s . : T h e M . I . T . P r e s s . C r u z e , E . & W e l d o n , J . ( 1 9 8 9 ) . M i n i t a b R e f e r e n c e M a n u a l :  R e l e a s e 1_. S t a t e C o l l e g e , P a . : M i n i t a b I n c . D i a m o n d , D. & S t u b b s , L . (1987) . H o c k e y : t w e n t y y e a r s . T o r o n t o : D o u b l e d a y . D i c k e y , G . ( 1 9 8 2 ) . T h e h i s t o r y o f p r o f e s s i o n a l b a s k e t b a l l  ( s i n c e 1896) . New Y o r k : S t e i n & D a y . E f r o n , B . & M o r r i s , C . ( 1 9 7 7 ) . S t e i n ' s p a r a d o x i n s t a t i s t i c s . S c i e n t i f i c A m e r i c a n , 2 3 6 , n o . 5 ( M a y ) , 1 1 9 - 1 2 7 . E l d e r t o n , W. P . ( 1 9 1 5 ) . C r i c k e t s c o r e s and some skew c o r r e l a t i o n d i s t r i b u t i o n s . J . R. S t a t i s t . S o c . A . , 1 0 8 , 1 - 1 1 . E l d e r t o n , W. P . ( 1 9 2 7 ) . 2nd e d . F r e q u e n c y c u r v e s and  c o r r e l a t i o n . L o n d o n : C h a r l e s & E d w i n L a y t o n . E l d e r t o n , W. P . & E l d e r t o n , E . M. ( 1 9 0 9 ) . P r i m e r o f s t a t i s t i c s . L o n d o n : B l a c k . E s k e n a z i , G . ( 1 9 7 2 ) . A t h i n k i n g m a n ' s g u i d e t o p r o h o c k e y . New Y o r k : D u t t o n . E s k e n a z i , G . ( 1 9 6 9 ) . H o c k e y . T o r o n t o : R u t l e d g e ( R y e r s o n ) . F e l l e r , W. ( 1 9 4 3 ) . An i n t r o d u c t i o n t o p r o b a b i l i t y t h e o r y and  i t s a p p l i c a t i o n s . V o l . 1 . New Y o r k : W i l e y & S o n s . F e l l e r , W. ( 1 9 5 0 ) . 2nd e d . An i n t r o d u c t i o n t o p r o b a b i l i t y t h e o r y and i t s a p p l i c a t i o n s . V o l . 1 . New Y o r k : W i l e y & S o n s . 185 REFERENCES ( c o n t i n u e d ) F e l l e r , W. (1968) . An i n t r o d u c t i o n t o p r o b a b i l i t y t h e o r y and  i t s a p p l i c a t i o n s . V o l . 1 1 . New Y o r k : W i l e y & S o n s . F i l l i b e n , J . J . ( 1 9 7 5 ) . T h e p r o b a b i l i t y p l o t c o r r e l a t i o n c o e f f i c i e n t t e s t f o r n o r m a l i t y . T e c h n o m e t r i c s , 1 7 , 1 1 1 . F i s c h l e r , S . & F i s c h l e r , S . ( 1 9 8 6 ) . B r e a k a w a y ' 8 6 : t h e h o c k e y  a l m a n a c . T o r o n t o : T o t e m . F i s k e , D. W. & J o n e s , L . V . ( 1 9 5 4 ) . S e q u e n t i a l a n a l y s i s i n p s y c h o l o g i c a l r e s e a r c h . P s y c h o . B u l l . , 5 1 , 2 6 4 - 2 7 5 . F r e e z e , R. A . ( 1 9 7 7 ) . M o n t e C a r l o a n a l y s i s o f b a s e b a l l b a t t i n g o r d e r . I n S . P . L a d a n y & R. E . M a c h o l ( E d . ) , S t u d i e s i n management s c i e n c e a n d s y s t e m s : v o l . 5 . o p t i m a l s t r a t e g i e s i n s p o r t s . A m s t e r d a m : N o r t h - H o l l a n d , p p . 6 3 - 6 7 . G i l o v i c h , T . , V a l l o n e , R. & T v e r s k y , A . ( 1 9 8 5 ) . T h e h o t h a n d i n b a s k e t b a l l : on t h e m i s p e r c e p t i o n o f r a n d o m s e q u e n c e s . C o g n i t i v e P s y c h o l o g y , 1 7 , 2 9 5 - 3 1 4 . G o o d m a n , M. L . ( 1 9 6 9 ) . On t h e i n c i d e n c e o f s w e p t d o u b l e - h e a d e r s . A m e r i c a n S t a t i s t i c i a n , v o l ? , 1 5 - 1 7 . G o u l d , S . J . ( 1 9 8 9 ) . T h e s t r e a k o f s t r e a k s . C h a n c e , 2 , 1 0 -1 6 . H a b e r m a n , S . J . ( 1 9 7 7 ) . A n a l y s i s o f s c o r e s o f I v y L e a g u e f o o t b a l l g a m e s . I n S . P . L a d a n y & R. E . M a c h o l ( E d . ) . S t u d i e s i n management s c i e n c e and s y s t e m s : v o l . 