An Empirical Study Comparing Various Methods Of Achievement Expectancy Bymeans Of Mental Ability

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Content AN EMPIRICAL STUDY COMPARING VARIOUS METHODS OF ACHIEVEMENT EXPECTANCY BY MEANS OF MENTAL ABILITY b y J e r r y C r a ig G a r lo c k A D i s s e r t a t i o n P r e s e n t e d t o th e FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA 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 ire m e n ts f o r t h e D eg ree DOCTOR OF PHILOSOPHY ( E d u c a tio n ) J u n e 1 9 5 9 UNIVERSITY OF SOUTHERN CALIFORNIA GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES 7. CALIFORNIA This dissertation, written by J e r r j C r a i g ^ r l o c k under the direction of h%&~.Dissertation C om mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of D O C T O R O F P H I L O S O P H Y Dean Date.. . T U t . ' E . 1 ^ 9 DISSERTATION COMMITTEE X. , ^ / , / ,T Chairman CvIJlLLu. 4 J - . '7 ( y A c k n o w le d g e m e n ts a r e due t o th e f o l l o w i n g f o r m ak in g p o s s i b l e t h e a v a i l a b i l i t y o f d a t a a n d i n f o r m a t i o n a l a n d m o n e ta r y a s s i s t a n c e : L os A n g e le s C o u n ty S u p e r i n t e n d e n t o f S c h o o ls O f fic e ,, D i v i s i o n o f R e s e a r c h a n d G u id a n c e ; C a l i f o r n i a T e s t B u r e a u ; a n d v a r i o u s s c h o o l d i s t r i c t s t h r o u g h o u t L os A n g e le s C o u n ty . TABLE O F CONTENTS CHAPTER PAGE I . THE PROBLEM.............................................................................. 1 I n t r o d u c t i o n ..................................................................... 2 The M e n ta l Age ..................................................... 2 The H orn F o r m u l a ..................................................... 3 E s t i m a t e d A c h ie v e m e n t b a s e d on t h e b a s i c r e g r e s s i o n e q u a t i o n . . . . . . . . . 4 The t r u e d i f f e r e n c e f o r m u la ........................... 4 The A n t i c i p a t e d A c h ie v e m e n t ........................... 5 L o c a l norm s . . . . . . . . • 5 p r e d i c t i o n a n d e x p e c t a n c y ............................ • 5 The p ro b le m 8 S p e c i f i c q u e s t i o n s t o be a n s w e r e d . . . 9 C r i t e r i a . . . . . 9 H y p o t h e s e s ..................................................................... 10 The s t u d y . . . . . . . . ................................ 10 Im p o rta n c e o f t h e s t u d y ...................................... 10 Need f o r t h e s t u d y .................................... 11 D e l i m i t a t i o n s o f th e s t u d y ........................... 11 O r g a n i z a t i o n o f th e s t u d y ...................................... 12 I I . REVIEW OF THE L IT E R A T U R E ............................................ 14 l v CHAPTER PAGE L i t e r a t u r e i n d i c a t i n g t h e v a r i a b l e s r e l a t e d t o i n t e l l i g e n c e a n d a c h ie v e m e n t . • • • 14 L i t e r a t u r e r e p o r t i n g t h e v a r i a b l e s u s e d i n p r e d i c t i n g a c h ie v e m e n t . . .. . •• 19 L i t e r a t u r e r e l a t i n g th e m e n ta l a g e a s a p r e d i c t o r • . , . • ........................................... • 24 L i t e r a t u r e r e p o r t i n g d i f f e r e n t i a l m e n ta l t e s t s a s p r e d i c t o r s ....................... . • • 26 L i t e r a t u r e r e l a t i n g t o th e " j a n g l e f a l l a c y " a n d r e l a t e d c o n t r o v e r s y ..................................... 27 L i t e r a t u r e i l l u s t r a t i n g th e e f f e c t s o f c h r o n o l o g i c a l a g e on m e n ta l a b i l i t y a n d a c h i e v e m e n t .......................................................... 31 L i t e r a t u r e r e p o r t i n g th e u s e o f t h e IQ, o v e r a p e r i o d o f t i m e ........................................... 34 L i t e r a t u r e d e s c r i b i n g d i f f e r e n c e s i n a b i l i t y a n d a c h ie v e m e n t b e tw e e n p o p u l a t i o n s ..................................................................... 37 L i t e r a t u r e r e p o r t i n g th e v a r i o u s m e a s u re s o f e x p e c ta n c y u s e d i n t h e s t u d y . . . . 39 The M A ............................................................................... 39 The H o rn F o r m u l a .................................... 41 CHAPTER PAGE The L o rd G row th F o rm u la • • • . . . • • 42 The McNemar e q u a t i o n ......................................... . 43 S u m m a r y .............................................. 44 I I I . EQUATIONS AND DERIVATION OF EQUATIONS . . . 46 D e r i v a t i o n o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a s a f u n c t i o n o f a b i l i t y a n d a c h ie v e m e n t ..................... 46 D e r i v a t i o n o f t h e e q u a t i o n o f t h e r e l i a b i l i t y f o r th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a s a f u n c t i o n o f a b i l i t y a n d a c h ie v e m e n t ..................................... 55 D e r i v a t i o n o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a s a f u n c t i o n o f a b i l i t y , a c h i e v e m e n t, a n d c h r o n o l o g i c a l a g e ........... 56 D e r i v a t i o n o f th e r e l i a b i l i t y e q u a t i o n f o r th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a s a f u n c t i o n o f a b i l i t y , a c h ie v e m e n t, a n d c h r o n o l o g i c a l a g e ........................... 61 D e r i v a t i o n o f th e r e l i a b i l i t y o f th e H orn F o rm u la • • • • . 62 Vi CHAPTER PAGE D e r i v a t i o n o f t h e r e l i a b i l i t y a n d e r r o r o f m e a su re m e n t o f t h e d i f f e r e n c e b e tw e e n a c u t a l a c h ie v e m e n t a n d p r e d i c e d a c h ie v e m e n t b a s e d on t h e u s u a l tw o - v a r i a b l e r e g r e s s i o n e q u a t i o n .................... 67 D e r i v a t i o n o f th e r e l i a b i l i t y a n d e r r o r o f m e a su re m e n t f o r t h e d i f f e r e n c e b e tw e e n a c h ie v e m e n t t e s t s c o r e s a n d i n t e l l i g e n c e t e s t s c o r e s ................................ 73 Summary ......................... 73 IV . PR O C E D U R E ............................................................................... 79 p o p u l a t i o n u s e d i n t h e s t u d y . . . . . 80 M e a s u rin g I n s t r u m e n t s u t i l i z e d a n d th e n a t u r e o f t h e g ra d e p la c e m e n t s c o r e s 80 N a tu re o f th e p o p u l a t i o n s ........................... 81 E s t i m a t e d d i f f e r e n c e e q u a t i o n s a c c o r d i n g t o d i s t r i c t s ..................................... 83 V a r io u s e x p e c t a n c i e s a p p l i e d t o e a c h p u p i l ..................................... 37 C o m p a riso n o f e x p e c ta n c y e q u a t i o n s • . 87 E x p e c t a n c i e s a s c a l c u l a t e d fro m r e g r e s s i o n e q u a t i o n s f o r d i f f e r e n t a b i l i t y g r o u p s ..................................................... 88 v i l CHAPTER PAGE The A n t i c i p a t e d A c h ie v e m e n t • . • • • 88 The c r i t e r i o n o f r e l i a b i l i t y . . . » 89 The c r i t e r i o n e q u a t i o n f o r e s t i m a t e d d i f f e r e n c e e q u a t i o n s ................................ 90 The c r i t e r i o n e q u a t i o n f o r e x p e c ta n c y e q u a t i o n s .......................................................... 90 S lo p e s a n d i n t e r c e p t s u s e d a s a m eth o d o f c o m p a ris o n • • • • • • • « • • • 91 S u m m a r y .................................................... 91 V. EQUATIONS OF ESTIMATED DIFFERENCES, THEIR SLOPES AND INTERCEPTS, AND THEIR R E L IA B IL IT IE S ......................................... 9 * D i f f e r e n c e s o f I n t e r p r e t a t i o n b e tw e e n e x p e c ta n c y e q u a t i o n s a n d d i f f e r e n c e e q u a t i o n s ......................................... 9^ C r i t e r i o n e s t i m a t e d d i f f e r e n c e e q u a t i o n ......................................... 97 C o m p a riso n o f th e s l o p e s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w i t h i n a n d b e tw e e n s e t s o f e q u a t i o n s 97 C o m p a riso n b e tw e e n t h e s l o p e s o f th e t w o - v a r i a b l e (X,Y ) e s t i m a t e d t r u e v i l l CHAPTER PAGE d i f f e r e n c e e q u a t i o n s a n d t h e t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s ..................................... 106 C o m p a riso n b e tw e e n t h e s l o p e s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d th e a p p r o x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s ..................................... 119 C o m p a riso n b e tw e e n t h e s l o p e s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d th e a l g e b r a i c d i f f e r e n c e s e q u a t i o n s ............................................................... 12 5 Summary c o m p a ris o n o f s l o p e s f o r th e v a r i o u s s t r a i g h t l i n e e x p e c ta n c y e q u a t i o n s ............................................................... 125 C o m p a riso n b e tw e e n th e i n t e r c e p t s o f t h e t w o - v a r i a b l e (X,Y) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d t h e t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w hen 6=0 . 131 C o m p a riso n b e tw e e n t h e i n t e r c e p t s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a n d th e a p p r o x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n l x CHAPTER PAGE w hen 6 » 0 ................................................................ 133 C o m p a riso n b e tw e e n th e i n t e r c e p t o f th e a r i t h m e t i c d i f f e r e n c e s a n d th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w hen 5* 0 . . . .......................................... 135 C o m p a riso n o f t h e r e l i a b i l i t i e s o f th e e s t i m a t e d d i f f e r e n c e e q u a t i o n s • . • 137 S u m m a r y ................................................................................ 143 V I . EXPECTANCY EQUATIONS, THEIR SLOPES AND INTERCEPTS, AND THEIR RELIABILITIES . . 144 C o m p a riso n o f th e s l o p e s o f t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n s b e tw e e n th e f o u r s e t s o f p o p u l a t i o n • • • • 144 C o m p a riso n o f th e s l o p e s b e tw e e n th e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n e q u a t i o n a n d t h e s i n g l e v a r i a b l e (X) r e g r e s s i o n e q u a t i o n • ................................ 153 C o m p a riso n o f t h e s l o p e s b e tw e e n th e th e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n e q u a t i o n a n d t h e H orn F o rm u la . . . 160 X CHAPTER PAGE C o m p a ris o n o f t h e I n t e r c e p t s o f t h e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n s w i t h i n g r a d e l e v e l s 163 C o m p a ris o n b e tw e e n t h e i n t e r c e p t s o f t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n a n d th e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n .••••• •• • 164 C o m p a ris o n b e tw e e n t h e I n t e r c e p t s o f th e t w o - v a r i a b l e s (X ,C ) r e g r e s s i o n e q u a t i o n a n d t h e H orn F o rm u la . • . • 156 C o m p a ris o n o f t h e V a r i o u s i n t e r c e p t s b a s e d on t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n b e tw e e n d i s t r i c t s . . • l o 8 C o m p a riso n o f t h e r e l i a b i l i t i e s o f th e s i n g l e a n d t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n s a n d t h e H o rn F o rm u la . . . 17 0 A n a l y s i s o f t h e p r e d i c t i v e v a lu e o f t h e r e g r e s s i o n e q u a t i o n w i t h t h e I n c l u s i o n o f t h e e f f e c t s o f c h r o n o l o g i c a l a g e . ...................................... 174 x i CHAPTER PAGE A n a l y s i s o f t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n b a s e d on t h r e e d i f f e r e n t g r o u p s a c c o r d i n g t o i n t e l l i g e n c e q u o t i e n t s . ........................... 178 R e l a t i o n s h i p s b e tw e e n r e g r e s s i o n e q u a t i o n s a n d A n t i c i p a t e d A c h ie v e m e n t . 187 S u m m a r y ..................................... 1 90 V I I . SUMMARY, CONCLUSIONS AND EDUCATIONAL IM PLIC A TIO N S.......................................................... 191 Summary o f p r o c e d u r e ........................................... 191 F i n d i n g s .......................................... 193 F i n d i n g s r e g a r d i n g t h e d i f f e r e n c e e q u a t i o n s a s o p p o s e d t o e x p e c t a n c y e q u a t i o n s ............................... 193 F i n d i n g s r e g a r d i n g t h e a p p l i c a b i l i t y o f u s i n g a s i n g l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n f o r G rad e a n d G rad e 6 R e a d in g a n d A r i t h m e t i c • • • • . • • 19^ F i n d i n g s r e g a r d i n g t h e i n c l u s i o n o f c h r o n o l o g i c a l a g e t o t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ........................... 195 x i l CHAPTER PAGE F i n d i n g s r e g a r d i n g th e u s e o f t h e a p p r o x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n I n p l a c e o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . 195 F i n d i n g s r e g a r d i n g th e c o m p a ris o n o f t h e a r i t h m e t i c d i f f e r e n c e a n d th e e s t i m a t e d t r u e d i f f e r e n c e ....................... 196 F i n d i n g s r e g a r d i n g th e c o m p a r is o n o f th e r e l i a b i l i t i e s o f e s t i m a t e d d i f f e r e n c e e q u a t i o n s • • • • • • • • 196 F i n d i n g s r e g a r d i n g th e u s e o f a s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n t o b e u s e d b e tw e e n g r a d e s a n d a c h ie v e m e n t a r e a s ...................................................... 1 9 ? F i n d i n g s r e g a r d i n g th e s i m i l a r i t y o f p r e d i c t i o n w i t h a n d w i t h o u t t h e u s e o f c h r o n o l o g i c a l a g e i n th e r e g r e s s i o n e q u a t i o n s ................................. 197 F i n d i n g s r e g a r d i n g th e u s e o f t h e H orn F o rm u la a s r e p r e s e n t a t i v e a n d a d e q u a te m e a su re o f e x p e c ta n c y . • • 198 x l l l CHAPTER PAGE F i n d i n g s r e g a r d i n g t h e u s e o f a s i n g l e r e g r e s s i o n e q u a t i o n a s a u n i v e r s a l e x p e c t a n c y e q u a t i o n f o r v a r i o u s d i s t r i c t s w hen g r a d e a n d a c h ie v e m e n t a r e a i s c o n s t a n t . • . ............................ 199 F i n d i n g s r e g a r d i n g t h e u s e o f r e g r e s s i o n e q u a t i o n s f o r e x p e c t a n c y w hen a b i l i t y g r o u p i n g s a r e c o n s i d e r e d , . 200 F i n d i n g s r e g a r d i n g t h e A n t i c i p a t e d A c h i e v e m e n t ........................................................... 201 C o n c l u s i o n s ...................................................................... 202 R e c o m m e n d a tio n s ...................... • • • • • • • 204 LIST OF TABLES TABLE PAGE I* Mean a n d S t a n d a r d D e v i a t i o n s o f Com m unity Mean S t a n d a r d S c o r e s , b y G r a d e s , on S u b t e s t s o f t h e M e t r o p o l i t a n A c h ie v e m e n t T e s t s B a se d on S e v e n ty C o m m u n itie s • . • . 40 I I , Summary o f Mean V a lu e s f o r C h r o n o l o g i c a l Age G rade P la c e m e n t (CAGP), M e n ta l Age G rade P la c e m e n t (MAGP), R e a d in g G rade P la c e m e n t (RGP) a n d A r i t h m e t i c G rad e P la c e m e n t (AGP) o f E ach G rade 4 P o p u l a t i o n ..................................... 82 I I I . Summary o f Mean V a lu e s f o r C h r o n o l o g ic a l Age G rad e P la c e m e n t (CAGP), M ental. Age G rade P la c e m e n t (MAGP), R e a d in g G rade P la c e m e n t (RGP), a n d A r i t h m e t i c G rad e P la c e m e n t (AGP) o f E ach G rade 6 P o p u l a t i o n . ........................... 83 IV . Summary o f F - R a t i o s a s a R e s u l t o f A n a l y s i s o f V a r ia n c e B e tw e e n D i s t r i c t s a n d W ith in D i s t r i c t s w i t h R e s p e c t to CAGP, MAGP, RGP, a n d AGP f o r G rade 4 • • • • . ........................... 84 V . Summary o f F - R a t l o s a s a R e s u l t o f A n a l y s i s o f V a r ia n c e B e tw ee n D i s t r i c t s a n d W ith in XV t a b l e pag e D i s t r i c t s w i t h R e s p e c t t o CAGP* MAGP, a n d AGP f o r G rade 6 ............................................ • • 66 V I . Summary o f t h e S lo p e s o f t h e R e g r e s s i o n L i n e s b a s e d u p o n t h e T w o -V a ria b le E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n f o r G rade 4 a n d 6 , R e a d in g a n d A r i t h m e t i c ..................................... 102 V I I . A Summary C o m p a riso n o f Mean S lo p e o f th e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s o f e a c h D i s t r i c t w i t h t h e T o t a l S lo p e o f a l l D i s t r i c t s f o r G rad e 4 a n d 6 , R e a d in g a n d A r i t h m e t i c , w hen E s t i m a t e d T ru e D i f f e r e n c e s a r e h e l d c o n s t a n t a s co m p u ted b y th e W alsh T e s t ................................................................ 104 V I I I . A Summary C o m p a riso n o f t h e S lo p e s o f E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s B e tw e e n V a r io u s C o m b in a tio n s o f A c h ie v e m en t (R e a d in g a n d A r i t h m e t i c ) a n d G rade ( F o u r t h a n d S i x t h ) b y M eans o f th e W alsh T e s t .......................................................................... 107 IX* E s t i m a t e d T ru e D i f f e r e n c e ( 6 ' ) E q u a t io n s B a s e d on G rade 4 R e a d in g (Y) a n d M e n ta l Age (X) D a ta f o r t h e D l B t r i c t B ...................... 108 x v i t a b l e p a g e X . E s t i m a t e d T ru e D i f f e r e n c e ( 6 1) E q u a t i o n s B a se d on G rade 4 A r i t h m e t i c (Y) a n d M e n ta l Age (X) D a ta f o r t h e D i s t r i c t s . • 109 X I . E s t i m a t e d T ru e D i f f e r e n c e ( 5 1) E q u a t io n s B a se d on G rad e 6 R e a d in g (Y) a n d M e n ta l Age (X) D a ta f o r t h e D i s t r i c t s . . 110 X I I . E s t i m a t e d T ru e D i f f e r e n c e ( 6 1) E q u a t io n s B a s e d on G rad e 6 A r i t h m e t i c (Y) a n d M e n ta l Age (X) D a ta f o r t h e D i s t r i c t . . I l l X I I I . E s t i m a t e d T ru e D i f f e r e n c e (& ') E q u a t io n s B a se d on G rade 4 R e a d in g (Y ), M e n ta l Age (X ), a n d C h r o n o l o g i c a l a g e (C) D a ta f o r t h e D i s t r i c t s ..................................................................... 112 X IV . E s t i m a t e d T rue D i f f e r e n c e (& ') E q u a t io n s B a se d on G rade 4 A r i t h m e t i c (Y ), M e n ta l Age (X ), a n d C h r o n o l o g i c a l Age (C) D a ta f o r t h e D i s t r i c t s .......................................................... 113 XV. E s t i m a t e d T ru e D i f f e r e n c e ( 5 1) E q u a t io n s B a se d on G rade 6 R e a d in g ( y ) , M e n ta l Age (X ), a n d C h r o n o l o g i c a l Age (C) D a ta f o r th e D i s t r i c t s . .................................. 114 X V I. E s t i m a t e d T ru e D i f f e r e n c e ( 6 1) E q u a t io n s B a se d on G rad e 6 A r i t h m e t i c (y ) , M e n ta l TABLE X V II. X V I I I . X IX . XX. X X I. X X II. Age (X ), a n d C h r o n o l o g i c a l Age (C) D a ta f o r t h e D i s t r i c t s Summary C o m p a riso n o f t h e S l o p e s o f th e R e g r e s s i o n L in e s B a se d u p o n t h e Two- V a r i a b l e (X,Y ) a n d t h e T h r e e - V a r i a b l e (X ,Y #C) E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r G rade 4 a n d G rad e 6 , R e a d in g a n d A r i t h m e t i c . . . . . . . . . . . . . . Summary C o m p a riso n B e tw e e n t h e S lo p e s o f t h e T w o -V a ria b le a n d th e T h r e e - V a r i a b l e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n f o r G rade 4 R e a d in g a n d A r i t h m e t i c a n d G rad e 6 R e a d in g a n d A r i t h m e t i c C o m b in a tio n s , . . A p p ro x im a te E s t i m a t e d T rue D i f f e r e n c e (D,„.) M E q u a t io n B a s e d on G rade 4 R e a d in g (Y) a n d M e n ta l Age (X) D a ta o f t h e D i s t r i c t s , . . A p p ro x im a te E s t i m a t e d T ru e D i f f e r e n c e (D ^ ,) E q u a t io n B a se d on G rade 4 A r i t h m e t i c (y ) a n d M e n ta l Age (X) D a ta o f t h e D i s t r i c t s . A p p ro x im a te E s t i m a t e d T ru e D i f f e r e n c e (Dm ( ) E q u a t i o n b a s e d on G rad e 6 R e a d in g (Y ) a n d M e n ta l Age (X) D a ta o f t h e D i s t r i c t s • • . A p p ro x im a te E s t i m a t e d T ru e D i f f e r e n c e (Dm i ) x v l l PAGE 115 116 118 120 121 122 x v i i i TABLE PAGE E q u a t i o n B a se d on G rade 6 A r i t h m e t i c (Y) a n d M e n ta l Age (X) D a ta o f th e D i s t r i c t s ........................ . . . . . . 123 X X I I I . Summary C o m p a riso n B e tw ee n t h e S lo p e s o f th e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n ( a f t e r L o rd ) a n d t h e A p p ro x im a te E s t i m a t e d T rue D i f f e r e n c e E q u a t i o n ( a f t e r McNemar) f o r G rade 4 R e a d in g a n d A r i t h m e t i c a n d G rad e 6 R e a d in g a n d A r i t h m e t i c . . . . . 124 XXIV. Summary o f t h e S lo p e s b y D e g re e s o f th e R e g r e s s i o n L in e s B a se d u p o n t h e Two- V a r i a b l e ( 6 * ) , th e T h r e e - V a r i a b l e “ if ( 5 ' Xy C) , th e A p p ro x im a te (Dm, ) T ru e D i f f e r e n c e s E q u a t i o n s a n d t h e A l g e b r a i c D i f f e r e n c e (D ), G rade 4 R e a d i n g ................................ 127 XXV. Summary o f t h e S lo p e s b y D e g re e s o f t h e R e g r e s s i o n L in e s B a s e d u p o n t h e Two- V a r i a b l e (&'_), t h e T h r e e - V a r i a b l e x y xyc t h e A p p ro x im a te (D ^ i) T rue D i f f e r e n c e E q u a t i o n s , a n d t h e A l g e b r a i c D i f f e r e n c e (D ), G rad e 4 A r i t h m e t i c ............................................... 128 XXVI. Summary o f t h e S l o p e s b y D e g re e s o f t h e R e g r e s s i o n L in e s B a s e d u p o n t h e Two- TABLE X X V II. X X V III. XXIX. V a r i a b l e ( 6 ' ) , t h e T h r e e - v a r i a b l e x y (5 1 Xy C )* t h e A p p ro x im a te (r > M , ) T ru e D i f f e r e n c e a E q u a t i o n s , a n d t h e A l g e b r a i c D i f f e r e n c e (D ), G ra d e 6 R e a d in g ...................... Summary o f t h e S l o p e s b y D e g r e e s o f t h e R e g r e s s i o n L i n e s B a s e d u p o n t h e T w o - v a r i a b l e (6 t h e T h r e e - v a r i a b l e ( 5 ‘ Ajr ^ j ^ t h e A p p ro x im a te (1> m i ) T ru e D i f f e r e n c e s E q u a t i o n s , a n d t h e A l g e b r a i c D i f f e r e n c e (D ), G rad e 6 A r i t h m e t i c . . . ...................... Summary C o m p a ris o n o f ( f i v e ) M e n ta l Age I n t e r c e p t s f o r R e a d in g a n d A r i t h m e t i c E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r V a r io u s D i s t r i c t s a n d G ra d e s b y M eans o f t h e t T e s t ........................... ..... Summary C o m p a ris o n B e tw e e n t h e f i v e M e n ta l Age I n t e r c e p t s o f t h e T w o -V a ria b le (X ,Y ) E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s a n d t h e T h r e e - v a r i a b l e (X ,Y ,C ) E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r G rad e 4 R e a d in g , G rad e 4 A r i t h m e t i c , G ra d e 6 R e a d in g , a n d G ra d e 6 A r i t h m e t i c b y m eans o f t h e t T e s t ...................................................................................... x l x PAGE 129 130 132 134 TABLE XXX. Summary c o m p a r is o n B e tw ee n t h e F i v e M e n ta l a g e I n t e r c e p t s o f t h e E s t im a te d T ru e D i f f e r e n c e E q u a t i o n s a n d th e A p p ro x im a te E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r G rad e 4 R e a d in g , G rad e 4 A r i t h m e t i c , G rad e 6 R e a d in g , a n d G rade 6 A r i t h m e t i c b y M eans o f t h e t T e s t • • XXXI. Summary C o m p a riso n B e tw e e n t h e F i v e M e n ta l a g e I n t e r c e p t s o f t h e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n a n d t h e A r i t h m e t i c D i f f e r e n c e f o r G rade 4 R e a d i n g , G rade 4 A r i t h m e t i c , G rade 6 R e a d i n g , a n d G rade 6 A r i t h m e t i c b y m eans o f t h e t T e s t . . . . . . . . . . . . . X X X II. Summary o f t h e R e l i a b i l i t y o f S c o r e s B a se d u p o n t h e T w o - v a r ia b le (X,Y ) E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n ( 6 ) , t h e T h r e e - V a r i a b l e (X ,Y ,C ) rfV J E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n x y c * 9 t h e A p p ro x im a te E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n (&') a n d t h e A r i t h m e t i c D i f f e r e n c e (D ), f o r V a r io u s D i s t r i c t G rade 6 R e a d in g TABLE X X X III. XXXIV. XXXV. x x i PAGE Summary o f t h e R e l i a b i l i t y o f S c o r e s B a se d upon, t h e T w o -V a ria b le (X,Y) E s t i m a t e d T ru e D i f f e r e n c e E q u a t io n ( 6x y ) , t h e T h r e e - v a r i a b l e (X ,Y ,C ) E s t i m a t e d T ru e D i f f e r e n c e E q u a t io n t h e A p p ro x im a te E s t i m a t e d xyc T ru e D i f f e r e n c e E q u a t i o n ( 6 1) a n d th e A r i t h m e t i c D i f f e r e n c e (D ), f o r v a r i o u s D i s t r i c t G ra d e 4 A r i t h m e t i c ........................... 140 Summary o f t h e R e l i a b i l i t y o f S c o r e s B a s e d u p o n t h e T w o - V a r ia b le (X,Y) E s t i m a t e d T ru e D i f f e r e n c e E q u a tio n t h e T h r e e - v a r i a b l e (X ,Y ,C ) E s t i m a t e d T ru e D i f f e r e n c e E q u a t io n (5 yw>) , t h e A p p r o x im a te E s t im a te d xyc T ru e D i f f e r e n c e E q u a t i o n ( 5 1) a n d th e A r i t h m e t i c D i f f e r e n c e (D), f o r v a r i o u s D i s t r i c t G ra d e 6 R e a d in g ................................ 141 Summary o f t h e R e l i a b i l i t y o f S c o r e s B a se d u p o n t h e T w o - V a ria b le (X,Y) E s t i m a t e d T r u e D i f f e r e n c e E q u a t io n th e T h r e e - V a r i a b l e (X ,Y ,C ) x y E s t i m a t e d T ru e D i f f e r e n c e E q u a t io n TABLE x x i i PAGE ( 6„ .r_ ) / The A p p ro x im a te E s t i m a t e d xy c T ru e D i f f e r e n c e E q u a t i o n (6 1) a n d t h e A r i t h m e t i c D i f f e r e n c e (D), f o r V a r io u s D i s t r i c t G rade 6 A r i t h m e t i c • • 142 XXXVI* S i n g l e v a r i a b l e R e g r e s s i o n E q u a t i o n s E s t i m a t i n g R e a d in g A c h ie v e m e n t f o r G rade 4 fro m M e n ta l Age f o r t h e D i s t r i c t s .......................................................................... 145 XXXVII. S i n g l e V a r i a b l e R e g r e s s i o n E q u a t i o n s E s t i m a t i n g A r i t h m e t i c A c h ie v e m e n t f o r G rad e 4 fro m M e n ta l Age f o r t h e D i s t r i c t s ...................................................... 146 X X X V III. S i n g l e v a r i a b l e R e g r e s s i o n E q u a t i o n s E s t i m a t i n g R e a d in g A c h ie v e m e n t f o r G rade 6 fro m M e n ta l Age f o r th e D i s t r i c t s ...................................................... 147 XXXIX. S i n g l e v a r i a b l e R e g r e s s i o n E q u a t io n s E s t i m a t i n g A r i t h m e t i c f o r G rade 6 fro m M e n ta l Age f o r t h e D i s t r i c t s . . . 148 XL. Summary o f th e S lo p e s ( i n D e g r e e s ) o f t h e G rad e 4 R e g r e s s i o n E q u a t i o n s . • . 154 X L I. Summary o f th e S lo p e s ( i n D e g r e e s ) o f th e G rad e 6 R e g r e s s i o n E q u a t i o n s . . . 155 x x i l l TABLE PAGE X L II* T w o -V a ria b le R e g r e s s i o n E q u a t i o n s E s t i m a t i n g H e a d in g A c h ie v e m e n t f o r G rade 4 fro m M e n ta l Age f o r th e D i s t r i c t s ............................... 13 6 X L I I I . T w o -V a ria b le R e g r e s s i o n E q u a t i o n s E s t i m a t i n g A r i t h m e t i c A c h ie v e m e n t f o r G ra d e 4 fro m M e n ta l Age f o r t h e D i s t r i c t s 137 XLIV. T w o - v a r ia b le R e g r e s s i o n E q u a t i o n s E s t i m a t i n g R e a d in g A c h ie v e m e n t f o r G rade 6 fro m M e n ta l Age f o r t h e D i s t r i c t s .......................................................................... 138 XLV* T w o -V a ria b le R e g r e s s i o n E q u a t i o n s E s t i m a t i n g A r i t h m e t i c A c h ie v e m e n t f o r G rade b fro m M e n ta l Age f o r t h e D i s t r i c t s .......................................................................... 139 XLVI. Summary C o m p a riso n b e tw e e n th e S l o p e s o f t h e s i n g l e V a r i a b l e a n d t h u T w o - V a ria b le R e g r e s s i o n E q u a t i o n s f o r G rad e 4 R e a d in g a n d A r i t h m e t i c a n d G rad e 6 R e a d in g a n d A r i t h m e t i c • • • • • . • • • • • • • • 161 X L V II. Summary o f t R a t i o s C o m p arin g V a r io u s I n t e r c e p t s o f t h e T w o - v a r i a b l e R e g r e s s i o n TABLE X L V III. XL IX . L • L I . L I I . x x iv PAGE E q u a tio n B a s e d o n R e a d in g a n d A r i t h m e t i c f o r G rad e 4 a n d G rad e 6 . . I 65 Summary o f b R a t i o s c o m p a r in g v a r i o u s I n t e r c e p t s o f t h e S i n g l e V a r i a b l e R e g r e s s io n E q u a t i o n t o t h e Two- V a r ia b l e R e g r e s s i o n E q u a t i o n w i t h R e s p e c t t o G ra d e 4 a n d 6 , R e a d in g a n d A r i t h m e t i c ..................................... 167 Summary o f t R a t i o s C o m p arin g V a r i o u s I n t e r c e p t s o f t h e T w o - V a r ia b le R e g r e s s i o n E q u a t io n t o t h e H o rn F o rm u la w i t h R e s p e c t to G ra d e 4 a n d 6 t R e a d i n g a n d A r i t h m e t i c ...................................................................... 169 Summary o f t h e R e l i a b i l i t y o f S c o r e s B a se d upo n t h e S i n g l e a n d T w o - v a r i a b l e R e g r e s s i o n E q u a t i o n s f o r t h e D i s t r i c t G rade 4 R e a d in g a n d A r i t h m e t i c • • • • 172 Summary o f t h e R e l i a b i l i t y o f S c o r e s B a se d upo n t h e S i n g l e a n d T w o - V a r ia b le R e g r e s s io n E q u a t i o n s f o r t h e D i s t r i c t G rade 6 R e a d in g a n d A r i t h m e t i c . . . . 173 Summary o f t h e C o r r e l a t i o n C o e f f i c i e n t s ( r ) b e tw e e n M e n ta l Age (X) a n d A c h i e v e - xy TABLE x x v PAGE L I I I • L IV . LV. L V I. m ent Age (Y) a n d M u l t i p l e C o r r e l a t i o n C o e f f i c i e n t s ( r ) b e tw e e n y . x c ' A c h ie v e m e n t A g e, M e n ta l A ge, a n d C h r o n o l o g i c a l Age f o r G rade 4..... 176 Summary o f t h e C o r r e l a t i o n C o e f f i c i e n t s ( r ) b e tw e e n M e n ta l Age (X) a n d A c h ie v e - x y m ent Age (Y) a n d M u l t i p l e C o r r e l a t i o n C o e f f i c i e n t s (R ) b e tw e e n A c h ie v e m e n t A g e , M e n ta l A g e, a n d C h r o n o l o g ic a l Age f o r G rade 6 . . .......................................................... 177 Summary o f th e S i n g l e v a r i a b l e R e g r e s s i o n E q u a t i o n s B a se d o n T h re e D i f f e r e n t A b i l i t y G ro u p s f o r G rade 4 R e a d in g a n d A r i t h m e t i c a n d G rad e 6 R e a d in g an d A r i t h m e t i c ..................................................................... 179 Summary o f th e D i f f e r e n c e s , i n M o n th s, b e tw e e n T o t a l I n t e r c e p t s a n d t h o s e f o r p u p i l s o f t h r e e IQ C a t e g o r i e s , f o r G rad e 4 R e a d in g a n d A r i t h m e t i c , w i t h F i v e M e n ta l Age G rade p la c e m e n t O r d i n a t e s .......................................................................... 185 Summary o f th e D i f f e r e n c e s , i n M o n th s, b e tw e e n T o t a l I n t e r c e p t s a n d t h o s e f o r x x v i TABLE PAGE p u p i l s o f T h r e e IQ C a te g o r ie s , f o r G r a d e 6 R e a d i n g an d A r ith m e tic w ith f i v e m e n t a l a g e G rade p lac em en t O r d i n a t e s ........................................................ 186 LIST OF FIGURES FIGURE PAGE 1 . S c h e m a tic p r e s e n t a t i o n o f th e I n t e r c e p t s 1 ^ , 1 ^ , 1 ^ , a n d 1^ f o r t h e R e s p e c t i v e M e n ta l A ges 2 . 1 , 4 . 1 , b . l , a n d 8 , 1 ..................................... 92 2 . L in e s R e p r e s e n t i n g th e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r th e D i s t r i c t s f o r G rade 4 R e a d in g When &1 ■ 0 ........................................................... 98 3 . L in e s R e p r e s e n t i n g t h e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r th e D i s t r i c t s f o r G rade 4 A r i t h m e t i c When 6 ' s 0 99 4 . L i n e s R e p r e s e n t i n g th e E s t i m a t e d T rue D i f f e r e n c e E q u a t i o n s f o r th e D i s t r i c t s f o r G rade 6 R e a d in g When 0 100 5 . L in e s R e p r e s e n t i n g th e E s t i m a t e d T ru e D i f f e r e n c e E q u a t i o n s f o r th e D i s t r i c t s f o r G rade 6 A r i t h m e t i c When &'■ 0 ......................................................... 101 6 . L in e s R e p r e s e n t i n g th e S i n g l e v a r i a b l e R e g r e s s i o n E q u a t i o n s f o r t h e D i s t r i c t s f o r G rade 4 R e a d i n g ................................................................................. 149 x x v i i i FIGURE PAGE 7 . L in e s R e p r e s e n t i n g th e S i n g l e v a r i a b l e R e g r e s s i o n E q u a t io n s f o r t h e D i s t r i c t s f o r G rade 4 A r i t h m e t i c ........................................... 150 8 . L in e s R e p r e s e n t i n g th e S i n g l e v a r i a b l e R e g r e s s i o n E q u a t io n s f o r th e D i s t r i c t s f o r G rade b R e a d in g .............................................. 