5 . O p t i m a l  s t r a t e g i e s i n s p o r t s . A m s t e r d a m : N o r t h - H o l l a n d , p p . 1 0 6 - 1 0 8 , K l e i n & R e i f ( 1 9 8 7 ) . T h e K l e i n a n d R e i f h o c k e y c o m p e n d i u m . T o r o n t o : M c C l e l l a n d & S t e w a r t . L a p i n , L . L . ( 1 9 8 4 ) . B u s i n e s s s t a t i s t i c s . New Y o r k : H a r c o u r t B r a c e J o v a n o v i c h . L a r k e y , P . D . , S m i t h , R. A . , & K a d a n e , J . B . ( 1 9 8 9 ) . I t ' s o k a y t o b e l i e v e i n t h e ' h o t h a n d . ' C h a n c e , 2_, 2 2 - 3 0 . L i n g r e n , B . W. ( 1 9 6 8 ) . S t a t i s t i c a l t h e o r y . 2 n d . e d . New Y o r k : M a c m i l l a n . L i n d s e y , G . R. ( 1 9 6 1 ) . T h e p r o g r e s s o f t h e s c o r e d u r i n g a g a m e . J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , 5 6 , 7 0 3 - 7 2 8 . 186 R E F E R E N C E S ( c o n t i n u e d ) M c F a r l a n e , B . (1967) . 5_0 y e a r s o f h o c k e y : a h i s t o r y o f t h e  N a t i o n a l H o c k e y L e a g u e . T o r o n t o : P a g u r i a n . M o s t e l l e r , F . ( 1 9 7 0 ) . C o l l e g i a t e f o o t b a l l s c o r e s , U . S . A . J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n , 6 5 , 3 5 - 4 8 . M u l l e t , G . M. ( 1 9 7 7 ) . S i m e o n P o i s s o n and t h e N a t i o n a l H o c k e y L e a g u e . The A m e r i c a n S t a t i s t i c i a n , 3 1 , 1 8 - 1 2 . N e l s o n , C . R. ( 1 9 7 3 ) . A p p l i e d t i m e s e r i e s a n a l y s i s f o r  m a n a g e r i a l f o r e c a s t i n g . S a n F r a n c i s c o : H o l d e n - D a y . N e m e c , D. ( 1 9 8 7 ) . G r e a t b a s e b a l l f e a t s , f a c t s a n d f i r s t s . New Y o r k : New A m e r i c a n L i b r a r y . O l i n i c k , M. ( 1 9 7 8 ) . An i n t r o d u c t i o n t o m a t h e m a t i c a l m o d e l s i n  t h e s o c i a l and l i f e s c i e n c e s . R e a d i n g , M a s s : A d d i s o n - W e s l e y . P e t e r s o n , A . V . J r . ( 1 9 7 7 ) . C o m p a r i n g t h e r u n - s c o r i n g a b i l i t i e s o f two d i f f e r e n t b a t t i n g o r d e r s : r e s u l t s o f a s i m u l a t i o n . I n S . P . L a d a n y & R. E . M a c h o l ( E d . ) . S t u d i e s i n management  s c i e n c e a n d s y s t e m s : v o l . 5 . o p t i m a l s t r a t e g i e s i n s p o r t s . A m s t e r d a m : N o r t h - H o l l a n d , p p . 3 6 - 3 8 . P o l l a r d , R. ( 1 9 7 3 ) . C o l l e g i a t e f o o t b a l l s c o r e s a n d t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n . J o u r n a l o f t h e A m e r i c a n S t a t i s t i c a l  A s s o c i a t i o n , 6 8 , 3 5 1 - 3 5 2 . P o l l a r d , R. ( 1 9 8 5 ) . G o a l - s c o r i n g and t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n . M a t h . G a z . , 6 9 , 4 5 - 4 7 . P o l l a r d , R. ( 1 9 8 6 ) . T h e a p p l i c a t i o n o f t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n t o s p o r t . R e s e a r c h B i - a n n u a l f o r M o v e m e n t , 4 , 3 5 2 - 3 6 1 . R e e p , C , P o l l a r d , R . , & B e n j a m i n , B . ( 1 9 7 1 ) . S k i l l a n d c h a n c e i n b a l l g a m e s . J o u r n a l o f t h e R o y a l S t a t i s t i c a l S o c i e t y , 1 3 4 , 6 2 3 - 6 2 9 . R o t h s c h i l d , V . & L o g o t h e t i s , N . ( 1 9 8 6 ) . P r o b a b i l i t y  d i s t r i b u t i o n s . New Y o r k : W i l e y & S o n s . R o l l i n , J . ( e d . ) . ( 1 9 8 8 ) . Ro thmans f o o t b a l l y e a r b o o k 1 9 8 8 - 8 9 . L o n d o n : Q u e e n Anne P r e s s . S c h m i d t , M. J . ( 1 9 7 9 ) . (2nd e d . ) . U n d e r s t a n d i n g a n d u s i n g  s t a t i s t i c s . L e x i n g t o n , M a s s . : D. C . H e a t h . 