151 9 . L in e s R e p r e s e n t i n g th e S i n g l e V a r i a b l e R e g r e s s i o n E q u a t io n s f o r t h e D i s t r i c t s f o r G rade 6 A r i t h m e t i c ......................................... 182 1 0 . L in e s R e p r e s e n t i n g th e A n t i c i p a t e d A c h ie v e m ent a n d R e g r e s s i o n E q u a t i o n s o f T h ree D i f f e r e n t A b i l i t y G roups f o r G rade 4 R e a d in g ............................................................................... 180 1 1 . L in e s R e p r e s e n t i n g th e A n t i c i p a t e d A c h ie v e m ent a n d R e g r e s s i o n E q u a t i o n s o f T h re e D i f f e r e n t A b i l i t y G roups f o r G rade 4 A r i t h m e t i c .......................................................................... 181 1 2 . L in e s R e p r e s e n t i n g th e A n t i c i p a t e d A c h i e v e m ent a n d R e g r e s s i o n E q u a t i o n s o f T h ree D i f f e r e n t A b i l i t y G roups f o r G rade 6 R e a d in g ....................................................... . . . . . 182 x x i x FIGURE PAGE 1 3 . L in e s R e p r e s e n t i n g th e A n t i c i p a t e d A c h ie v e m e n t a n d R e g r e s s i o n E q u a t io n s o f T h re e D i f f e r e n t A b i l i t y G ro u p s f o r G rad e 6 A r i t h m e t i c . . . . 183 14-, S c h e m a tic p r e s e n t a t i o n o f a Recommended P r o c e d u r e f o r I n d i c a t i n g D e v i a t i o n fro m E x p e c t a n c y .................................................................................... 207 CHAPTER I THE PROBLEM F o r w h o m so ev er much I s g i v e n , o f h im s h a l l be much r e q u i r e d , • • , (Luke 1 2 : 4 8 ) i s a b i b l i c a l q u o t a t i o n . The f o l l o w i n g i s a r e c e n t q u o t a t i o n : The d i v i s i o n o f R e s e a r c h a n d G u id a n c e o f t h e O f f i c e s o f t h e L os A n g e le s C o u n ty S u p e r i n t e n d e n t o f S c h o o ls i n c l u d e s a s one o f i t s f u n c t i o n s t h a t o f a s s i s t i n g c o u n ty s c h o o l d i s t r i c t s i n t h e i n t e r p r e t a t i o n o f t h e i r t e s t a n d m e a su re m e n t f i n d i n g s . F o r some y e a r s t h e s e i n t e r p r e t a t i o n s h av e s t r e s s e d a c h i e v e m e n t i n r e l a t i o n t o a b i l i t y t o a c h i e v e , e m p h a s i z i n g th e I m p o r ta n c e o f c u r r i c u l a r e n v ir o n m e n t f o r t h e b r i g h t c h i l d a n d t h e re m o v a l o f g r a d e a c h ie v e m e n t p r e s s u r e s f ro m t h e d u l l c h i l d , , • ( 8 9 :1 2 7 ) B o th o f t h e s e s t a t e m e n t s , a l t h o u g h made h u n d r e d s o f y e a r s a p a r t , e x p r e s s a s i m i l a r p r o b le m . T h is p ro b le m o f r e l a t i n g c a p a c i t y m e a s u re m e n t t o a c h ie v e m e n t m e a s u re m e n t I n v o l v e s a m ore t e d i o u s a p p r o a c h a s a t t e m p t s a r e made t o r e f i n e t h e d e g r e e o f c a p a c i t y a n d i t s r e l a t i o n t o f i n e r d e g r e e s o f a c h i e v e m e n t . 2 I . INTRODUCTION T h is s t u d y I s c o n c e r n e d v l t h a n t i c i p a t i n g how w e l l t h e c h i l d w i l l do i n s c h o o l b y u s e o f t h e d a t a o f t h e s c h o o l t e s t i n g p ro g ra m * M eth o d s o f r e f i n i n g t h e e x p e c t e d l e v e l o f a n i n d i v i d u a l ' s a c h ie v e m e n t i n r e l a t i o n t o h i s a b i l i t y w e re i n v e s t i g a t e d i n t h i s s t u d y . A b r i e f d e s c r i p t i o n o f t h e v a r i o u s m e th o d s f o r r e f i n i n g t h e l e v e l o f a t t a i n m e n t fro m a b i l i t y m e a s u r e s i s now g i v e n . The M e n ta l A g e . The M A ( m e n ta l a g e ) h a s b e e n u s e d r a t h e r r o u t i n e l y a s t h e v a lu e o f a b i l i t y t o w h ic h a c h i e v e m en t i s c o m p a re d , i t i s a ssu m e d t h a t a n i n d i v i d u a l o b t a i n i n g a c e r t a i n l e v e l o f c o m p e te n c y on a n i n t e l l i g e n c e t e s t w i l l b e e x p e c t e d to a c h i e v e a t a s i m i l a r l e v e l o f c o m p e te n c y on a n a c h ie v e m e n t t e s t . W h e th e r o r n o t t h e r e i s s u f f i c i e n t j u s t i f i c a t i o n f o r t h i s a s s u m p t i o n h a s l o n g b e e n a r g u e d a n d may c o n t i n u e t o be a r g u e d i n t h e f u t u r e a t v a r y i n g r a t e s o f i n t e n s i t y w i t h t i m e . A d i s c u s s i o n o f t h i s c o n t r o v e r s y i s g i v e n i n C h a p te r I I . The AQ (A c h ie v e m e n t Q u o t i e n t ) f o l l o w e d t h e M A a s a m eth o d o f e x p e c t a n c y . T h is m eth o d o f r e p o r t i n g c o m p a r is o n s f l o u r i s h e d d u r i n g t h e t h i r d a n d f o u r t h d e c a d e s o f t h i s 3 c e n t u r y * S in c e t h e AQ d i d n o t im p ro v e on a n y o f t h e c h a r a c t e r i s t i c s w h ic h w e re t h e b a s e s o f c r i t i c i s m s f o r th e u s e o f t h e M A a s a n e x p e c t a n c y , th e AQ w as n o t i n v e s t i g a t e d i n t h i s s tu d y * The H orn F o r m u l a , H orn h a s ( 8 : 6 5 ) d i s c u s s e d th e u s e o f t h e M A a n d t h e AQ a s m e th o d s o f e x p e c t a n c y . She s t a t e d t h a t t h e a c h ie v e m e n t q u o t i e n t w as a n e a r l y a n s w e r t o t h e e v a l u a t i o n o f " t h e c o m p le te n e s s o r s a t l s f a c t o r l - n e s s o f a n a c h ie v e m e n t i n c o m p a r is o n w i t h n a t i v e a b i l i t y " ( 8 : 3 9 ) . A lth o u g h t h e AQ i s se ld o m u s e d now, H orn ( 8 :4 0 ) h a s g i v e n e v id e n c e o f i t s u s e i n 19^1* She c o n c lu d e d t h a t b o t h p r o c e d u r e s c o u ld b e im p ro v e d by t a k i n g i n t o a c c o u n t r e l i a b i l i t y a n d r e g r e s s i o n e f f e c t s . T h en , by m eans o f a r e g r e s s i o n e q u a t i o n , H orn d e r i v e d h e r c e l e b r a t e d f o r m u la b a s e d on r e g r e s s i o n c o e f f i c i e n t s w h ic h sh e o b t a i n e d fro m s e t s o f d a t a f o r v a r i o u s a g e g ro u p s * When c o m p u tin g t h e r e g r e s s i o n e q u a t i o n s sh e made c e r t a i n a s s u m p t io n s b e y o n d t h o s e o f M A e x p e c t a n c y a s s u m p t i o n s , w h ic h m ig h t w e l l b e q u e s t io n e d * T h ese a s s u m p t io n s a r e d i s c u s s e d i n C h a p te r I I a n d V I . 4 E s t i m a t e d a c h ie v e m e n t b a s e d o n th e b a s i c r e g r e s s i o n e q u a t i o n . S in c e H orn d e r i v e d h e r f o r m u l a s f r o m t h e b a s i c r e g r e s s i o n e q u a t i o n , w h e re o n e v a r i a b l e i s p r e d i c t e d fro m a s e c o n d , t h i s m ore b a s i c f o r m u l a w as a l s o i n v e s t i g a t e d i n t h e p r e s e n t s t u d y . I n a d d i t i o n t o r e a d i n g , e q u a t i o n s w e re d e r i v e d on t h e b a s i s o f a r i t h m e t i c s c o r e s . The e f f e c t s o f c h r o n o l o g i c a l a g e w ere i n v e s t i g a t e d , g i v i n g two e q u a t i o n s to be i n v e s t i g a t e d , (1 ) t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n (X: m e n ta l a g e ) a n d , ( 2 ) t h e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n (X: m e n ta l a g e a n d C: c h r o n o l o g i c a l a g e ) . The t r u e d i f f e r e n c e f o r m u l a . R e c e n t l y , L o rd (9 2 ) h a s i n v e s t i g a t e d th e t r u e g a i n b e tw e e n a n i n i t i a l a n d f i n a l a c h ie v e m e n t t e s t . S in c e b o t h i n i t i a l a n d f i n a l t e s t s w e re a c h ie v e m e n t t e s t s , L o r d ’ s f o r m u la c a n n o t b e c o m p a re d w i t h a n y o f th e p r e v i o u s e x p e c t a n c i e s . H ow ever, C a f f r e y , i n a n u n p u b l i s h e d c o m m u n ic a tio n (1 3 2 ) h a s s u g g e s t e d t h e u s e o f a n e q u a t i o n s i m i l a r to t h a t o f L o r d 's w h ic h may a p p l y t o e x p e c t a n c y . E v en th o u g h t h e s e two e q u a t i o n s a r e v e r y s i m i l a r i n a p p e a r a n c e t h e y m u st be d e r i v e d s e p a r a t e l y . The d e r i v a t i o n o f t h e e x p e c t a n c y a d a p t a t i o n o f t h e L o r d 's f o r m u la b y t h e p r e s e n t i n v e s t i g a t o r i s g i v e n i n C h a p te r I I I . 5 The a p p r o x im a te t r u e d i f f e r e n c e e q u a t i o n . McNemar, a f t e r r e a c t i n g t o L o r d 's f o r m u l a , s u g g e s t e d t h e u s e o f a m ore s im p le e q u a t i o n . He recom m ended t h i s f o r m u l a t o t h e p r a c t i c a l w o r k e r I n t h e f i e l d s i n c e i t i s l e s s t e d i o u s , y e t , a n a p p r o x i m a t i o n t o L o r d 's f o r m u l a . The a n t i c i p a t e d a c h i e v e m e n t . A n o th e r ty p e o f e x p e c t a n c y w h ic h h a s r e c e n t l y b e e n d e v e l o p e d i s t h e " A n t i c i p a t e d A c h i e v e m e n t." The A n t i c i p a t e d A c h ie v e m e n t i s a ty p e o f e x p e c t a n c y t h a t i s o b t a i n e d fro m t a b l e s b a s e d on a c t u a l g r a d e p l a c e m e n t , m e n ta l a g e , a n d c h r o n o l o g i c a l a g e ( i f " a t g r a d e " ) . The t a b l e s w e re d e v e l o p e d b y t h e p r o c e d u r e o f d u a l n o rm ln g o f t h e C a l i f o r n i a A c h ie v e m e n t T e s t a n d t h e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y . L o c a l n o rm s . L o c a l n orm s o r l o c a l s t a t u s quo s i t u a t i o n w i t h r e s p e c t t o a n i n d i v i d u a l ' s a c h i e v e m e n t , w hen g i v e n h i s a b i l i t y on a n o t h e r i n s t r u m e n t , a r e s t i l l a n o t h e r ty p e o f e x p e c t a n c y w h ic h w as i n v e s t i g a t e d i n t h i s s t u d y . P r e d i c t i o n a n d e x p e c t a n c y . "No t o p i c i n e d u c a t i o n a l l i t e r a t u r e h a s w i d e r p o p u l a r a p p e a l t h a n p r o g n o s i s . " ( 1 0 :6 7 4 ) I t s fo r m h a s r a n g e d fr o m f o r t u n e - t e l l i n g t o s c i e n t i f i c r e s e a r c h . C o u r t i s h a s s t a t e d , " P r e d i c t i o n i s 6 t h e m o st I m p o r ta n t a s p e c t o f s c i e n t i f i c a c t i v i t y * " ( 5 2 :4 8 2 ) E v e ry m an, woman, b o y a n d g i r l I s a t one tim e o r a n o t h e r a b s o r b i n g l y i n t e r e s t e d i n h i s f u t u r e a s w e l l a s th o s e w i t h whom he I s a s s o c i a t e d , p a r e n t s make f r e q u e n t p r e d i c t i o n s o f g ro w th a n d a c h ie v e m e n t o f t h e i r o f f s p r i n g . B oys a n d g i r l s a t p l a y p r e d i c t w h ic h o f t h e i r num ber i s m o st a p t t o l e a d them t o v i c t o r y i n t h e i r g a m e s. T e a c h e r s p r e d i c t w h ic h o f t h e i r p u p i l s w i l l s u c c e s s f u l l y do t h e w o rk o f t h e n e x t g r a d e o r t h e n e x t t h r e e y e a r s w i t h p o s s i b l e i m p l i c a t i o n s f o r a d m i s s io n to h i g h e r i n s t i t u t i o n s . T hey a r e v i r t u a l l y f o r e c a s t i n g th e f u t u r e o f t h e s e p e r s o n s . E m p lo y e rs do t h e same w i t h t h e i r e m p lo y e e s , p r e d i c t i o n s a r e made i n e v e r y a r e a o f l i v i n g . I t i s n o t s u r p r i s i n g , t h e r e f o r e , t o f i n d p r o g n o s i s a m a jo r e d u c a t i o n a l p r o b le m . B e f o r e th e d e v e lo p m e n t o f v a r i o u s a p t i t u d e t e s t s , n u m ero u s m e th o d s f o r p r e d i c t i o n o f a c h ie v e m e n t w ere a d v a n c e d . I n some q u a r t e r s p h y sio g n o m y a n d p h r e n o lo g y s t i l l p r e s i s t . I n s p i t e o f t h e f a c t t h a t i n v e s t i g a t i o n s h a v e g i v e n l i t t l e b a s i s f o r r e l y i n g u p o n su c h m e th o d s , t e a c h e r s a n d s c h o o l a d m i n i s t r a t o r s so m e tim e s f a l l I n t o t h e e r r o r o f p r e d i c t i n g t h e f u t u r e o f th e c h i l d on 7 o u tw a rd a p p e a r a n c e s . G a in , M i c h a e l i s , a n d E u r i c h ( 1 0 :8 7 4 ) s t a t e t h a t , " i f i n v e s t i g a t i o n s h a v e show n a n y t h i n g , t h e y h a v e r e v e a l e d t h a t ju d g m e n ts b a s e d u p o n s u c h o b s e r v a t i o n s a r e f a u l t y a n d m i s l e a d i n g , " They f u r t h e r s t a t e t h a t , As a s c i e n t i f i c p r o c e d u r e i n e d u c a t i o n p r o g n o s i s i n v o l v e s th e p r e d i c t i o n o f f u t u r e g r o w th , s u c c e s s , o r a d j u s t m e n t o f I n d i v i d u a l s a n d o f g r o u p s fro m a c c u r a t e r e c o r d s o f c l e a r l y d e f i n e d b e h a v i o r . The p rim e p u rp o s e o f su c h p r e d i c t i o n i s t o p r o v i d e i n f o r m a t i o n w h ic h may b e u s e d i n g u id a n c e a n d c o u n s e l i n g o f i n d i v i d u a l s , ( 1 0 :8 7 4 ) The te rm " p r e d i c t i o n " i s e a s y t o u s e i n c o n v e r s a t i o n b u t th e p e r s o n l i s t e n i n g u s u a l l y h a s d i f f i c u l t y i n k no w in g j u s t w h a t i s m ean t b y i t s u s e , ( 5 2 ;4 8 1 ) T h ere may b e a s i m i l a r dilem m a i n th e w ord " e x p e c t a n c y ," p r e d i c t i o n a n d e x p e c ta n c y a r e s i m i l a r c o n c e p ts i n many r e s p e c t s b u t f o r t h e p u r p o s e o f t h i s s t u d y e a c h h a s a d i s t i n c t m e a n in g , p r e d i c t i o n i m p l i e s f o r e t e l l i n g o r p r o p h e s y i n g , w h e r e a s , e x p e c ta n c y I m p l i e s l o o k i n g f o r s o m e th in g w i t h c o n f i d e n c e . A p p l i e d t o a p r a c t i c a l s i t u a t i o n , t h e d i f f e r e n c e b e tw e e n t h e s e two w o rd s m ig h t b e c l a r i f i e d a s f o l l o w s . H a r r y 's p r e d i c t e d 8 s c o r e o f 5 * ° m eans t h a t we now f o r e c a s t H a r r y 's s c o r e a t a l a t e r tim e t o b e 5*0* D i c k 's e x p e c ta n c y s c o r e o f 4 . 0 m eans t h a t w i t h r e f e r e n c e to D i c k 's c u r r e n t a b i l i t y we f e e l t h a t he I s c a p a b le o f a t t a i n i n g a s c o r e o f 4 . 0 . T h u s, th e m e n ta l ag e may n o t n e c e s s a r i l y b e a n i n d i v i d u a l ' s b e s t p r e d i c t e d a c h ie v e m e n t l e v e l b u t may b e u s e d a s a n e x p e c ta n c y v a l u e . L a n tz ( 8 9 :1 2 9 ) h a s w r i t t e n t h a t th e c o n c e p t o f e x p e c ta n c y e m p h a s iz e d f i r s t , a c h ie v e m e n t o f th e i n d i v i d u a l s o r t h e g ro u p i n r e l a t i o n t o t h e i r a b i l i t y to a c h i e v e ; a n d s e c o n d , a c h ie v e m e n t i n r e l a t i o n t o th e g r a d e l e v e l . E x p e c ta n c y , t h e n , may b e d e f i n e d a s a c o n c e p t w h ic h i m p l i e s a c e r t a i n l e v e l o f a c h ie v e m e n t w h ic h a n i n d i v i d u a l i s c a p a b le o f a t t a i n i n g , a s i n d i c a t e d b y th e e v id e n c e o f h i s a b i l i t y t o p e r f o r m a t t h a t l e v e l . I I . THE PROBLEM I n g e n e r a l , t h i s s t u d y w as c o n d u c te d i n o r d e r ( 1 ) t o a n a l y z e v a r i o u s m e th o d s o f o b t a i n i n g a n " e x p e c te d " a c h ie v e m e n t l e v e l a n d (2 ) t o co m p are t h e r e s u l t s o f c o r r e s p o n d i n g v a l u e s o b t a i n e d fro m t h e s e m e th o d s . The m e th o d s o f o b t a i n i n g e x p e c ta n c y w h ic h a r e i n v e s t i g a t e d a r e l i s t e d a b o v e . How v a r i o u s l y o b t a i n e d e x p e c t a n c i e s d i f f e r 9 w i t h r e g a r d t o th e e f f e c t s o f d i f f e r i n g p o p u l a t i o n s , c h r o n o l o g i c a l a g e s , m e n ta l a b i l i t y a n d m eth o d i s th e p ro b le m w i t h w h ic h t h i s s t u d y i s c o n c e r n e d . S p e c i f i c q u e s t i o n s t o be a n s w e r e d . The p u r p o s e o f t h i s s t u d y w as t o make a n e m p i r i c a l i n v e s t i g a t i o n o f (1 ) t h e e f f e c t o f v a r i o u s p o p u l a t i o n s on th e e s t i m a t e d t r u e d i f f e r e n c e e x p e c ta n c y f o r m u l a ; ( 2 ) t h e e f f e c t o f c h r o n o l o g i c a l a g e o n e x p e c t a n c y ; (3 ) t h e r e l a t i o n s h i p b e tw e e n t h e v a r i o u s m e th o d s o f e x p e c ta n c y a n d , i n t u r n , t h e i r r e l a t i o n s h i p t o t h e a c t u a l a c h ie v e m e n t a c c o r d i n g to a b i l i t y a n d c h r o n o l o g i c a l a g e ; (4 ) th e a d e q u a c y o f t h e r e l i a b i l i t i e s o f th e e s t i m a t e d s c o r e s b a s e d on th e v a r i o u s e x p e c ta n c y e q u a t i o n s ; a n d ( 5 ) th e e f f e c t o f e x p e c ta n c y e q u a t i o n s o n v a r i o u s p o p u l a t i o n s w hen, I n f a c t , t h e s e e q u a t i o n s w e re d e r i v e d fro m d i f f e r e n t p o p u l a t i o n s . C r i t e r i a . E ac h e x p e c ta n c y c o n c e p t h a s I t s d i s t i n c t u n d e r l y i n g t h e o r y a n d a s s u m p t i o n . I n o r d e r t o e v a l u a t e t h e s e d i f f e r e n t e x p e c t a n c i e s on a common b a s i s , c e r t a i n c r i t e r i a w ere e s t a b l i s h e d . T h ese b a s i c c r i t e r i a a r e a s f o l l o w s : 1 . The r e l i a b i l i t y o f t h e s c o r e s b a s e d on v a r i o u s e x p e c t a n c i e s . 2 . The u s e o f r e g r e s s i o n e f f e c t s . 10 3 . The v a l i d i t y o f th e a s s u m p tio n s u s e d i n o b t a i n i n g v a r i o u s e x p e c ta n c y m e a s u re s * H y p o th e s e s * A lth o u g h s p e c i f i c h y p o t h e s e s w e re i n v e s t i g a t e d a n d a r e r e p o r t e d i n C h a p te r IV , t h e f o l l o w i n g i s a n o u t l i n e o f g e n e r a l h y p o t h e s e s a b o u t w h ic h t h i s s t u d y i s c o n c e r n e d . 1 . T h ere a r e d i f f e r e n c e s b e tw e e n p r o p e r t i e s ( s l o p e a n d I n t e r c e p t ) o f e q u a t i o n s b a s e d u p o n v a r i o u s m e th o d s o f e x p e c ta n y f o r a g iv e n p o p u l a t i o n . 2 , T h e re a r e no s i g n i f i c a n t d i f f e r e n c e s b e tw e e n p r o p e r t i e s ( s l o p e a n d i n t e r c e p t ) o f th e v a r i o u s e x p e c ta n c y e q u a t i o n s b a s e d up on d a t a fro m d i f f e r e n t p o p u l a t i o n s ( s c h o o l d i s t r i c t s ) . 3 . The r e l i a b i l i t y c o e f f i c i e n t s o f th e s c o r e s b a s e d u p o n th e v a r i o u s e x p e c t a n c i e s a r e s u f f i c i e n t l y h i g h so t h a t t h e y may be u s e d w i t h c o n f i d e n c e . 4 , T h e re a r e s i g n i f i c a n t d i f f e r e n c e s b e tw e e n e x p e c t a n c i e s f o r d i f f e r e n t a b i l i t y l e v e l s o f p u p i l s co m p u ted b y e x p e c ta n c y m e th o d s . I I I . THE STUDY I m p o r ta n c e o f th e S t u d y . The f a c t t h a t e d u c a t o r s , e d u c a t i o n a l p s y c h o l o g i s t s , a n d lay m an h a v e l o n g b e e n c o n c e r n e d w i t h t h e p ro b le m o f how w e l l a n i n d i v i d u a l i s a c h i e v i n g w i t h r e s p e c t t o c a p a c i t y h a s a l r e a d y b e e n 11 e m p h a s iz e d . S t a n d a r d i z e d a c h ie v e m e n t t e s t s a n d i n t e l l i g e n c e t e s t s a r e a s com m onplace i n th e e d u c a t i o n a l p ro g ra m to d a y a s th e incom e t a x i s i n t h e A m e ric a n hom e. The e x p e c ta n c y c o n c e p t h a s I m p l i c a t i o n s w h e r e e v e r a c h ie v e m e n t t e s t s a r e I n t e r p r e t e d . N eed f o r th e s t u d y . D i f f e r e n t e x p e c t a n c i e s a n d e x p e c ta n c y c o n c e p t s a r e u s e d b y t e a c h e r s a n d g u id a n c e w o r k e r s . I t i s i m p o r t a n t , t h e n , t h a t t h e s e e x p e c t a n c i e s , t o g e t h e r w i t h o t h e r s , be i n v e s t i g a t e d i n o r d e r t o i s o l a t e t h e m ore d e s i r a b l e e x p e c t a n c i e s a n d t o i d e n t i f y a r e a s o f s t r e n g t h s a n d w e a k n e s s e s o f t h e s e v a r i o u s m e th o d s . E x p e c t a n c i e s a r e o f t e n t a k e n a t f a c e v a l u e . T h e re a r e , h o w e v e r, l i m i t a t i o n s t o t h e i r a p p l i c a t i o n . T h ere i s , t h e n , a n e e d f o r ( 1 ) a n I n v e s t i g a t i o n o f th e c o m p a ris o n o f e x i s t i n g e x p e c t a n c i e s a s w e l l a s ; ( 2 ) a c o m p a ris o n o f o t h e r m eth o d s o f o b t a i n i n g e x p e c t a n c i e s w h ic h a r e a d a p t a t i o n s o f o t h e r c o n c e p ts a n d h a v e n e v e r b e e n i n v e s t i g a t e d a s e x p e c t a n c i e s ; a n d (3 ) a n i n v e s t i g a t i o n o f t h e l i m i t a t i o n s o f v a r i o u s e x p e c t a n c i e s . D e l i m i t a t i o n s o f t h e s t u d y . The sc o p e o f t h i s s t u d y w as r e s t r i c t e d t o a n i n v e s t i g a t i o n o f th e e x p e c t a n c i e s o f f o u r t h g r a d e a n d s i x t h g ra d e p u p i l s w i t h r e s p e c t 12 t o r e a d i n g a n d a r i t h m e t i c . A l l t h i r t e e n d i s t r i c t s u s e d w e re d i s t r i c t s o f L os A n g e le s C o u n ty , C a l i f o r n i a , O nly t h o s e s t u d e n t s w e re u s e d i n t h e s t u d y f o r whom c o m p le te d a t a w e re a v a i l a b l e (CA, MA, r e a d i n g a n d a r i t h m e t i c ) . A l l i n t e l l i g e n c e a n d a c h ie v e m e n t d a t a w e re c o n f i n e d t o two t e s t s ( t h e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y E le m e n t a r y , 1 9 5 0 , S h o r t f o r m , a n d th e C a l i f o r n i a A c h ie v e m e n t T e s t , E le m e n t a r y , Form A A ). IV . ORGANIZATION OF THE STUDY The remainder of the d is s e r ta tio n i s organized as fo llo w s; Chapter II i s a review of the lite r a tu r e on v a ria b les which e f f e c t expectancy and the various methods of c a lc u la tin g expectancy. Chapter I I I con tain s the d eriv a tio n s o f estim ated true d iffer en ce expectancy equations which, h ith e r to , have not been in v e s tig a te d . D erivation s o f r e l i a b i l i t i e s have a lso been derived fo r a l l expectancy equations used in the study. C h a p te r IV d i s c l o s e s th e r e s e a r c h d e s i g n , a d e s c r i p t i o n o f t h e p o p u l a t i o n s , a n d k i n d s o f c o m p a r is o n s o f v a r i o u s e x p e c t a n c i e s w h ic h w e re i n v e s t i g a t e d . 13 C h a p te r V c o n t a i n s th e f i n d i n g s o f t h e i n v e s t i g a t i o n w i t h r e g a r d t o e s t i m a t e d d i f f e r e n c e e q u a t i o n s . The r e s u lts of various s t a t i s t i c a l a n alyses and a d isc u ssio n o f the resu ltB are p resen ted . C h a p te r V I p r e s e n t s a f o r m a t s i m i l a r t o C h a p te r V b u t c o m p a re s r e g r e s s i o n e q u a t i o n e x p e c t a n c i e s a s w e l l a s th e a n t i c i p a t e d a c h i e v e m e n t . C h a p te r V I I c o n s i s t s o f t h e sum m ary, c o n c l u s i o n s , a n d r e c o m m e n d a tio n s . The bibliography concludes the d is s e r ta tio n . CHAPTER I I REVIEW OF THE LITERATURE Although th is study i s concerned s p e c if ic a lly w ith an a lyses and comparisons among various expectancy eq u ation s, the lite r a t u r e reviewed here covers both the equations and the v a ria b les r e la te d to expectancy, the la t t e r coming f i r s t . I . LITERATURE INDICATING THE VARIABLES RELATED TO INTELLIGENCE AND ACHIEVEMENT The v a ria b les in th is study are lim ite d to those o f ch ron ological age, grade, in t e lle c t u a l a b i li t y , reading achievem ent and a rith m etic achievem ent. I t Is im portant, however, to p o in t out th at there are a large number of other v a ria b les th at c o r r e la te w ith achievement and in te llig e n c e t e s t Bcores which have been in v e stig a te d and that they may w e ll serve to f a c i l i t a t e p r e d ictio n or expectancy. The p resent in v e s tig a to r reco g n izes the p o te n tia l th at th ese v a ria b les have w ith regard to th e ir congruence to the corresponding v a ria b les w ith in the 1 5 In d ivid u al th at con trib u te to h is a ch iev in g a t a c e r ta in le v e l of attain m en t. These v a r ia b le s then, are sim ply mentioned or d escrib ed b r ie f ly in r e la tio n to th e ir connection w ith m ental age and achievem ent. As a r e s u lt , the next sev e ra l s e c tio n s h elp to recogn ize the lim ita tio n s o f the p resen t study by p resen tin g the scope o f th ese v a r ia b le s as a r e fe r e n tia l b a B ls. P aterson, in h is book physique and I n t e lle c t (1 3 ) wrote on the r e la tio n s h ip between p h y sica l growth and in t e llig e n c e . This study was a forerunner of other stu d ie s in v e s tig a tin g p h y sica l c h a r a c t e r is t ic s . Many p h y sic a l c o r r e la te s have been In v e stig a te d w ith re sp e c t to I n te llig e n c e and achievem ent although some of them have low fa ce v a lid it y . For example, N ally ( 1 0 1 :1 5 ^ ) found th at development in h eigh t appears to bear a p r e d ic tiv e r e la tio n s h ip to reading ln c ip ie n c y . Olson la t e r m aintained th a t, The i n d i v i d u a l c a n n o t b e d i v i d e d e x p e c t i n a v e r y a r b i t r a r y f a s h i o n a n d t h a t t h e r e i s a n i n t i m a t e r e l a t i o n s h i p i n t h e f u n c t i o n i n g o f a l l a s p e c t s o f g r o w t h . ( 1 2 :1 6 3 ) Gleason and Klausmeier have r e c e n tly summarized th is area of in v e s tig a tio n by the sta tem en t: 1 6 Much r e s e a r c h h a s b e e n c a r r i e d on d u r i n g t h e p a s t f o u r d e c a d e s t o d e te r m in e t h e r e l a t i o n s h i p b e tw e e n v a r i o u s m e a s u re s o f i n t e l l i g e n c e a n d o f a c a d e m ic a c h ie v e m e n t • • • T h is r e s e a r c h h a s show n r a t h e r c o n s i s t e n t low p o s i t i v e c o r r e l a t i o n s b e tw e e n m e a s u re s o f p h y s i c a l g ro w th a n d o f m e n ta l a g e o r i n t e l l i g e n c e * (6 7 :6 2 1 ) Many s t u d i e s c o m p a rin g p e r s o n a l i t y v a r i a b l e s a n d i n t e l l i g e n c e t e s t s c o r e s h a v e b e e n u n d e r t a k e n . F o r e x a m p le , some c l i n i c a l I n v e s t i g a t o r s ( 2 3 :3 4 6 , 3 8 :8 3 ) h a v e f o u n d r e l a t i o n s h i p s b e tw e e n t h e R o r s c h a c h a n d I n t e l l i g e n c e , w h i l e o t h e r s ( 6l *.197 ) ( 2 6 :3 2 4 ) h av e n o t . I n o t h e r a r e a s , H i l l a n d Rogge ( 8 0 :6 4 7 ) fo u n d s l i g h t ( b u t p o s i t i v e ) c o r r e l a t i o n s b e tw e e n i n t e l l i g e n c e t e s t s c o r e s a n d i n t e r e s t t e s t s c o r e s , w i e n e r ( 1 2 5 : 1 2 9 ) show ed t h a t d i s t r u s t f u l p r e d i s p o s i t i o n s a r e c o r r e l a t e d w i t h i m p a i r e d p e rf o r m a n c e on a n i n t e l l i g e n c e t e s t . T a y l o r (1 2 0 :2 7 0 ) fo u n d s i g n i f i c a n t r e l a t i o n s h i p s b e tw e e n s o c i a l a c c e p t a n c e a n d m e n ta l a g e . A p p a r e n t l y many o f t h e v a r i a b l e s w h ic h h a v e b e e n i n v e s t i g a t e d , a r e r e l a t i v e l y u n s t a b l e . F o r e x a m p le , a l t h o u g h G r ic e 7 5 :7 3 ) a n d K e r r i c k ( 8 5 :7 7 ) show ed t h a t s i g n i f i c a n t n e g a t i v e c o r r e l a t i o n s e x i s t b e tw e e n a n x i e t y s c a l e s a n d i n t e l l i g e n c e s c o r e s , o t h e r s (1 1 3 , 9 6 :4 0 3 , 1 1 1 ; 2 2 4 , 9 3 :2 0 4 ) d i d n o t r e p l i c a t e t h e s e f i n d i n g s . 17 The p u r p o s e o f t h i s d i s c u s s i o n i s n o t t o r e p o r t on a l l e x i s t i n g s t u d i e s r e l a t i n g i n t e l l i g e n c e t o o t h e r v a r i a b l e s , b u t t o i n d i c a t e t h e sc o p e o f t h e s e v a r i a b l e s a n d t h e i r p o t e n t i a l a s c o n t r i b u t o r s to a m ore c o m p re h e n s i v e i n t e r p r e t a t i o n o f i n t e l l i g e n c e w h ic h m ig h t b e u s e d a s a b a s i s f o r e x p e c ta n c y a n a l y s i s * F o r t h e p u r p o s e o f p l a c i n g th e p r e s e n t s t u d y i n a p r a c t i c a l p e r s p e c t i v e i t i s i m p o r t a n t t o e m p h a s iz e t h a t t h e s e f i n d i n g s , r e g a r d i n g th e r e l a t i o n s h i p o f t h e s e " p o t e n t i a l " v a r i a b l e s w i t h i n t e l l i g e n c e , f l u c t u a t e w i t h c o n d i t i o n s , a n d a r e o f t e n i n c o n s i s t e n t . I m p o r t a n t fro m a p r a c t i c a l s t a n d p o i n t i s t h a t m o st o f t h e s e d a t a a r e n o t r o u t i n e l y f o u n d f o r e a c h p u p i l on t y p i c a l s c h o o l r e c o r d s . S i m i l a r t y p e s o f c o r r e l a t e s a r e fo u n d w i t h r e s p e c t t o s t u d i e s o f a c h i e v e m e n t . S t a g n e r ( l i d ) h a s c o n c lu d e d t h a t a l t h o u g h e a r l i e r s t u d i e s show ed o n ly n e g l i b l e r e l a t i o n s h i p s b e tw e e n p e r s o n a l i t y t e s t s a n d a c h ie v e m e n t, l a t e r s t u d i e s h a v e i n d i c a t e d m ore p o s i t i v e r e s u l t s . P e r s i s t e n c e i s n o t c o r r e l a t e d h i g h l y w i t h a l l a r e a s o f a c h i e v e m e n t , b u t E d m ls to n ( 5 9 :5 0 ) f o u n d c e r t a i n a r e a s o f a c h ie v e m e n t t o c o r r e l a t e s i g n i f i c a n t l y w i t h p e r s i s t e n c e . Bond a n d C lym er (3 5 ) r e p o r t e d t h a t p e r s e v e r a r t i o n c o r r e l a t e d s i g n i f i c a n t l y ( a n d n e g a t i v e l y ) w i t h 18 measures of read in g, F rederiksen and M e lv ille ( 6 4 :6 5 0 ) r e la te d com pulsiveness to achievem ent t e s t s c o r e s . Harder ( 7 8 :2 1 3 ) concluded as a r e s u lt o f h is stu d ie s comparing academic groups th a t d iffe r e n c e s in achievem ent could be a ttr ib u te d to d iffe r e n c e s in m o tiv a tio n , Malpass ( 9 5 : 4 8 0 ) found r e la tio n sh ip between stu d en ts' p ercep tion s o f sch ool and th e ir achievem ent. Other workers have stu d ied achievem ent w ith resp ect to the e f f e c t s o f l a t e r a li t y ( 1 1 3 : 3 2 1 ) , d ise a se ( 1 2 4 : 3 5 4 ) , a n x iety s c a le s ( 3 1 : 5 7 7 , 1 0 4 , 1 0 9 : 4 8 9 ) / the Rorschach ( 3 2 : 7 4 ) , the M M P I ( 2 0 : 3 2 7 ) , and s o c ia l m aturity ( 1 0 3 : 8 5 ) , Drews and Teahan ( 5 7 : 3 3 1 ) found th at d iffe r e n c e s between academic overach ievers and underachievers were r e la te d to p aren tal a ttit u d e s w h ile Brown and Holtzaman ( 4 1 :8 3 ) found that p u p il a ttitu d e s toward studying can p la y a s u b sta n tia l r o le in subsequent academic achievem ent and In tr o v e r sio n , dominance, s e lf - s u f f ic ie n c y , good m otiva tio n , lib e r a l s o c ia l a t t it u d e s , and la ck o f m aladjustm ent. Other p r e d ic tiv e fa c to r s have been review ed by McQuary ( 9 8 : 2 1 6 ) , These areas of in v e s tig a tio n in clu d ed the occu p ation al le v e l of the fa th e r , s iz e o f community, ed u cation al l e v e l o f the fa th e r , ed u cation al l e v e l of the 19 m o th e r , num ber o f s i b l i n g s , p o s i t i o n am ong s i b l i n g s , num ber o f i l l n e s s e s a n d d i s e a s e s , h o u r s t u d i e d p e r w e ek , s o c i a l i n t r o v e r s i o n - e x t r o v e r s i o n , a n d f o r e i g n - b o r n p a r e n t s . I t i s s e e n fro m t h e s e s t u d i e s t h a t t h e v a r i a b l e s o f hum an g ro w th a n d b e h a v i o r , w h ic h a r e f o u n d t o b e r e l a t e d t o a c h ie v e m e n t a r e b r o a d i n s c o p e , v a r y w i t h c o n d i t i o n s , a r e o f t e n i n c o n s i s t e n t , a n d t h e y a r e n o t u s u a l l y f o u n d on t h e s c h o o l r e c o r d s f o r e a c h p u p i l , I I . LITERATURE REPORTING THE VARIABLES USED IN PREDICTING ACHIEVEMENT The p r e v i o u s d i s c u s s i o n h a s d e a l t w i t h c e r t a i n v a r i a b l e s a n d t h e i r r e l a t i o n s h i p s t o i n t e l l i g e n c e a n d a c h i e v e m e n t . The n e x t a r e a o f e x p l o r a t i o n h a s to do w i t h some o f t h e s e v a r i a b l e s a n d t h e i r r o l e a s p r e d i c t o r s . I t w i l l b e show n t h a t o t h e r v a r i a b l e s ( o t h e r t h a n t h e o n e s u s e d I n t h i s s t u d y ) w hen i n c l u d e d i n m u l t i p l e r e g r e s s i o n e q u a t i o n s may w e l l c o n t r i b u t e t o t h e a c c u r a c y t o p r e d i c t i o n , Gough h a s s t a t e d th e m a t t e r c o n c i s e l y : R e c o u rs e h a s o r d i n a r i l y b e e n h a d t o t e s t s o f a b i l i t y a n d c a p a c i t y , b u t r e c e n t l y t h e r e h a s b e e n a g ro w in g r e a l i z a t i o n t h a t o t h e r , 20 n o n - i n t e l l e c t u a l , f a c t o r s m u st b e a s s e s s e d i f t h e m a r g in o f p r e d i c t i v e e r r o r i s t o b e d i m i n i s h e d a p p r e c i a b l l y . ( 6 9 : 6 5 ) The l i t e r a t u r e i n v e s t i g a t i n g t h e e f f e c t s o f t h e s e v a r i a b l e s i s su m m a riz e d a t t h i s tim e i n o r d e r t o r e c o g n i z e t h e sc o p e o f p r e d i c t i v e v a r i a b l e s a s a b a c k g ro u n d t o t h e p r e s e n t , r a t h e r l i m i t e d s t u d y ; a n d t o p l a c e them i n t h e i r p r o p e r p e r s p e c t i v e . I t i s s e e n t h a t t h e r e a r e m any c o n t r i b u t o r s t o t h e p r e d i c t i o n o f a c h ie v e m e n t b u t t h a t many o f t h e s e c o n t r i b u t o r s a r e d i f f i c u l t t o o b t a i n a s a r o u t i n e m e a s u r e , t h a t t h e r e s u l t s w i t h r e g a r d t o t h e c o n s l s t a n c y a n d r e l a t i v e i m p o r ta n c e o f t h e s e v a r i a b l e s a r e n o t c o n c l u s i v e i n a n y s i n g l e d i r e c t i o n . P r e d i c t i o n , i n a n d o f I t s e l f , h o w e v e r, c a n be d a n g e r o u s i f q u a l i f i c a t i o n s a r e n o t g i v e n a n d c a r e o f i n t e r p r e t a t i o n I s n o t m a i n t a i n e d . K l a u s m e i e r , B eem an, Lehm ann ( 8 6 : 1 8 6 ) , f o r e x a m p