187 REFERENCES ( c o n t i n u e d ) S c h u t z , R. W. ( 1 9 8 2 ) . An i n t r o u c t i o n t o m a t h e m a t i c a l m o d e l i n g . P a p e r g i v e n a t c o n f e r e n c e a t U n i v e r s i t e L a v a l , F e b . 1 2 , 1 9 8 2 . S c h u t z , R. W. ( 1 9 7 0 ) . S t o c h a s t i c p r o c e s s e s : t h e i r n a t u r e a n d u s e i n t h e s t u d y o f s p o r t a n d p h y s i c a l a c t i v i t y . R e s e a r c h  Q u a r t e r l y , 4 1 , 2 0 5 - 2 1 2 . S c h u t z , R. W. & J a n s s e n , C . T . L . ( 1 9 8 7 , F e b . ) P r o b a b i l i t y a n d  d e c i s i o n m a k i n g i n s p o r t : o n t h e c h a n g e o f c o a c h d e c i s i o n i n  h o c k e y . P a p e r p r e s e n t e d a t t h e m e e t i n g o f t h e A m e r i c a n A s s o c i a t i o n f o r H e a l t h , P h y s i c a l E d u c a t i o n , R e c r e a t i o n a n d D a n c e , A p r i l 1 3 - 1 7 , 1 9 8 7 . L a s V e g a s , N e v . S h a p i r o , S . S . & W i l k , M. B . ( 1 9 6 5 ) . An a n a l y s i s o f v a r i a n c e t e s t f o r n o r m a l i t y ( c o m p l e t e s a m p l e s ) . B i o m e t r i k a , 5 2 , 5 9 1 . S h o r t , T . & W a s s e r m a n , L . ( 1 9 8 9 ) . S h o u l d we be s u r p r i s e d by t h e s t r e a k o f s t r e a k s ? C h a n c e , 2 , 1 3 . S i m o n , W. ( 1 9 7 7 ) . " B a c k - t o - t h e - w a l l e f f e c t " : 1976 p e r s p e c t i v e . I n S . P . L a d a n y & R. E . M a c h o l ( E d . ) . S t u d i e s i n management  s c i e n c e and s y s t e m s : v o l . 5 . o p t i m a l s t r a t e g i e s i n s p o r t s . A m s t e r d a m : N o r t h - H o l l a n d , p p . 4 6 - 4 7 . S m i t h , J . H . ( 1 9 7 7 ) . A d j u s t i n g b a s e b a l l s t a n d i n g s f o r s t r e n g t h o f teams p l a y e d . In S . P . L a d a n y & R. E . M a c h o l ( E d . ) . S t u d i e s i n management s c i e n c e and s y s t e m s : v o l . 5 . o p t i m a l  s t r a t e g i e s i n s p o r t s . A m s t e r d a m : N o r t h - H o l l a n d , p p . 4 8 - 4 9 . S n e d e c o r , G . W. & C o c h r a n , W. G . ( 1 9 6 7 ) . S t a t i s t i c a l m e t h o d s . A m e s , I o w a : Iowa S t a t e U n i v . P r e s s . T h o r n & P a l m e r , P . ( 1 9 8 9 ) . T o t a l B a s e b a l l . New Y o r k , N . Y . : W a r n e r . T v e r s k y , A . & G i l o v i c h , T . ( 1 9 8 9 ) . T h e c o l d f a c t s a b o u t t h e ' h o t h a n d ' i n b a s k e t b a l l . C h a n c e , 2 , 1 6 - 2 1 . T v e r s k y , A . & K a h n e m a n , D. ( 1 9 7 3 ) . A v a i l a b i l i t y : a h e u r i s t i c f o r j u d g i n g f r e q u e n c y a n d p r o b a b i l i t y . C o g n i t i v e P s y c h o l o g y , 5 , 2 0 7 - 2 3 2 . Wood, G . H . ( 1 9 4 5 ) . C r i c k e t s c o r e s and g e o m e t r i c a l p r o g r e s s i o n . J . R. S t a t i s t . S o c . A . , 1 0 8 , 1 2 - 2 2 . Z a r , J . H . ( 1 9 8 4 ) . 2nd e d . B i o s t a t i s t i c a l A n a l y s i s . E n g l e w o o d C l i f f s , N . J . : P r e n t i c e - H a l l . T h e B e l l i n g h a m H e r a l d  T h e H o c k e y News T h e V a n c o u v e r Sun 

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