le , f o u n d t h a t s t a n d a r d m anagem ent o f v a r i a b l e s i n th e m u l t i p l e r e g r e s s i o n e q u a t i o n p r e d i c t i n g a c h ie v e m e n t b e t t e r t h a n t h e " O rg a n ls m ic a g e , " w h ic h I n c l u d e s p o o r l y r e l a t e d p h y s i c a l m e a s u r e s . S t r o u d (1 1 9 ) h a s e v e n I n d i c a t e d , "we do n o t im p r o v e , b u t l o w e r , p r e d i c t i o n o f a c h ie v e m e n t b y c o m b in in g p h y s i c a l s c o r e s w i t h M A . s c o r e s . " An e x a m p le o f m i s p r e d i c t i o n w h ic h a t t r a c t e d n a t i o n a l fam e i s t h e 1 9 ^ 6 p r e s i d e n t i a l 21 p o l l i n g p r e d i c t i o n s o f G a l l u p , C r o s s l e y , a n d R o p e r ( 8 4 : 1 5 ) . R a u s c h , w i t h r e g a r d t o c a r e o f i n t e r p r e t a t i o n f o r p r e d i c t i v e p u r p o s e , c a u t i o n s a s f o l l o w s ; • . • A s c o r e o f 50 o n e a c h o f two s e p a r a t e t e s t s d o e s n o t h a v e t h e same s i g n i f i c a n c e a s r e s p e c t i v e s c o r e s o f 70 a n d 30 o n t h e same t e s t s , a l t h o u g h t h e a v e r a g e s c o r e w o u ld be i d e n t i c a l . A s a r e s u l t , a n a v e r a g e s c o r e on s e p a r a t e t e s t s o r on d i f f e r e n t p a r t s o f t h e same g e n e r a l t e s t d o e s n o t fo r m a r e l i a b l e b a s i s f o r i n d i v i d u a l g u i d a n c e , • • T e a c h e r s a n d c o u n s e l o r s m u st b e c o n c e r n e d w i t h a l l o f t h e s e p a r a t e m e a s u r e s o f t h e i n d i v i d u a l t h a t may h a v e b e e n o b t a i n e d , ( 1 0 6 :4 7 8 ) An e x a m i n a t i o n now f o l l o w s t h e l i t e r a t u r e c o n c e r n i n g v a r i o u s a r e a s t h a t h a v e b e e n f o u n d t o b e i m p o r t a n t i n p r e d i c t i n g a c h i e v e m e n t , S h in n ( 1 1 3 :2 2 0 ) f o u n d t h a t t h e PMA h a s h i g h p r e d i c t i v e v a l u e f o r a c h i e v e m e n t . By a d d i n g t h e e f f e c t s o f t h e K u d e r P r e f e r e n c e R e c o r d , S h in n c o n c lu d e d t h a t t h e K u d e r p r e f e r e n c e R e c o r d may m ake a s u b s t a n t i a l c o n t r i b u t i o n w hen u s e d w i t h o t h e r t e s t s i n p r e d i c t i n g s u c c e s s i n h i g h s c h o o l f o r b o y s a n d g i r l s B e l e c t e d a c c o r d i n g t o s e x a n d g r a d e l e v e l ( 1 1 3 : 2 2 0 ) . B a rc h ( 2 8 :3 5 8 ) n o t o n l y f o u n d t h a t p e r s i s t e n c e t o c o m p le te a n a c h ie v e m e n t t e s t w as f o u n d t o b e s i g n i f i c a n t l y r e l a t e d t o t e s t a c h ie v e m e n t 22 b u t t h a t t h e p e r s i s t e n c e m e a s u re s i g n i f i c a n t l y im p ro v e d m u l t i p l e p r e d i c t i o n f o r a c a d e m ic a c h i e v e m e n t . O th e rs h av e fo u n d d i f f e r e n t a p t i t u d e t e s t s a s s u c c e s s f u l p r e d i c t o r s o f a c h ie v e m e n t ( 5 5 : 2 1 4 ) . Brogden has shown th e o r e tic a lly th at the use o f d iffe r e n tia l p red ictors w ill provide su b sta n tia l Increase in s e le c tiv e e f f ic ie n c y over that obtained w ith the use o f a sin g le p red ictor ( 3 6 : 1 8 2 ) . A ssociated w ith stu d ie s o f these types are corresponding a ttitu d e s towards m u ltip le measurement. In th is regard, Doppelt has sta te d , I t i s true that u sers of t e s t s have in crea sin g ly accepted the p rin cip le that m u ltip le measurement i s a more e f fe c t iv e aid to evalu ation and cou n selin g than a sin g le sco re. (56:261) Cain, M ichaeliB, and Eurich m aintain th a t, General m e n ta l-a b ility t e s t s are w idely used to p red ict a c h ild 's a b ilit y to succeed in d iffe r e n t elem entary school su b je c ts. Mental t e s t s alo n e, however, are Inadequate as a means o f p red ictin g achievem ent. ( 1 0 :8 7 8 ) Bond and Clymer have sta te d , I n m e n ta l t e s t i n g t h e r e h a s b e e n a g ro w in g d i s s a t i s f a c t i o n w i t h a s i n g l e m e a su re o f g e n e r a l m e n ta l a b i l i t y . T h is t r e n d h a s p ro d u c e d t h e A m e ric a n C o u n c il on E d u c a ti o n p s y c h o l o g i c a l E x a m in a tio n , t h e C a l i f o r n i a T e s t o f M e n ta l~ M a tu r lty T ~and many o t h e r t e s t s w h ic h p r o v i d e two o r m ore q u o t i e n t s 23 o r m e a s u r e s o f a b i l i t y . To e la te , t h e m o s t a m b i t i o u s o f a l l t h e m u l t i p l e m e a su re i n t e l l i g e n c e t e s t s a r e th o s e p r o d u c e d b y L .L . a n d T*G. T h u r s t o n e . ( 3 5 :1 3 2 ) W h ile t h e r e h a s b e e n t h e o r i z i n g a b o u t t h e im p o r ta n c e o f t h e p a r t - s c o r e s o f v a r i o u s t e s t s , c o m p a r t i v e l y l i t t l e u s e h a s b e e n made o f p a r t - s c o r e s o r o f f a c t o r e d t e s t s i n t h e p r e d i c t i o n o f a c h i e v e m e n t . S u c h v a r i a b l e s a s p ro b le m s o l v i n g ( 5 3 : 1 9 8 ) / s t u d y h a b i t s ( 4 5 : 1 2 6 ) , a n d s e n t e n c e c o m p le tio n t e s t s ( 5 1 :3 5 9 ) h a v e a l s o b e e n u s e d i n t h e a t t e m p t t o im p ro v e p r e d i c t e d a c h i e v e m e n t . I n g e n e r a l , m u l t i p l e a n d d i f f e r e n t i a l p r e d i c t i o n s h a v e n o t a d d e d a g r e a t d e a l t o m e n ta l a g e a s a p r e d i c t o r , a s f o r p h y s i c a l v a r i a b l e s , t h e summary b y S t r o u d may be q u o t e d : We a l s o know fro m a n a b u n d a n c e o f r e s e a r c h t h a t i n t e l l i g e n c e s c o r e s a n d e d u c a t i o n a l a c h ie v e m e n t s c o r e s a r e r a t h e r c l o s e l y i n t e r - c o r r e l a t e d , a n d t h a t n e i t h e r o f t h e s e i s c l o s e l y I n t e r c o r r e l a t e d w i t h m e a s u r e s o f p h y s i c a l g r o w th . M o re o v e r, we c a n im p ro v e p r e d i c t i o n o f e d u c a t i o n a l a c h ie v e m e n t som ew hat b y c o m b in in g i n t e l l i g e n c e a n d p a s t a c h i e v e m e n t . ( 1 1 9 :7 8 ) The w o rk o f G a te s ( 6 5 :3 5 7 ) show ed m ore t h a n t w e n t y - f i v e y e a r s a g o , t h a t p r e d i c t i o n i s n o t im p ro v e d by c o m b in in g i n t e l l i g e n c e s c o r e s w i t h v a r i o u s m e a s u re s o f p h y s i c a l 24 g r o w th , R ie s c h ( 1 0 7 :5 4 ) a f t e r u s i n g a m u l t i v a r i a b l e a p p r o a c h w i t h t e s t s o f p e r s o n a l i t y , a c h i e v e m e n t , a t t i t u d e , m e n ta l a b i l i t y , a n d a t t i t u d e t o p r e d i c t g a i n i n a c h ie v e m e n t, c o n c lu d e d t h a t " I t i s i m p o s s i b l e t o p r e d i c t g a i n i n a c h ie v e m e n t w i t h a n y a c c e p t a b l e d e g r e e o f a c c u r a c y " b y m eans o f m u l t i p l e r e g r e s s i o n e q u a t i o n s , I I I . LITERATURE RELATING THE MENTAL AGE AS A PREDICTOR N ot a l l w r i t e r s a g r e e on t h e I m p o r ta n c e o f th e m e n ta l a g e r e l a t i v e to e x p e c ta n c y o f a c h i e v e m e n t . Some I n v e s t i g a t o r s h a v e f o u n d r e l a t i v e l y low c o r r e l a t i o n s b e tw e e n m e n ta l a b i l i t y a n d c e r t a i n a r e a s o f a c h i e v e m e n t . E x a m p le s o f t h i s a r e g i v e n I n t h e s t u d i e s o f H a r r i n g t o n ( 7 9 :3 8 0 ) i n t h e a r e a o f l e a r n i n g t o r e a d , A l l e n (2 1 ) i n th e a r e a o f a r i t h m e t i c c o m p u t a t i o n , a n d Assum a n d Levy ( 2 6 :3 1 0 ) i n t h e a r e a o f a v e r a g e c o m p r e h e n s io n . R a m a se sh a n h a s s u g g e s t e d t h a t t h e r e a r e s y s t e m a t i c q u a l i t a t i v e d i f f e r e n c e s b e tw e e n b r i g h t a n d d u l l c h i l d r e n o f t h e same M A a n d t h a t t h e " v a l i d i t y , m e a n in g g e n e r a l i t y , o f t h e m e n ta l a g e c o n c e p t i s o p e n t o some q u e s t i o n , " ( 1 0 5 :5 8 ) 25 More ty p ic a l, however, are stu d ie s in d ic a tin g the in flu en ce th at M A has on achievem ent, Moreau ( 9 9 :1 7 6 ) found M A . to be a s lig h t ly b e tte r p red ictor o f reading than the Lee-Clark rea d in ess t e s t . Cross and Gaier (5 3 * 2 0 5 ) g ive evidence th at the in d iv id u a l who tend3 to use general p r in c ip le s would be superior to those depending upon the use of s p e c ific fa ctB , in school achievem ent. Thus, the mental t e s t , which presumably measures broad general a b i l i t i e s , would be an important Instrument in a n tic ip a tin g measuring school achievem ent, Stroud has sta te d h is p o s itio n rather fo r c e fu lly when he sa id , "I do mean to defend the M A as a h igh ly u se fu l u n it of measurement, 1 1 ( 1 1 9 :7 8 ) K olstoe ( 8 7 : 1 6 7 ) , a fte r in v e s tig a tin g the mental age as a u n it of measurement, supported oth er fin d in g s in of the u sefu ln e ss of the M A as a measure of le v e l o f mental a b i l i t y . B liesm er ( 3 3 :3 2 8 ) supported K o lsto e fs fin d in g s (although only p a r t ia lly w ith resp ect to su b sc o res). Brown and Bryan ( 3 9 :1 3 7 ) however, have found high c o r r e la tio n s between part scores and t o t a l mental a b i li t y sc o r e s. They found c o r r e la tio n s of .8 7 between IQ's and a ltitu d e q u o tien ts on the W echsler-B ellevue• 2 6 IV . LITERATURE REPORTING DIFFERENTIAL MENTAL TESTS AS PREDICTORS The techniques of d iffe r e n tia l p red ictio n have been developed as a means of fo r e c a stin g a stu d en t's r e la tiv e fu tu re sta tu s in sev era l areas of endeavor. In the l a s t tw en ty-five years considerable research has been done in an attem pt to I s o la te such mental t r a it s as memory, p ercep tion , verbal a b i lit y , and o th e r s. The vorkB o f Thurstone ( 1 7 ; l 8 9 , 2 3 6 ) , H olzinger ( 1 2 9 ) , Spearman ( 1 5 ) , Thomson ( 1 6 ) , G uilford ( 7 6 ) , and others have been concerned w ith the th e o r e tic a l is o la t io n of these a b i l i t i e s through the technique of fa c to r a n a ly s is . In regard to t h is , G uilford and Zimmerman have sta te d , W e b e lie v e th at the ap titu d es required fo r doing s u c c e ss fu lly the many kinds of task s in a complex s o c ie ty are much more numerous and varied than has g en era lly been supposed. ( 7 6 :2 4 ) Factor a n a ly sis has had great In fluence and undoubtedly w ill continue to have con siderab le in flu en ce on the problem o f p red ictio n , but fu rth er evidence i s e s s e n tia l to a sc e r ta in i t s r e a l v a lu e. To th is end, Doppelt has sta te d , 27 F a c t o r a n a l y s i s i s a v e r y p o w e r f u l t o o l f o r a r r i v i n g a t t e s t s w h ic h a r e r e l a t i v e l y i n d e p e n d e n t • • • b u t • • • a n o t h e r a d v a n c e i n t h e m e a su re m e n t o f a b i l i t i e s h a s b e e n t h e r e a l i z a t i o n t h a t f a c t o r a n a l y s i s a l o n e i s n o t t h e s o l u t i o n t o t h e p ro b le m o f o b t a i n i n g e f f e c t i v e d i f f e r e n t i a l t e s t s . We m u st s t i l l do c o r r e l a t i o n s t u d i e s f o r t h e p o p u l a t i o n s i n w h ic h we a r e i n t e r e s t e d b e f o r e we c a n b e r e a s o n a b l e s u r e o f I n d e p e n d e n c e o f m e a s u r e m e n t. ( 5 9 :2 6 1 ) I t i s n o t t h e p u r p o s e o f t h i s s t u d y t o e n t e r t h e c o n t r o v e r s y o f s i n g l e v e r s e s m u l t i p l e t e s t i n g . M u l t i p l e a p t i t u d e a n a l y s i s i s r e c o g n i z e d a s c o n t r i b u t i n g much t o t h e f i e l d o f m e n t a l t e s t i n g . H ow ever, s i n c e th e u s e o f t h e s e t e s t s i s r e l a t i v e l y r e c e n t , c o m p a red t o s i n g l e s c o r e t e s t s , a n d s i n c e t h e y a r e u s e d much l e s s t h a n s i n g l e s c o r e t e s t s , i t w o u ld f o l l o w t h a t t h e c o n t r i b u t i o n o f a n i n v e s t i g a t i o n u s i n g t h e s i n g l e s c o r e t e s t e x p e c t a n c i e s w o u ld b e n e f i t m ore c o n su m e rs a t t h i s t i m e . V. LITERATURE RELATING TO THE "JANGLE FALLACY" AND RELATED CONTROVERSY A lth o u g h t h e M A h a s b e e n u s e d a s a m e th o d o f e x p e c ta n c y f o r many y e a r s , t h e r e a r e many who h a v e a c c e p t e d t h i s p r o c e d u r e a n d o t h e r s who h a v e i n d i c a t e d t h a t t h e M A s e r v e s o n ly a s a d u p l i c a t e o f a c h ie v e m e n t 28 t e s t s c o r e s . Trum an K e l l e y , a s e a r l y a s 1 9 2 4 , w as c o n c e r n e d a b o u t t h e q u e s t i o n , "How much o f a c h ie v e m e n t i s I n t e l l i g e n c e ? " (9 ) The a t t e m p t t o a n s w e r t h e s e q u e s t i o n s l e d him t o c o i n h i s fam o u s p h r a s e " j a n g l e f a l l a c y . " He show ed t h a t w hen t h e c o r r e l a t i o n b e tw e e n a c h ie v e m e n t a n d i n t e l l i g e n c e w as o f t y p i c a l m a g n itu d e s , "no l e s s t h a n 90 p e r c e n t o f one I s th e same I n n a t u r e a s t h e o t h e r . " ( 9 : 6 3 ) He f u r t h e r s t a t e d t h a t s i n c e t h e co m m u nity b e tw e e n t h e s e two f u n c t i o n s I s n i n e t im e s a s g r e a t a s t h e d i s p a r i t y b e tw e e n th em , a n y ju d g m e n t o f d i f f e r e n c e b e tw e e n a c h ie v e m e n t a n d I n t e l l i g e n c e muBt b e b a s e d u p o n t h e 10 p e r c e n t o f e a c h n o t r e p r e s e n t e d i n t h e o t h e r , o r i t i s a s p u r i o u s ju d g m e n t. T h e r e f o r e K e lle y , m a i n t a i n s t h a t , The u s e o f tw o s e p a r a t e w o rd s o r e x p r e s s i o n s c o v e r i n g I n f a c t th e same b a s i c s i t u a t i o n , b u t s o u n d in g d i f f e r e n t , a s th o u g h t h e y w e re i n t r u t h d i f f e r e n t . • • i s c a l l e d th e " j a n g l e F a l l a c y . " ( 9 : 6 4 ) A lth o u g h K e l l e y d o e s n o t recom m end o r u s e th e a c c o m p lis h m e n t q u o t i e n t i n h i s t e x t he d o e s a c c e p t i t s u s e u n d e r tw o c o n d i t i o n s , (1 ) w hen th e d i f f e r e n c e b e tw e e n a p e r s o n 's t r u e a c h ie v e m e n t a n d h lB t r u e i n t e l l i g e n c e i s v e r y g r e a t a n d ( 2 ) i f t h e a c h ie v e m e n t 2 9 c a p a c i t y I s n o t g e n e r a l b u t s p e c i a l , e . g . , m u s ic c o m p u ta t i o n , s p e l l i n g , h a n d w r i t i n g , e t c . T h is j a n g l e t h e o r y h a s b e e n r e - e m p h a s i z e d a n d r e s t a t e d v a r i o u s t im e s t h r o u g h o u t t h e y e a r s s i n c e K e l l e y 's o r i g i n a l d i s c u s s i o n o f t h e s u b j e c t . C olem an a n d C u r e to n ( 4 7 :3 4 7 ) h a v e w r i t t e n r e c e n t l y on t h e s u b j e c t . T hey h a v e n o t o n l y c o n f i r m e d K e l l e y ' s p r e v i o u s f i n d i n g s b u t s u g g e s t t h a t t h e o v e r l a p b e tw e e n g r o u p - t e s t e d i n t e l l i g e n c e a n d a c h ie v e m e n t i s c l o s e r t o 95 p e r c e n t . M u l l i n s c o n c l u d e d , a f t e r o b t a i n i n g a h i g h d e g r e e o f r e l a t i o n s h i p b e tw e e n s p e e d o f r e a d i n g a n d q u a n t i t a t i v e s c h o l a s t i c a p t i t u d e , t h a t " i t i s e n t i r e l y p o s s i b l e t h a t some o r a l l o f t h e c o r r e l a t i o n c a n p e r h a p s b e t r a c e d t o g e n e r a l m e n t a l a b i l i t y . " ( 1 0 0 : 1 9 0 ) O th e r e d u c a t o r s a n d p s y c h o l o g i s t s r e g a r d th e M A a s s o m e th in g d i f f e r e n t f ro m a c h i e v e m e n t . ( 6 0 : 6 2 2 ) W ith r e s p e c t t o m e n t a l a b i l i t y , S t r o u d h a s s t a t e d t h a t " B i n e t r e g a r d e d t h e M A a s a g e n e r a l a v e r a g e . " ( 8 2 ; 7 7 ) S t r o u d a l s o a n s w e r s t h e a t t i t u d e e x p r e s s e d b y K e l l e y e t a l b y s t a t i n g , I t h a s b e e n s u g g e s t e d t h a t i n v iew o f t h e h i g h c o r r e l a t i o n s o b t a i n e d b e tw e e n g e n e r a l a p t i t u d e t e s t s a n d i n t e l l i g e n c e s c o r e s a n d 30 a l s o b e tw e e n g e n e r a l a c h ie v e m e n t t e s t s a n d i n t e l l i g e n c e s c o r e s , t h e i n t e l l i g e n c e t e s t i s no l o n g e r r e q u i r e d i n s c h o o l p r a c t i c e I n s o f a r a s p r e d i c t i o n o f a c h ie v e m e n t i s c o n c e r n e d . I t i s t r u e t h a t p r i o r a c h i e v e m ent p r e d i c t s f u t u r e a c h ie v e m e n t a t l e a s t a s w e l l a s d o e s t h e g e n e r a l i n t e l l i g e n c e t e s t . B u t i t i s e q u a l l y t r u e t h a t t h e i n t e l l i g e n c e t e s t w h ic h c a n b e a d m i n i s t e r e d i n a f r a c t i o n o f t h e tim e a n d a t a f r a c t i o n o f t h e c o s t p r e d i c t s f u t u r e a c h ie v e m e n t a l m o s t a s w e l l . The m e r i t s o f t h e a rg u m e n t m u st r e s t p r e t t y l a r g e l y u p o n t h e n a t u r e o f t h e e v id e n c e o f d i f f e r e n t i a l p r e d i c t i o n . T h is b r i n g s u s b a c k t o t h e I n t e l l i g e n c e s c o r e , , • P e r h a p s i t i s t r u e t h a t t h e g e n e r a l a c h ie v e m e n t t e s t a n d t h e g e n e r a l I n t e l l i g e n c e t e s t e a c h t e l l s u s s o m e th in g u n iq u e a b o u t t h e p u p i l a n d t h a t a r g u m e n ts a b o u t o n e * s r e p l a c i n g th e o t h e r c o u ld w e l l b e f o r g o t t e n , ( 1 1 9 :7 7 ) A lth o u g h t h e s t u d i e s o f N ash ( 1 0 2 : 2 6 ) , S t r a n g ( 1 1 8 : 4 4 0 ) , G re e n ( 7 2 :1 9 9 )# a n d o t h e r s h a v e r e v e a l e d a c e r t a i n l a c k o f r e l a t i o n s h i p b e tw e e n v a r i o u s i n t e l l i g e n c e t e s t m e a s u r e s , F i n d l e y h a s c o n c lu d e d : L e t u s s t i l l u s e o t h e r m e a s u re s o f r e a d i n e s s t o l e a r n i n t h e e a r l y s c h o o l g r a d e s t o h e l p u s a s c e r t a i n t h e c a u s e s o f l e a r n i n g d i f f i c u l t y a n d t h e p r o s p e c t s o f I n d i v i d u a l im p ro v e m e n t, b u t l e t u s r e c o g n i z e t h a t we c a l l v a r i o u s l y a c h i e v e m e n t , a b i l i t i e s a p t i t u d e s , i n t e l l i g e n c e , a s c o m p e te n c e s i n b a s i c m e d ia o f th o u g h a n d u t i l i z e m e a s u r e s o f them f o r a l l o f t h e i r p r o v e n f u n c t i o n s , ( 6 3 :2 5 8 ) K ra th w o h l ( 8 8 ) s u p p o r t s t h i s c o n t e n t i o n b y reco m m en d in g t h a t a n y t e s t f o r w h ic h t e s t s c o r e s h a v e some c o r r e l a t i o n c o u ld b e u s e d a s a b a s i s f o r e x p e c t a n c y . 31 A lth o u g h t h e " j a n g l e F a l l a c y " h a s b e e n c o n t r o v e r s i a l f o r m ore t h a n t h r e e d e c a d e s , t h e c o n f l i c t d o e s n o t h a v e d i r e c t c o n c e r n t o t h i s s t u d y . The r e a s o n f o r t h i s i s t h a t a n u m e r i c a l v a l u e i s u s e d a s a n e x p e c t a n c y , w h ic h i s i n d e p e n d e n t o f a n i n t e r p r e t a t i o n o f t h a t v a l u e . T h a t i s , i f some f u n c t i o n o f t h e M A i s u s e d a s a m e a s u re o f e x p e c t a n c y , w h e t h e r o r n o t t h i s M A i s m e a s u r in g " g e n e r a l a b i l i t y " o r " a c h ie v e m e n t a b i l i t y " l i e s i n i n t e r p r e t a t i o n a n d i s n o t i n h e r e n t i n t h e e x p e c ta n c y e q u a t i o n . V I . LITERATURE ILLUSTRATING THE EFFECTS OF CHRONOLOGICAL AGE ON MENTAL ABILITY AND ACHIEVEMENT S in c e CA i s one o f t h e v a r i a b l e s u s e d i n t h i s s t u d y , a r e v ie w o f t h e l i t e r a t u r e r e v e a l i n g th e r e l a t i o n s h i p o f CA t o M A a n d t o a c h ie v e m e n t w as I n v e s t i g a t e d . H orn (8 : 6 8 ) s t a t e s t h a t t h e a c h ie v e m e n t o f a c h i l d t e n d s t o r e g r e s s to w a r d t h e n a t u r a l m ean o f t h e g r o u p o r t h e l e v e l o f h i s own c h r o n o l o g i c a l a g e w hen u s e d i n c o n j u n c t i o n w i t h t h e r e g r e s s i o n e q u a t i o n i n v o l v i n g a c h i e v e m e n t a n d i n t e l l i g e n c e . She e q u a t e d t h e CA t o YQ cl ( t h e n a t u r a l m ean f o r r e a d i n g f o r t h e c h i l d ) . She a l s o 32 e q u a t e d t h e CA t o X„ ( t h e n a t u r a l m ean o f th e m e n ta l a g e f o r th e c h i l d ) • One r e p o r t s u p p o r t i n g H o r n 's t h e o r i e s , im p ly in g t h a t CA e f f e c t s a c h ie v e m e n t f o r a g iv e n g ra d e l e v e l , s t a t e s t h i s : E d u c a to r s h a v e l o n g b e e n a w a re t h a t c h r o n o l o g i c a l a g e a l s o a f f e c t s a c h i e v e m e n t . They h a v e n o t e d t h a t o l d e r p u p i l s g e n e r a l l y h a v e l o n g e r a t t e n t i o n s p a n s , b e t t e r c o o r d i n a t i o n , m ore p e r s i s t i n g I n t e r e s t , a n d g r e a t e r e m o tio n a l m a t u r i t y t h a n y o u n g e r p u p i l s o f i d e n t i c a l m e n ta l a g e s . T hese r e l a t i o n s h i p s o f c h r o n o l o g i c a l a g e t o a c t u a l a c h ie v e m e n t h a v e so u n d p s y c h o l o g i c a l b a s i s . ( 1 3 1 :4 8 ) C a r t e r ( 4 4 :1 0 2 ) a l s o s u p p o r t s t h i s t h e s i s . He fo u n d t h a t t h e c h r o n o l o g i c a l l y o l d e r c h i l d a p p e a r s t o h a v e t h e a d v a n ta g e i n a c a d e m ic a c h ie v e m e n t o v e r t h e y o u n g e r c h i l d w hen g i v e n th e same s c h o o l e x p e r i e n c e s . H ow ever, i t i s p o i n t e d o u t t h a t t h e s e d i f f e r e n c e s b e tw e e n a g e s o f c h i l d r e n , a s t h e y r e l a t e t o a c h ie v e m e n t, do n o t y i e l d c o n s t a n t e f f e c t s f o r v a r i o u s a g e s . T hese d i f f e r e n c e s a r e b o r n o u t i n B a e r s ' s t u d y o n u n d e r a g e a n d o v e ra g e s t u d e n t s w h ic h g i v e s e v id e n c e a n d d i f f e r e n c e s b e tw e e n t h e a v e r a g e a n d th e u n d e ra g e s t u d e n t s t e n d t o d e c r e a s e w i t h h i g h e r g ra d e l e v e l s . ( 2 7 :1 9 ) O th e r s t u d i e s do n o t s u p p o r t t h e h y p o t h e s i s t h a t 33 t h e o l d e r c h i l d t e n d s t o a c h i e v e b e t t e r t h a n t h e y o u n g e r f o r a g i v e n g r a d e o r a g i v e n m e n ta l a g e , A l l e n ( 2 1 : 2 3 7 ) , f o r e x a m p le , o b t a i n e d a m ore r e l i a b l e p r e d i c t i o n o f a c h ie v e m e n t b y u s i n g t h e M A r a t h e r t h a n t h e IQ . S t r a n g fo u n d t h a t c h r o n o l o g i c a l a g e w i t h i n a g r a d e l e v e l i s n o t a s s o c i a t e d w i t h r e a d i n g t e s t s c o r e s , e x c e p t s l i g h t l y i n th e c a s e o f t h e m a rk e d ly o v e ra g e p u p i l , S t r o u d ( 1 1 9 ) , a l t h o u g h a w a re t h a t o l d e r s t u d e n t s may h a v e m ore e x p e r i e n c e b e c a u s e t h e y h a v e s p e n t m ore tim e i n s c h o o l , m a i n t a i n e d t h a t , • , • Y oung, b r i g h t p u p i l s w o u ld s u r p a s s ( i n a c h ie v e m e n t) o l d e r , d u l l p u p i l s o f c o m p a ra b le M A 's, They a r e l i k e l y t o b e b e t t e r m o t i v a t e d , t h e y h a v e b e e n e x p e c t e d t o do w e l l i n s c h o o l , t h e y a r e m ore l i k e l y t o b e a c c u s to m e d t o s u c c e s s . I n g e n e r a l t h e y h a v e h a d b e t t e r e a r l y home t r a i n i n g a n d h a v e h a d m ore e n c o u ra g e m e n t a n d s t i m u l a t i o n a t hom e. (1 1 9 :8 1 ) F u r t h e r s u p p o r t f o r n e g a t i v e c o r r e l a t i o n s b e tw e e n c h r o n o l o g i c a l a g e a n d m e n ta l a g e a n d a c h ie v e m e n t i s g i v e n b y T a y l o r ( 1 2 0 :2 6 9 ) who fo u n d c o r r e l a t i o n s r a n g i n g fro m - . 1 ^ t o - . 3 4 b e tw e e n CA a n d M A i n a g i v e n g r a d e . H orn h a s d e m o n s tr a te d th e r e l a t i o n s h i p s among c h r o n o l o g i c a l a g e , m e n ta l a g e , a n d a c h i e v e m e n t , d ra w in g a t t e n t i o n t o t h e f a c t t h a t o l d e r c h i l d r e n w i t h l e s s a b i l i t y t e n d t o a c h i e v e b e y o n d t h e i r m e n ta l a g e e x p e c t a n c y ; y o u n g e r 34 c h i l d r e n w i t h s u p e r i o r a b i l i t y t e n d t o a c h i e v e b e lo w t h e i r m e n ta l a g e e x p e c t a n c y . I t may now be s e e n t h a t t h i s c o n c e p t b y H orn i n v o l v e s some r e d u n d a n c y s i n c e f o r a g i v e n g r a d e , o l d e r c h i l d r e n t e n d t o h a v e l e s s a b i l i t y a n d y o u n g e r c h i l d r e n t e n d t o h a v e m ore a b i l i t y . C h r o n o l o g i c a l a g e , t h e n , f o r a n y g iv e n g ra d e may n o t be a s a n I m p o r t a n t a f a c t o r a s h a s b e e n p r e v i o u s l y t h o u g h t . V I I . LITERATURE REPORTING THE USE OF THE IQ OVER A PERIOD OF TIME A r e v ie w o f t h e l i t e r a t u r e i n v e s t i g a t i n g t h e c o n s t a n c y o f t h e M A d i v i d e d by t h e CA i s p r e s e n t e d b e c a u s e i t I s I m p o r t a n t t o know w h e th e r o r n o t (1 ) e x p e c t a n c i e s c a n b e b a s e d on th e r e s u l t s o f I n t e l l i g e n c e t e s t s g i v e n a t a tim e c o n s i d e r a b l y p r i o r t o th e tim e a t w h ic h t h e m e a su re o f e x p e c ta n c y I s a p p l i e d , a n d w h e th e r ( 2 ) e x p e c t a n c i e s b a s e d u p o n one a b i l i t y t e s t g iv e s i m i l a r r e s u l t s t o t h o s e b a s e d u p o n a n o t h e r a b i l i t y t e s t s . H orn ( 8 : 8 7 ) , a f t e r t h o r o u g h l y r e v i e w i n g th e l i t e r a t u r e o f t h e 1 9 2 0 ‘s a n d 1 9 3 0 's c o n c lu d e d t h a t a l t h o u g h t h e p ro b le m o f t h e c o n s t a n c y o f t h e I n t e l l i g e n c e q u o t i e n t h a s b e e n fo u n d t o b e c o m p l i c a t e d a n d t h e r e a r e 35 c o n t r a d i c t i o n s i n t h e l i t e r a t u r e , , i t i s p o s s i b l e t o s t a t e t h a t t h e i n t e l l i g e n c e q u o t i e n t f o r a l l c l a s s e s o f c h i l d r e n i s r e m a r k a b ly c o n s t a n t # T h ese c o n c l u s i o n s a r e g e n e r a l l y s u p p o r t e d by s u c c e e d i n g s t u d i e s . A lth o u g h A l l e n f o u n d low c o r r e l a t i o n s b e tw e e n M A o r IQ i n G rad e I a n d G rade IV ( 2 2 : 2 5 4 ) , B a y le y f o u n d c o r r e l a t i o n s b e tw e e n i n t e l l i g e n c e s c o r e s e a r n e d a t e i g h t a n d a t e i g h t e e n y e a r s o f a g e t o r u n a r o u n d . 8 5 . ( 3 0 ;7 8 ) Gehman a n d M a ty as ( 6 6 :1 5 2 ) f o u n d B l n e t a n d WISC t e s t s a d m i n i s t e r e d a t G rade V a n d a g a i n a t G rad e IX t o t h e same p u p i l s y i e l d e d r e s u l t s " t h a t w e re r e l a t i v e l y c o n s t a n t a n d show ed e q u a l s t a b i l i t y o v e r t h i s i n t e r v a l o f tim e " a s d e te r m i n e d b y c o m p a r is o n s b e tw e e n m ean s c o r e s , t e s t e d - r e t e s t e d c o r r e l a t i o n s , a n d i n t e r c o r r e l a t i o n s o f m a jo r p a r t s # B ro w n 's s t u d y s u p p o r t s t h i s f i n d i n g b y o b t a i n i n g r e s u l t s o f a s t o n i s h i n g s t a b i l i t y o f I Q 's f o r 51 c h i l d r e n f i r s t t e s t e d i n th e f i r s t g r a d e w i t h t h e 1 9 1 6 S t a n f o r d R e v i s i o n o f t h e B i n e t S c a l e a n d r e t e s t e d a s c o l l e g e f r e s h m e n 12 y e a r s l a t e r w i t h e i t h e r t h e Army A lp h a o r t h e O tis * D i f f e r e n c e s b e tw e e n I Q 's f o r B ro w n 's s u b j e c t v a r i e d fro m 0 t o j u s t 9 , w i t h a m ean o f 3 * 8 8 . S t a n l e y 's s t a t i s t i c a l a n a l y s i s o f B r o w n 's s t u d y l e d him t o b e l i e v e 36 t h a t t h e r e s u l t s o f B ro w n 's s t u d y " a r e f r e a k r e s u l t s , " S t a n l e y p o i n t s o u t t h a t "much m ore i n s t a b i l i t y i s u s u a l l y f o u n d . " ( 1 1 7 :5 ^ 6 ) Honzlk, MacFarlane, and A llen , however, summarized a study they conducted by s ta tin g , "Mental t e s t constancy fo r the age period 21 months to 18 years i s markedly dependent upon the age a t te s tin g and the In terv a l between t e s t s ." That i s , group p red ictio n i s good over short age p erio d s, and mental t e s t scores become in c r e a sin g ly p r e d ic tiv e a fte r the preschool y ea r s. Thus, in gen eral, the IQ i s reported to be co n sta n t, but the degree of constancy reported by various stu d ie s d i f f e r s . A p recaution th at should be noted i s th a t. Whereas the r e s u lts fo r the group su ggest mental t e s t s t a b ilit y between 6 and 18 years, the observed flu c tu a tio n s in the scores of In d ivid u al ch ild ren In d icate the need fo r the utmost cau tion in the p r e d ic tiv e use of a sin g le t e s t sco r e, or even two such sc o r e s. ( 8 2 :3 1 5 ) This fin d in g seems o f s p e c ia l importance sin ce many plans fo r in d iv id u a l ch ild ren are made by sc h o o ls, ju v en ile c o u rts, and mental hygiene c lin ic s on the b a s is of sin g le mental t e s t sc o r e . S p e c ific a lly , i t could be noted that 37 a p r e d i c t i o n b a s e d on a 6 y e a r t e s t w o u ld b e w ro n g t o th e e x t e n t o f 20 IQ p o i n t s f o r one o u t o f t h r e e c h i l d r e n b y t h e a g e o f 18 y e a r s , a n d t o t h e e x t e n t o f 15 IQ, p o i n t s f o r a p p r o x i m a t e l y s i x o u t o f t e n c h i l d r e n . (8 2 ) T h e re i s e v id e n c e t o j u s t i f y t h e a s s u m p t i o n t h a t e x p e c t a n c i e s may b e co m p u ted I n d e p e n d e n t o f t h e s p e c i f i c i n t e l l i g e n c e t e 3 t u s e d . F o r e x a m p le , B a r r a t t a n d B a u m g a rte n ( 2 9 : 1 ^ ) f o u n d i n a s t u d y u s i n g t h e S t a n f o r d - B i n e t a n d t h e WISC, t h a t w i t h i n a c o m p a ra b le s c h o o l r e f e r r a l s e t t i n g , "The WISC a n d CTMM a r e m a rk e d ly c o m p a ra b le a s t o g ro u p a s s e s s m e n t a n d r o u g h l y c o m p a r a b le a s t o i n d i v i d u a l s c o r e s . " M a n o la k e s ( 1 1 2 :5 0 0 ) f o u n d t h a t " t h e r e i s n o t s i g n i f i c a n t d i f f e r e n c e b e tw e e n t h e m eans o f th e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y a n d t h e B i n e t . " A lth o u g h t h e s e s t u d i e s I n d i c a t e s i m i l a r i t i e s b e tw e e n c e r t a i n i n t e l l i g e n c e t e s t s , i t w o u ld be e r r o n e o u s t o g e n e r a l i z e t h e s e f i n d i n g s t o a l l i n t e l l i g e n c e t e s t s , V I I I . LITERATURE DESCRIBING DIFFERENCES IN ABILITY AND ACHIEVEMENT BETWEEN POPULATIONS T h a t d i f f e r e n t p o p u l a t i o n s s h o u ld be i n v e s t i g a t e d I n a n y s t u d y o f e x p e c t a n c y i s c o n f ir m e d b y s e v e r a l s t u d i e s 3 8 w h ic h i n v e s t i g a t e v a r i o u s c u r r i c u l a r a r e a s i n w h ic h p o p u l a t i o n s d i f f e r . S a x a n d O t t i n a ( 1 1 0 :1 8 ) f o u n d s i g n i f i c a n t d i f f e r e n c e s i n a r i t h m e t i c a c h ie v e m e n t b e tw e e n s t u d e n t s i n " c o n v e n t i o n a l " s c h o o l s a n d t h o s e i n " p r o g r e s s i v e " s c h o o l s . S a n d e r s , O s b o rn e , a n d G re e n e ( 1 0 8 : 1 8 9 ) f o u n d d i f f e r e n c e s b e tw e e n r u r a l a n d u r b a n s t u d e n t s w i t h r e s p e c t t o a c h i e v e m e n t . G ough ( 6 8 :5 5 7 ) f o u n d t h a t s o c i o - e c o n o m i c s t a t u s h a s a s l i g h t p o s i t i v e r e l a t i o n s h i p t o a c a d e m ic a c h i e v e m e n t . D a v i s , E e l l s , H a v i g h u r s t , H a g g a rd , e t a l ( 1 2 8 ) , c o n c l u d e d t h a t m ean d i f f e r e n c e s i n w h a t t h e y b e l i e v e d w as " i n n a t e " t e s t e d a b i l i t y b e tw e e n c h i l d r e n o f h i g h a n d low s o c i o - e c o n o m i c s t a t u s a r e i n s i g n i f i c a n t . B u t C o lem an a n d W ard ( 4 8 :4 6 7 ) t a k e e x c e p t i o n t o t h i s c o n c l u s i o n . T h e re i s some d e f i n i t e e v i d e n c e (4 ) t h a t t h e c h i e f r e a s o n f o r t h e s t a t u s d i f f e r e n c e s i n I Q 's may b e t h e d i f f e r e n t o p p o r t u n i t i e s w h ic h p u p i l s f r o m h i g h a n d low s t a t u s l e v e l s h a v e f o r f a m i l i a r i t y w i t h t h e k i n d s o f c u l t u r a l m a t e r i a l s a n d p r o c e s s e s r e p r e s e n t e d b y t h e u s u a l t e s t s . W h e th e r o r n o t t h e s e d i f f e r e n c e s a r e d u e t o t h e d i f f e r e n c e s i n t h e g r o u p s o r d i f f e r e n c e s i n t h e t e s t s i s 39 n o t t h e i s s u e o f t h i s s t u d y , s i n c e m e a s u re s o f e x p e c ta n c y t o d a t e a r e r e s u l t s o f m e a s u re d b e h a v i o r a l o u tc o m e s , n o t c a u s e s . T h a t a n a n a l y s i s o f e x p e c ta n c y s h o u ld I n v o l v e d i f f e r e n t p o p u l a t i o n s i s s u p p o r t e d b y L e n n o n . I n a t t e m p t i n g t o a n s v e r a num ber o f q u e s t i o n s , L e n n o n , r e g a r d i n g a c h ie v e m e n t a n d a b i l i t y w i t h r e s p e c t to c o m m u n itie s , c o n c l u d e d t h e f o l l o w i n g : 1 . " T h e re i s a m a rk e d te n d e n c y f o r c o m m u n itie s w h ic h a r e a b o v e a v e r a g e w i t h r e s p e c t t o IQ t o be a b o v e a v e r a g e i n a c h ie v e m e n t a s m e a s u re d b y t h e s e r e s p e c t i v e t e s t s . " 2 . "The v a r i a b i l i t y o f com m unity m ean I Q 's r e m a in s q u i t e c o n s t a n t t h r o u g h o u t t h e g r a d e s , t o b e s u r e , b u t t h e r e i s a c l e a r te n d e n c y f o r th e c o m m u n itie s t o becom e m ore v a r i a b l e i n a c h i e v e m en t a t t h e h i g h e r g r a d e s . " (S ee l a s t ro w s o f T a b le I . ) 3 . " F o r c o m m u n itie s , a s f o r i n d i v i d u a l s , t h e r e a r e m a rk e d d i f f e r e n c e s am ong s u b j e c t s i n a l l g r a d e l e v e l s . " ( 9 1 :3 0 6 ) IX . LITERATURE REPORTING THE VARIOUS MEASURES OF EXPECTANCY USED IN THE STUDY The MA. A r e v ie w o f t h e l i t e r a t u r e r e g a r d i n g t h e M A h a s b e e n I n h e r e n t I n t h e d i s c u s s i o n o f t h e p r e s e n t c h a p t e r . T h e r e f o r e , n o s p e c i f i c r e f e r e n c e i s made t o t h e l i t e r a t u r e a t t h i s t i m e . I t s h o u l d b e n o t e d , h o w e v e r, TABLE I MEAN AND STANDARD DEVIATIONS OP COMMUNITY MEAN STANDARD SCORES, BY GRADES, ON SUBTESTS OP THE METROPOLITAN ACHIEVEMENT TESTS BASED ON SEVENTY COMMUNITIES A f t e r L ennon (9 1 ) S t a n d a r d S c o r e s f o r A c h ie v e m e n t G rade R e a d in g Word M eaning A r l t h . F u n d . A r l t h . P r o b . S p e l l i n g IQ Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD I I 1 1 2 .5 5 . 8 1 1 2 1 .2 6 .2 1 3 7 .2 C V I C M . C O 1 0 0 .7 7 .2 I I I 1 4 1 .2 5 .7 1 4 2 .6 5 . 9 1 3 7 .8 2 . 5 1 4 9 .4 2 . 6 1 4 2 .7 6 .7 1 0 0 .7 5 .5 IV 1 6 4 .9 7 .1 1 6 5 .7 6 .6 1 5 4 .7 5 .1 I 6 5 .8 4 .1 1 6 3 .7 7 .7 9 8 .3 6 .9 V 1 8 0 .7 8 .1 1 7 9 .2 7 .6 1 7 1 .6 6 .3 1 7 8 .6 * . 9 1 7 9 .6 6 .2 1 0 0 ,7 6 .0 VI 1 9 4 .7 7 .4 1 9 6 .0 8 . 0 1 9 5 .4 1 0 .4 1 9 8 .2 6 . 8 1 9 6 .6 6 .3 9 9 .5 6 . 0 V II 2 0 4 .3 7 .5 2 0 4 .7 7 .3 2 2 0 .2 9 .4 2 1 1 .2 7 .2 2 1 0 .1 6 .9 9 8 ,9 5 .9 V I I I 2 1 4 .9 7 .1 2 1 6 .0 7 .2 2 4 2 .7 1 2 .3 2 2 3 .8 9 .3 2 2 8 .6 8 .4 9 9 .5 5 . 9 IX 2 2 1 .2 6 .4 2 2 4 .2 6 . 8 2 5 7 .4 1 4 .5 2 3 3 .3 9 .4 2 3 9 .0 9 .3 1 0 0 .0 5 .5 T Word R e c o g n i t i o n T e s t 2 N um bers T e s t 41 th a t the M A i s b a sic to a l l ex p ecta n cies used in t h is Btudy. The H o rn F o r m u l a . H o rn (8.*72) w as one o f t h e f i r s t t o p r o d u c e p r a c t i c a l f o r m u l a s f o r a p p l i c a t i o n o f e x p e c t a n c y t o c h r o n o l o g i c a l a g e , y i e l d i n g a n e x p e c t e d a c h i e v e m e n t (XA) w h ic h c a n a l s o b e e x p r e s s e d i n g r a d e p l a c e m e n t u n i t s (XAGP). ( 6 l 4 ; 4 8 ) H orn F o rm u la t a b l e s h a v e b e e n c o m p ile d a n d u s e d b y s e v e r a l s c h o o l s y s te m s (1 3 4 , 1 3 5 , 1 3 6 ) . A lth o u g h H orn u s e d a r e g r e s s i o n e q u a t i o n t o t a k e i n t o a c c o u n t r e g r e s s i o n e f f e c t s , s h e made c e r t a i n q u e s t i o n a b l e a s s u m p t i o n s i n h e r d e r i v a t i o n o f t h e f o r m u la XA« r x ^ ‘ + (l* * r^ y ) CA, w h ic h w as d e r i v e d f r o m t h e s o r i g i n a l e q u a t i o n , XA-Mx = (Yc -M y) r x y ~ • The f o u r y a s s u m p t i o n s i n q u e s t i o n a r e a s f o l l o w s ; 1 . The a s s u m p t i o n t h a t 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 r e a d i n g a g e i s e q u a l t o 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 m e n t a l a g e . 2 . The m ean a b i l i t y o f t h e g r o u p i s e q u a l t o t h e a g e o f t h e i n d i v i d u a l t o whom t h e e q u a t i o n i s a p p l i e d . 3 . The m ean r e a d i n g a c h ie v e m e n t o f t h e g r o u p i s e q u a l t o t h e a g e o f t h e i n d i v i d u a l t o whom t h e e q u a t i o n i s a p p l i e d . 4 . The c o e f f i c i e n t o f c o r r e l a t i o n b e tw e e n r e a d i n g a g e a n d m e n t a l a g e w h ic h i s u s e d f o r r e g r e s s i o n c o e f f i c i e n t s c a n b e u s e d I n d e p e n d e n t o f t h e 42 p o p u l a t i o n s w h ic h s u p p l i e d t h e c o e f f i c i e n t s o f c o r r e l a t i o n * T h a t t h e u s e o f H o r n 's c o e f f i c i e n t s a r e n o t u n i v e r s a l l y a p p l i c a b l e t o a l l ( o r c e r t a i n ) p o p u l a t i o n s h a s b e e n i n d i c a t e d b y S i s t e r J o s e p h l n a , C*S*J* ( 1 1 4 :8 6 7 ) Although Horn's form ulas u t i l i z e c e r ta in e f f e c t s o f r e g ressio n , another need th at she sta te d as being im portant, which was not taken in to account, was th at of r e l i a b i l i t y . ( 8 : 6 5 ) The L o rd G row th F o r m u la . W h ile i n v e s t i g a t i n g t h e t r u e g a i n b e tw e e n a c h ie v e m e n t t e s t s c o r e s , L o rd ( 9 2 :4 2 1 ) d e r i v e d a n e q u a t i o n i n w h ic h he u t i l i z e d t h e e f f e c t s o f t e s t r e l i a b i l i t y i n a d d i t i o n t o r e g r e s s i o n . He m a i n t a i n s t h a t i f t e s t s w e re c o m p l e t e l y r e l i a b l e t h e c o r r e c t p r o c e d u r e f o r o b t a i n i n g t h e t r u e g a i n o f t e s t s c o r e s w o u ld b e t o s im p ly s u b t r a c t t h e two s c o r e s . H ow ever, a s h e sh o w s, t h e f a c t t h a t " t e s t s c o r e s a r e n o t p e r f e c t l y r e l i a b l e o f t e n m akes t h i s 'o b v io u s * p r o c e d u r e p ro d u c e a b s u r d r e s u l t s , 1 1 a s a r e s u l t , L o rd i n d i c a t e s t h a t t h e r e i s a n e e d f o r a so u n d m eth o d f o r e s t i m a t i n g t h e " t r u e g a in " o f e a c h p u p i l , i . e . , t o o b t a i n a n e s t i m a t e o f h i s g a i n fro m w h ic h a n y b i a s due t o e r r o r s o f m e a su re m e n t h a s b e e n e l i m i n a t e d I n s o f a r a s p o s s i b l e , ( 9 2 :4 2 4 ) When t h e s e 43 same c o n c e p ts a r e a p p l i e d t o e x p e c ta n c y r a t h e r t h a n t o t h e d i f f e r e n c e b e tw e e n tw o a c h ie v e m e n t t e B t s , H o r n 's c o n c e r n a b o u t p r e v i o u s d e f i c i e n c i e s i n th e u s e o f r e l i a b i l i t y i n e x p e c ta n c y m e th o d s w o u ld b e r e a l i z e d , C a f f r e y ( 4 3 :1 7 8 ) s u p p o r t e d L o r d 's w o rk a n d l a t e r (1 3 3 ) a p p l i e d a s i m i l a r p r o c e d u r e w h ic h L o rd u s e d t o two d i f f e r e n t t e s t B ( a b i l i t y a n d a c h i e v e m e n t ) , c a f f r e y a l s o i n c l u d e d c h r o n o l o g i c a l a g e i n a n a d d i t i o n a l e q u a t i o n . T h ese e q u a t i o n s a r e d e r i v e d i n C h a p te r I I I , L o r d 's e s t i m a t e d t r u e g a i n e q u a t i o n t a k e s i n t o a c c o u n t t h e s e v e r a l I m p o r t a n t f e a t u r e s o f t e s t r e l i a b i l i t y , r e g r e s s ion*, a n d t h e p o p u l a t i o n o f w h ic h a p e r s o n I s a p a r t . The McNemar E q u a t i o n . McNemar ( 9 7 :4 7 ) h a s c r i t i c i z e d L o r d 's a r t i c l e on th e b a s i s o f (1 ) a d e r i v a t i o n o f th e b a s i c f o r m u l a s w i t h o u t t h e u n r e a l i s t i c a n d r e s t r i c t i v e a s s u m p tio n o f e q u a l i t y o f e r r o r v a r i a n c e f o r i n i t i a l a n d f i n a l s c o r e s , a n d ( 2 ) t h e c o m p l e x i t y o f L o r d 's d e r i v a t i o n a n d r e s u l t i n g f o r m u l a . McNemar a l s o c o n f ir m s C a f f r e y 's u s e o f a b i l i t y a n d a c h ie v e m e n t t e s t s I n t h e same d i f f e r e n c e r e g r e s s i o n e q u a t i o n w hen he (1 ) d e r i v e d a n e q u a t i o n s i m i l a r t o L o r d 's b u t n o t m a k in g t h e a s s u m p t io n o f e q u a l i t y o f e r r o r v a r i a n c e , a n d (2 ) r e f e r r e d t o m e n ta l g r o w th , e d u c a t i o n a l g r o w th , a n d g r o w th 44 d u e t o t r i a l a n d e r r o r l e a r n i n g a s b e i n g m e r g e a b le i n t o on e e q u a t i o n . A s im p le a p p r o x i m a t i o n t o L o r d 's f o r m u la h a s b e e n g i v e n b y McNemar ( 9 ? : 5 0 ) w h ic h may a l s o b e a p p l i e d t o e x p e c t a n c y . T h is e q u a t i o n i s d i s c u s s e d i n C h a p te r I I I . L o rd ( 9 3 ;4 3 9 ) l a t e r a g r e e d w i t h McNemar w i t h r e s p e c t t o t h e a s s u m p t i o n o f e q u a l i t y o f e r r o r v a r i a n c e , a c c e p t i n g M cN em ar's f o r m u la s a s p r e f e r a b l e t o t h e p r a c t i c a l w o r k e r . SUMMARY T h is c h a p t e r su m m a riz e s t h e l i t e r a t u r e r e l a t i n g t o t h e r e l a t i v e e f f e c t i v e n e s s o f m e n ta l a g e a s a b a s i s f o r e x p e c t a n c y . The f i r s t p a r t o f t h e c h a p t e r r e c o g n i z e s a l a r g e nu m b er o f v a r i a b l e s t o g e t h e r w i t h m e n ta l a g e w h ic h c o u l d c o n t r i b u t e t o t h e p r e d i c t i o n a n d e x p e c ta n c y o f a c h ie v e m e n t t e s t s c o r e B . I t i s e v i d e n t h o w e v e r, t h a t m any o f t h e s e s t u d i e s w e re c l i n i c a l l y o r i e n t e d a n d n o t a lw a y s d i r e c t l y a p p l i c a b l e t o t h e s c h o o l s e t t i n g , e s p e c i a l l y s i n c e m any o f t h e v a r i a b l e s t h e m s e lv e s a r e r e l a t i v e l y u n s t a b l e , f l u c t u a t e w i t h c o n d i t i o n s , a n d a r e n o t t y p i c a l l y f o u n d i n s c h o o l s i t u a t i o n s . The v e r y n a t u r e o f e x p e c t a n c y b y u s i n g a m e n t a l 45 a b i l i t y t e s t a s a b a s i s f o r a n t i c i p a t i n g a c h ie v e m e n t l e v e l s h a s b e e n a t t a c k e d b y some i n v e s t i g a t o r s , o t h e r w o r k e r s i n t h i s a r e a , h o w e v e r, h a v e t a k e n a n o p p o s in g p o s i t i o n . A lth o u g h t h e l i t e r a t u r e i n t h i s som ew hat c o n t r o v e r s i a l a r e a w as p r e s e n t e d , i t i s som ew hat a c a d e m ic i n t h a t th e c o n t r o v e r s y i s p r i m a r i l y a n a t t a c k on th e nam es a t t r i b u t e d to t e s t s . F o r th e p u r p o s e o f t h i s s t u d y th e t e s t s c o u ld b e l a b e l e d T e s t 1 a n d T e s t 2 w i t h o u t a n y r e f e r e n c e t o th e t y p e s o f t e s t s , t h u s s i d e s t e p p i n g t h e j a n g l e f a l l a c y , a t l e a s t u n t i l f i n a l i n t e r p r e t a t i o n . Although ch ron ological age i s q u ite d ir e c tly r e la te d to mental age over a long range of a g es, there i s evidence to the contrary fo r any given grade. There i s a lso evidence th at fo r any given grade ch ron ological age has l i t t l e re la tio n sh ip to achievement t e s t sc o r e s. Although the IQ i s r e la t iv e ly sta b le over a period of tim e, i t s co n sisten cy decreases w ith tim e. Thus, i f mental age i s to be used as a fu n ctio n o f expectancy, in general i t would have maximum u t i l i t y immediately fo llo w in g te s tin g w ith i t s accuracy d ecreasing w ith tim e. CHAPTER I I I EQUATIONS AND DERIVATION CP EQUATIONS T h is c h a p t e r i s c o n c e r n e d o n l y v l t h t h e e q u a t i o n s u s e d i n th e p r e s e n t s t u d y , t h e d e r i v a t i o n o r t h e s e e q u a tio n s a n d a s s u m p tio n s n e c e s s a r y t o d e v e l o p th e m . A l l o f th e e x p e c ta n c y e q u a t i o n s h a v e t h e u n d e r l y i n g a s s u m p tio n t h a t d i f f e r e n t l e v e l s o f a c h i e v e m e n t c a n h e " e x p e c te d ” fro m p u p i l s h a v i n g v a r i o u s a b i l i t y l e v e l s a s i n d i c a t e d fro m th e r e s u l t s o f a n i n t e l l i g e n c e t e s t , I , DERIVATION OF THE ESTIMATED TRUE DIFFERENCE EQUATION AS A FUNCTION OF ABILITY AND ACHIEVEMENT The a s s u m p tio n s m ade f o r t h e f o l l o w i n g d e v e l o p m e n t a r e t h a t (L o rd , 9 2 ; 4 2 4 ) , 1 , The a c h ie v e m e n t t e s t s c o r e (Y ) a n d t h e a b i l i t y t e s t (X) a r e e x p r e s s e d i n c o m p a r a b l e u n i t s . 2 , The e r r o r s o f m e a s u r e m e n t f o r t h e a c h i e v e m e n t t e s t (Y) a n d t h e a b i l i t y t e s t ( X ) : (a ) Have z e r o m e a n I n t h e g r o u p t e s t e d , 1*7 (b ) A re u n c o r r e l a t e d w i t h e a c h o t h e r a n d v l t h t r u e s c o r e s (x* a n d Y*) on e i t h e r t e s t , 2 X*e 2 X*e — " ft’ = ------- ft— = N N 2 Y*e,r 2 Y*e 2 e e — ^ » 0 , — — = 0 , * 0 , N N N 3 . a n d t h a t t h e c o n d i t i o n s , s 2 ■ s 2 (1 - r _ _ ) a n d s 2 = s 2 ( l " r m ,) e x x v x x e y y x y y 7 a r e c h a r a c t e r i s t i c s o f t h e t e s t s . The t r u e d i f f e r e n c e ( 6 ) f o r a p u p i l i s d e f i n e d a s t h e d i f f e r e n c e b e t v e e n h i s t r u e s c o r e on t h e a b i l i t y t e s t a n d h i s t r u e s c o r e o n t h e a c h ie v e m e n t t e s t . I f XQ i s Q» t h e o b s e r v e d s c o r e on t h e a b i l i t y t e s t , a n d X* i s t h e c L t r u e s c o r e on t h e a b i l i t y t e s t , th e c o r r e s p o n d i n g e r r o r o f m e a su re m e n t (e ) i s d e f i n e d a s * a V ■ Xa ‘ x 2 * W a S i m i l a r l y , i f Y_ i s t h e o b s e r v e d s c o r e o n t h e a c h ie v e m e n t c L t e s t , t h e c o r r e s p o n d i n g e r r o r o f m e a su re m e n t (e ) i s ya d e f i n e d a s e ya - Ya - X* - Ba (2 ) I f e q u a t i o n ( 1 ) I s summed o v e r a l l p u p i l s a n d d i v i d e d b y t h e num ber o f p u p i l s (N ), I t I s s e e n t h a t 2 e x ZX ZX*. ( 3 ) N * N " N P rom t h i s a n d f r o m t h e a b o v e a s s u m p t io n t h a t Ze = 0 , I t I s c l e a r t h a t N X* s X , (4 ) w h e re a " b a r " I s u s e d t o i n d i c a t e t h e m ean v a l u e . S i m i l a r l y , I f e q u a t i o n (2 ) i s summed o v e r a l l p u p i l s a n d d i v i d e d b y t h e num ber o f p u p i l s (N ), i t I s s e e n t h a t Z ey ZY Z5 . (5 ) N * N N " N I t fo llo w s from th is and from the above assum ptions t h a t _ _ _ _ _ _ 6 ■ Y - X* ■ Y - X . ( 6 ) E q u a t i o n ( 1 ) a n d ( 2 ) c a n e a s i l y b e w r i t t e n I f t h e r i g h t a n d l e f t h a n d m em bers o f e q u a t i o n (7 ) a r e s q u a r e d , a s v e i l a s t h o s e o f e q u a t i o n (8 ) , a n d i f t h e r i g h t a n d l e f t h a n d m em bers o f e q u a t i o n s (7 ) a n d ( 8 ) a r e m u l t i p l i e d t o g e t h e r , t h e p r o d u c t s becom e x a = x f + 2XS \ + ( » ) A ' 4 + A + \ + 2X2 6a + 2Xa e ya + 2 6 a e ya ' (9 b ) a n d 2 X Y r X* + X*B + X*e + X*e + e e • ( 9 c ) a a a a a a v a x x y„ v ' rfa a a ' a I f t h e e q u a t i o n s ( 9 ) a r e summed o v e r a l l p u p i l s a n d d i v i d e d b y t h e s q u a r e o f t h e num ber o f p u p i l s i t i s e v i d e n t t h a t ZX2 ™ «2 2 SX*e JC 2 e x a n d (1 0 ) A Z £ X * + 1 A ■ + -7 5 s - N N N ZXY SC*2 , SX.6 , « • » , , ^ * ex , 26ex , ‘ V 1 = " A1 + * A" - + 1 1 A1 *■ + Tj + A— + * 'g " " i r IT N N N N I t i s a l s o s e e n fro m e q u a t i o n (4 ) a n d ( 6 ) t h a t 50 a n d X Y ■ B X* + X*2 , (1 3 ) From t h e a s s u m p t io n s g i v e n a b o v e a n d t h e d e f i n i t i o n s o f v a r i a n c e a n d c o v a r i a n c e a n d a f t e r s u b t r a c t i n g e q u a t i o n s ( 1 1 ) , (1 2 ) a n d (1 3 ) f ro m e a c h s i d e o f e q u a t i o n (1 0 ) (w h ere a p p r o p r i a t e ) , i t i s a p p a r e n t t h a t 2 2 2 s * s _ + s . x x* e * s y ’ + *6 + Be + 2fW a n d (1 4 ) y 2 s * * s „ + s „6 + s s • x y x * x * e e x y S in c e t h e r e a r e t h r e e e q u a t i o n s i n v o l v i n g 2 2 (s s c . a n d s „ _ ) , t h e s e te r m s c a n b e s o l v e d f o r t h r e e ' X* 0 X *D unknow ns b y s t r a i g h t f o r w a r d a l g e b r a i c m a n i p u l a t i o n t o g iv e 2 2 2 4 ■ * 1 - v (1 5 ) A i o o P 2 P s f ■ B - 2 s + a - * - s , (1 6 ) 6 x x y y e y ' s ■ s - s 2 + s 2 . (1 7 ) x *6 x y x e K J 5 1 The g e n e r a l f o r m u la ( 7 : 2 6 ) f o r r e l i a b i l i t y I s d e f i n e d a s 2 r u u ■ 1 ' ~ t * ( l 8 ) u where ruu 13 a gen eral exp ression of r e l i a b i l i t y . 2 2 The value of s and s can be found r e s p e c tiv e ly , when e x y th e r e l i a b i l i t i e s o f t h e a b i l i t y t e s t ( r ^ ) a n d th e a c h ie v e m e n t t e s t ( r ) a r e know n a s w e l l a s t h e v a r i a n c e s ' yy o f e a c h o f t h e s e t e s t s . I t fo llo w s th at s e * s x <1 9 > and sey = Sy ( ^ y y ) - < 20> I t i s now e v i d e n t t h a t fro m e q u a t i o n s ( 5 ) , ( 6 ) , ( 1 5 ) , ( 1 6 ) , ( 1 7 ) , ( 1 9 ) , a n d (2 0 ) t h a t m e a n s , v a r i a n c e s a n d c o v a r i a n c e s o f th e q u a n t i t i e s X* a n d 6 _ c a n now b e ft ft 2 2 2 2 2 e x p r e s s e d e n t i r e l y i n te r m s o f s , s , s , s , s , x ex y ey *y r , and r • These equations can then be used to ob tain x x yy fro m t h e a b i l i t y a n d o b s e r v e d a c h ie v e m e n t s c o r e a l e a s t s q u a r e l i n e a r r e g r e s s i o n e s t i m a t e o f t h e t r u e d i f f e r e n c e 52 o f e a c h p u p i l , p ro m L o rd ( 9 2 ;4 2 5 ) i t i s s e e n t h a t i f X *, 6 , e a n d e a r e j o i n t l y n o r m a l ly d i s t r i b u t e d , t h e x y r e g r e s s i o n o f t r u e d i f f e r e n c e on t h e o b s e r v e d s c o r e s w i l l a c t u a l l y b e l i n e a r . A ls o i t i s n o t e d t h a t t h i s p r o c e d u r e d o e s n o t a ssu m e l i n e a r i t y o r n o r m a l i t y s i n c e t h e l i n e a r r e g r e s s i o n e s t i m a t e w i l l o f t e n p r o v i d e a u s e f u l a p p r o x i m a t i o n , e v e n i f t h e t r u e r e g r e s s i o n i s n o t s t r i c t l y l i n e a r . The p a r t i a l r e g r e s s i o n c o e f f i c i e n t s n e c e s s a r y f o r t h e r e g r e s s i o n e q u a t i o n w h ic h w o u ld p r e d i c t a n a c h i e v e m e n t s c o r e fr o m a n a b i l i t y s c o r e a r e g i v e n a s ( 5 :4 2 1 ) s r r r 6 6x - 6y x y \ - 7T • ------------o — * ( 2 1 ) B x .y s x 1 _r a xy V .* = £ • ^ 9 * ^ • < 22> y x y Some o f t h e q u a n t i t i e s o f t h e s e e q u a t i o n s a r e g i v e n i n t r u e s c o r e t e r m s . I n o r d e r t o e x p r e s s t h e s e q u a n t i t i e s i n te r m s o f m ore c o n v e n i e n t raw s c o r e d a t a , f u r t h e r e x p r e s s i o n s n e e d t o b e o b t a i n e d . E q u a t i o n s (7 ) a n d (8 ) a r e m u l t i p l i e d b y 6 _ t o ct g iv e Ba Xa ■ 6a x2 + 6a e x <2 3 ) a 53 a n d 2 5a Ya * 5a Xa + 6a + 5a e v • r e s p e c t i v e l y , (2 4 ) B . I f e q u a t i o n s (2 3 ) a n d (2 4 ) a r e summed o v e r a l l p u p i l s a n d d i v i d e d b y t h e num ber o f p u p i l s , i t 1 b s e e n t h a t S6x = S6x* (25) s6y ■ S6x* + s6 < 26> From e q u a t i o n s ( 2 5 ) , ( 1 7 ) , ( 2 6 ) , a n d (1 6 ) i t i s a p p a r e n t t h a t b6x a sxy - sx + B t < 27> Jv a n d s_ = s 2 - s - s 2 (2 8 ) 6y y x y e y v ' E q u a t i o n (2 1 ) a n d ( 2 2 ) , co m b in e w i t h e q u a t i o n s (2 7 ) a n d ( 2 8 ) , a n d w i t h t h e c o e f f i c i e n t o f c o r r e l a t i o n f o r m u la w hose g e n e r a l fo rm may b e w r i t t e n s mn 1 1 1 1 1 ” s s ' m n t o p r o v i d e a m ore c o n v e n i e n t fo rm o f t h e p a r t i a l r e g r e s s i o n c o e f f i c i e n t : 2 2 2 S 8 + 8 B e_ _ y ev xy ’ 'J v - ,* ‘ 1 (2 9 ) x y xy 5 4 6y . x 2 2 s S + s B eX XJ eY X s 1 I s — s 2 - 8 S - S x y x y (3 0 ) The e q u a t i o n f o r t h e b e s t l i n e a r e s t i m a t e o f t h e t r u e d i f f e r e n c e (B1) b e tw e e n th e s c o r e on a n a b i l i t y t e s t a n d a s c o r e on a n a c h ie v e m e n t f o r a n y p u p i l ( 6 1) u n d e r t h e g i v e n c o n d i t i o n s i s g i v e n b y t h e r e g r e s s i o n e q u a t i o n . ( 9 2 :4 2 7 ) 8a ■ 8 + b 6x . y < x # > + b 6y . x <3 1 > = b8x.y Xa + b 6y . x Ya + K' v h e r e K . Y < l - b By. x ) - X (1 + t>6 x . y ) . I t i s n o t e d i n p a s s i n g t h a t th e m ean e s t i m a t e d t r u e d i f f e r e n c e f o r a p o p u l a t i o n c a n b e e x p r e s s e d a s 6 * = 6 = Y - X, (3 2 ) when the mean in te llig e n c e t e s t scoreB and the mean achievem ent t e s t scores are used in equation ( 3 1 ) . 55 I I . DERIVATION OF THE EQUATION OF THE RELIABILITY FOR THE ESTIMATED TRUE DIFFERENCE EQUATION AS A FUNCTION OF ABILITY AND ACHIEVEMENT The r e l i a b i l i t y o f e s t i m a t e d t r u e d i f f e r e n c e ( r e f e l ) w i l l b e e q u a l t o t h e s q u a r e o f t h e c o r r e l a t i o n 0 0 b e tw e e n t r u e d i f f e r e n c e s a n d e s t i m a t e d t r u e d i f f e r e n c e s ( r 5 5 , ) ( 9 2 :4 2 9 ) a n d ( 6 : 4 7 8 ) . r 6 ' 6 ' s R B .* y (33 I t f o l l o w s t h a t 5 ' i s t h e l e a s t s q u a r e e s t i m a t e f o r p r e d i c t i n g 6 a n d x a n d Y* From t h e b a s i c f o r m u la o f m u l t i p l e c o r r e l a t i o n i t i s known ( 5 :4 2 5 ) t h a t r 6 , 6 * s R5 .x y * b 6x . y r &x s ^ + b 6 y . x r &y s^ - * E x p r e s s i n g t h e c o r r e l a t i o n c o e f f i c i e n t s i n te r m s o f v a r i a n c e s a n d c o v a r i a n c e s a n d s i m p l i f y i n g t h e r e s u l t s i t i s s e e n t h a t (3 5 ) From O u l l l k s o n ( 7 : 3 5 2 ) i t f o l l o w s t h a t 2 ( 3 5 a ) s 56 2 w h e re s i s t h e s t a n d a r d e r r o r o f m e a su re m e n t o f th e V t r u e d i f f e r e n c e . I I I . DERIVATION OF THE ESTIMATED TRUE DIFFERENCE EQUATION AS A FUNCTION OF ABILITY, ACHIEVEMENT, AND CHRONOLOGICAL AGE The p r e c e d i n g d e r i v a t i o n o f t h e m u l t i p l e r e g r e s s i o n e q u a t i o n i n v o l v e d t h e M A a n d t h e a c h ie v e m e n t t e s t s c o r e . The CA may a l s o h e a d d e d t o t h i s e q u a t i o n b y u s i n g t h e r e g u l a r p r o c e d u r e f o r a t h r e e v a r i a b l e m u l t i p l e r e g r e s s i o n e q u a t i o n . By u s i n g t h e same e q u a t i o n a n d d e r i v a t i o n s (up t o e q u a t i o n ( 3 1 ) , e x c e p t f o r e q u a t i o n (2 1 ) , (2 2 ) , (2 9 ) , a n d (3 0 ) a s t h o s e u s e d i n t h e d e v e lo p m e n t o f th e p r e c e d i n g r e g r e s s i o n e q u a t i o n t h e f o l l o w i n g e s t i m a t e d d i f f e r e n c e e q u a t i o n ( i n v o l v i n g MA, CA a n d a c h ie v e m e n t t e s t s c o r e s ) w o u ld h o l d ; 6a « b6*.yc x + b6y.*c Y + b 6c.xy C + K < 36> where K . ? (l-DBy>xo) - X (1+ t> 6x<yc) - 0 (b60>xy). C , o f c o u r s e , i s t h e sym bol f o r c h r o n o l o g i c a l a g e a n d C, i s t h e a v e r a g e c h r o n o l o g i c a l a g e f o r t h e g r o u p . 57 Since V . y c ' b6y.xc' and b6c.xy are aot e x p r e s s e d i n te r m s o f known q u a n t i t i e s , t h e y w i l l be d i s c u s s e d a t t h i s t i m e . E q u a t i o n s w h e re e a c h o f t h e s e b ' s a r e r e p r e s e n t e d a s f u n c t i o n s o f known o r a v a i l a b l e q u a n t i t i e s a r e o b t a i n e d a s f o l l o w s : The g e n e r a l e x p r e s s i o n f o r b_ i s g i v e n a s ox#yc ( 5 :4 2 1 ) . (3 7 ) Bx.yc Bx.yo a^ B e c a u se o f th e r e l a t i o n s h i p s r 6x.yc * " r6C‘y r ™ ’ 7 ' (5:415) (38) y ) ( 1 - r ^ c# y ) S6 .y c “ s 6 \ R 7 n /1 ’ 1,6 0 .y • ‘5 :4 1 8 ) (3 9 ) s s s x .y c x \ / l - r 2 \ l 1 - r 2 , (4 0 ) V x y y x c . y ' ' ' a n d , i n g e n e r a l , *12.3 “ r 12 ~ r 1 3 r 23 (4 1 ) 58 e q u a tio n . (3 7 ) may b e w r i t t e n , b &x c e q u a l s yc B6 ^ * * r c y ^ r 6x r 6 y r x y ^ " S6^r &c r 6yr c y ^ r xc r x y r cy^ s ( 1 - r 2 - r 2 - r 2 + 2 r r r ) x v x y xc yc yc x y x c ' E q u a t i o n (4 2 ) c a n be e x p r e s s e d b 6x e q u a l s , 2 2 2 2 2 \ s (s s s K - s s_ s - s s R - s s s_ + s s s_ + s s c s ) x v y c Bx c By x y c y 6x y xc Be x y c y By xc By c y ' 2 p 2 2 2 2 2 2 2 „ e , (s s s - s s - s s - s s + 2s s s ) v x y c c x y y xc x yc yc xy x c ' (4 3 ) b y th e s u b s t i t u t i o n o f v a r i a n c e s a n d c o v a r i a n c e s i n p l a c e o f c o r r e l a t i o n s b y m eans o f t h e r e l a t i o n s h i p ( e x p r e s s e d i n g e n e r a l te r m s ) a s s nm r nm * s s * n m The v a lu e s ^ c i s n o t e x p r e s s e d i n te r m s o f known o r e a s i l y a t t a i n a b l e q u a n t i t i e s . I t i s n e c e s s a r y , t h e n a t t h i s t i m e , t o a t t a i n a n e x p r e s s i o n i n v o l v i n g s &c w h ic h i s e x p r e s s e d i n te r m s o f q u a n t i t i e s known f o r a g iv e n p o p u l a t i o n . 59 I f e a c h s i d e o f e q u a t i o n (7 ) a n d (8 ) a r e m u l t i p l i e d b y C , r e s p e c t i v e l y , th e e q u a t i o n s a r e t h e n e x p r e s s e d a s S I ° a V C aX l + C a ex and (W) a Ca Ya ■ ° a X£ + Ca 6a + Ca e y * ( « > o > By sum m ing, r e s p e c t i v e l y , e q u a t i o n (4 4 ) a n d (4-5) o v e r a l l p u p i l s a n d d i v i d i n g b y t h e num ber o f p u p i l s , i t f o l l o w s ( w ith th e a s s u m p t io n s o f z e r o c o r r e l a t i o n s b e tw e e n t h e e r r o r v a r i a b l e a n d o t h e r v a r i a b l e s ) t h a t s c x * s 3c x a n d <4 6 ) s c 6 " s oy ' a o x * ' E lim inating s from equations (4 b ) and (4 7 ) gives ex* th e r e s u l t i n g e q u a t i o n o f s_ «s-s„ . (4 8 ) c 6 cy c x v 1 I f e q u a t i o n s ( 4 8 ) , (2 5 ) a n d (2 6 ) a r e s u b s t i t u t e d i n e q u a t i o n (4 3 ) a n d a f t e r s t r a i g h t - f o r w a r d b u t r a t h e r t e d i o u s a l g e b r a i c m a n i p u l a t i o n t h e r e s u l t may be w ritten Se xB ( = 0 + Sey D(* y ) V . y c ■ -5 ^ ----------------------- 1. (*9) 60 w h e re D(x x ) - a V - b ^ , (5 o ) 2 D/ \ ■ s s - s s a n d (5 1 ) (x y ) c xy xc yc v ' Q O O ? P P P P P D s a s s + 2 a s s - a s - s S - s S * . (5 2 ) xyc x y c xy yc xc c xy y xc x yc ' ' By a s i m i l a r p r o c e d u r e i t i s f o u n d t h a t 2 2 SexV y ) + se D(yy) b By>xc - 1 --------------- *---------- a n d ( 5 3 ) xyc s exD(xc) + *e D(ye) b 6 c .x y “ ------------ ' x y c w h e re D(y y ) . a V - a ^ . 2 D/ v * s a - a s . a n d (5 6 ) (x c ) x y yc xc y ' w ' D/ \ a a2 s - s a . (5 7 ) ( y c ) x yc xc x y v ' E q u a t i o n s ( 3 6 ) , ( 4 9 ) , ( 5 3 ) , a n d (5 4 ) y i e l d a n e q u a t i o n w h e re 61 2 2 & ■ = f (XjY.C.s s , s , s , s , s , s , a n d s ) , (5 8 ) ua ^ x y ' c x y ' x c ' y c ' e ' e " v ' x y t h a t I s , 6& i s a f u n c t i o n o f e a c h o f t h e s e q u a n t i t i e s ( i n s i d e t h e p a r e n t h e s i s a n d i f t h e y a r e known, i t i s p o s s i b l e t o c a l c u l a t e 5 1• £l IV . DERIVATION OP THE RELIABILITY EQUATION FOR THE ESTIMATED TRUE DIFFERENCE EQUATION AS A FUNCTION OF ABILITY, ACHIEVEMENT, AND CHRONOLOGICAL AGE The d e r i v a t i o n o f th e e q u a t i o n o f th e r e l i a b i l i t y f o r t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a s a f u n c t i o n o f t h r e e v a r i a b l e s ( a b i l i t y , a c h ie v e m e n t, a n d c h r o n o l o g i c a l a g e ) i s s i m i l a r t o t h a t o f e q u a t i o n (3 5 ) w h ic h i s a f u n c t i o n o f two v a r i a b l e s ( a b i l i t y a n d a c h i e v e m e n t ) . I t I s known ( 9 2 :4 2 9 , b :4 7 8 ) t h a t ^ ' B 1 = R& .x y c ^ 9 ) a n d ( 5 : 4 2 5 ) t h a t R6 .x y c * b 6x . y c r 6x s^~ + b 6 y . x c r 6y + ^Bc.xy^^dc s^~* 62 E x p r e s s i n g t h e c o r r e l a t i o n c o e f f i c i e n t s i n te r m s o f v a r i a n c e s a n d c o v a r i a n c e s a n d s i m p l i f y i n g t h e r e s u l t s i t i s s e e n t h a t s b + s b + s b r » 5x s x .y c By B y .x c 5c 5c .x y ^ SB Once a g a i n , i t f o l l o w s t h a t a e_ , ' a 6 ( 6 l a ) 0 V. DERIVATION OF THE RELIABILITY OF THE HORN FORMULA I t w as n e c e s s a r y f o r t h e p r e s e n t i n v e s t i g a t o r to d e r i v e a n e q u a t i o n f o r t h e r e l i a b i l i t y o f t h e H orn f o r m u l a • The g e n e r a l fo rm o f th e H orn F o rm u la i s g i v e n a s (8 :68) ax_ + bn X - " a TV ' < 6 2 > G b w h e re Y„ i s t h e e x p e c t e d v a lu e o f a n i n d i v i d u a l ' s Ha a c h ie v e m e n t t e s t s c o r e , Xa a n d Ca a r e h i s m e n ta l a g e a n d c h r o n o l o g i c a l a g e , r e s p e c t i v e l y , a n d a a n d b a r e c o n s t a n t s . 63 Horn, h a s s u g g e s t e d a c h a n g e o f c o n s t a n t s w i t h c h r o n o l o g i c a l a g e . F o r c o n v e n ie n c e s a k e , e q u a t i o n (6 2 ) c a n b e w r i t t e n Yh = mxa + n ca , ( 6 3 ) € 1 w h e re m = ^ , (6 4 ) a n d n - — . (6 5 ) I t i s a p p a r e n t t h a t m + n m l , (6 6 ) The p r e s e n t s t u d y i s c o n c e r n e d w i t h d i f f e r e n c e s a s w e l l a s s c o r e s . The H orn F o rm u la w as c o n c e r n e d o n ly w i t h t h e l a t t e r . The o b s e r v e d d i f f e r e n c e (Ha ) , b e tw e e n a n o b s e r v e d a c h ie v e m e n t s c o r e , (YQ) , a n d a n e x p e c t e d cl a c h ie v e m e n t s c o r e ( f o r a n y i n d i v i d u a l ) a c c o r d i n g t o th e H orn F o rm u la c a n b e e x p r e s s e d a s Ya " nCa = Ha + ( o 7 > I f e a c h s i d e o f e q u a t i o n (6 7 ) i s s q u a r e d , i t may t h e n be e x p r e s s e d a s ^ + n 2 ° a = Ha + 2mlJa x a + ' t 6 8 ) From e q u a t i o n (6 7 ) i t i s a l s o s e e n t h a t V a - “ V s. ■ Xa Ha + "“ a ^ 64 a n d Y - nC s H + mX* (7 0 ) A f t e r sum m ing e a c h o f t h e te r m s i n e q u a t i o n (6 8 ) , s u b s t i t u t i n g e q u a t i o n ( 6 9 ) , d i v i d i n g e a c h te rm i n t h e e q u a t i o n b y N, a n d s u b t r a c t i n g e q u a t i o n (7 0 ) f ro m e a c h s i d e , t h e n e q u a t i o n ( 6 8 ) may be w r i t t e n i n te r m s o f v a r i a n c e a n d c o v a r i a n c e a s SH = By + + “ 2 s o - 2m8x y ‘ 2 n 8 y c + 2mn3x o * <7 1 > The a d a p t e d H orn F o rm u la d i f f e r e n c e e q u a t i o n i s m ore g e n e r a l t h a n t h e c o r r e s p o n d i n g H orn E x p e c ta n c y F o r m u la , s i n c e , w hen Y« 0 t h e c o r r e s p o n d i n g H orn F o rm u la (Y s mx + n c ) r e s u l t s * The H o rn F o rm u la t o t a l v a r i a n c e , it t h e n , b ecom es S2 = m2 S2 + 2mnSxo + n2S2 . (7 2 ) By u s i n g e q u a t i o n s (1 ) a n d ( 1 4 ) , Xa « XS + e x ' C k 2 2 2 a n d s x « 3x # + B , X a n d fr o m e q u a t i o n (6 7 ) i t i s s e e n t h a t t h e t r u e d i f f e r e n c e (h ) b e tw e e n a t r u e a c h ie v e m e n t s c o r e a n d t r u e e x p e c t e d a c h ie v e m e n t s c o r e ( f o r a n y i n d i v i d u a l ) a c c o r d i n g t o t h e 65 Horn Formula can be e x p r e s s e d as ha = Ya * - n0a + V <7 3 > o r Ya - n ca - mX* + ha - e y . (7 4 ) £ k A g a in , s u b s c r i p t ”a " r e f e r s t o t h e i n d i v i d u a l ' s s c o r e . S q u a r i n g e a c h s i d e o f ( 7 4 ) , summing a l l v a l u e s t h r o u g h o u t t h e g r o u p , a n d d i v i d i n g e a c h s i d e o f th e e q u a t i o n b y N, a n d u s i n g t h e b a s i c e r r o r a s s u m p t i o n , t h e r e s u l t i n g e q u a t i o n i n te r m s o f v a r i a n c e s a n d c o v a r i a n c e s a n d becom es 2 3 y " yc I f t h e c o r r e s p o n d i n g s i d e s o f e q u a t i o n s ( 1 ) a n d (7 4 ) 4 - 2n\ c * “ 2 s x = “ 2 s x * + s h + al + 3mBh x . - y a r e m u l t i p l i e d t o g e t h e r t h e p r o d u c t becom es X Y -nX C =mx*2+h „X*e +mx*e +h e -e e . (7 6 ) a a a a a a a y a x n a x x_ y Q v a a a a a R e p e a ti n g t h e a b o v e p r o c e d u r e t h e e q u a t i o n i n te rm s o f v a r i a n c e s a n d c o v a r i a n c e s i s w r i t t e n a s sxy * B8xc “ “ x . + shx*- < 77> When e q u a t i o n (7 7 ) i s m u l t i p l i e d b y 2m a n d s u b t r a c t e d fro m (7 5 ) a n d e q u a t i o n (1 4 ) i s s u b s t i t u t e d th e r e s u l t i s 66 s 2 - s 2 - 2 n s +n2 s 2 -2m sv +2mns +m2 s 2 -m2 s 2 - s 2 (7 8 ) h y y c c x y xc x e e ' ' x y The t r u e v a r i a n c e o f th e c o r r e s p o n d i n g H orn F o rm u la (w hen Y z 0 ) i s g i v e n a s s | > m2S® + 2ran Sxo + n 2s | - m2S2 . (7 9 ) in x S in c e t h e v a r i a n c e o f t r u e s c o r e s , o b s e r v e d s c o r e s a n d e r r o r s c o r e s a r e r e l a t e d b y t h e e q u a t i o n ( 7 :2 6 ) 2 2 2 SH = 3h + 3e h - <8 0 > When e q u a t i o n s (7 2 ) a n d (7 8 ) a r e s u b s t i t u t e d i n e q u a t i o n (8 0 ) t h e r e s u l t i n g e q u a t i o n o f e r r o r v a r i a n c e i s e x p r e s s e d a s s 2 = m 2 s 2 + s 2 . (8 1 ) e h x y 2 2 2 W ith t h e a b o v e e x p r e s s i o n s s , s , a n d s H h e h 2 2 g i v e n i n te r m s o f n , m, s , s , a n d th e v a r i a n c e s a n d 6 _ 6 x y c o v a r i a n c e s o f s . y , a n d c , t h e r e l i a b i l i t y ( r ) o f th e J ill H orn F o rm u la c a n a l s o be e x p r e s s e d i n t h e s e te r m s s i n c e 2 2 r e l i a b i l i t y i s a f u n c t i o n o f s h a n d s ^ , 2 rH H * Hr • < 82> H 67 By + m 8x + “ 2 s o ' 2msx y ' 2 n s yc + 2mn s xo > (8 3 ) a n d r s 1 (8 3 a ) f o r t h e c o r r e s p o n d i n g H orn F o rm u la r e l i a b i l i t y . An a l t e r n a t e s o l u t i o n f o r t h e r e l i a b i l i t y may b e o b t a i n e d by m eans o f s e t t i n g th e r e l i a b i l i t y e q u a l t o th e s q u a r e o f t h e m u l t i p l e c o r r e l a t i o n ( t h e c o r r e l a t i o n b e tw e e n H a n d t h e b e s t c o m b in a tio n o f X, Y, a n d C a c c o r d i n g t o th e H orn F o r m u l a .) The r e l i a b i l i t i e s o f t h e s e two m e th o d s y i e l d r e s u l t s w h ic h a r e n u m e r i c a l l y s i m i l a r . V I . DERIVATION OF THE RELIABILITY AND ERROR OF MEASUREMENT OF THE DIFFERENCE BETWEEN ACTUAL ACHIEVEMENT AND PREDICTED ACHIEVEMENT BASED ON THE USUAL TWO-VARIABLE REGRESSION EQUATION H orn (8 : 6 8 ) g i v e s th e e x p e c ta n c y f o r m u la b y u s i n g i n t e l l i g e n c e a n d a c h ie v e m e n t t e B t s c o r e s i n t h e r e g r e s s i o n 68 e q u a t i o n s “ - \ * <x o ' «Yn ) r x y i * • <8 4 ) T r a n s l a t i n g t h i s e q u a t i o n i n t o sy m b o ls w h ic h a r e m ore a p p r o p r i a t e w hen d e r i v i n g e q u a t i o n s a n d i n te r m s o f sy m b o ls w h ic h a r e c o n g r u e n t w i t h t h o s e u s e d i n o t h e r p a r t s o f t h i s s t u d y , t h e e q u a t i o n s c a n b e e x p r e s s e d a s Ya « <Xa - r x y I * + Y & ) o r Y ' = TX - TX + Y, (8 6 ) cl d s w h e re T « r ^ • (8 7 ) x y s I f e a c h s i d e o f e q u a t i o n (8 6 ) i s s u b t r a c t e d f ro m y d * Ye Q > a t h e r e s u l t i n g d i f f e r e n c e e q u a t i o n b eco m es = Y a - ix a + E C - Y, (8 8 ) w h e re « Ya - y £ . (8 9 ) E q u a t i o n (8 8 ) c a n b e e x p r e s s e d i n te r m s o f d i f f e r e n c e fro m t h e m ean v a l u e s a s Ya * Da + “ a* <9 0 > I f e a c h Bide o f e q u a t i o n (8 9 ) i s squared and summed f o r 69 a l l t h e p u p i l s i n t h e g ro u p a n d d i v i d e d b y N th e r e s u l t i n g e q u a t i o n c a n be w r i t t e n i n te r m s o f v a r i a n c e s a n d c o v a r i a n c e s a s s y « 4 < * 21 V x + ^ x • I f a f t e r e a c h s i d e o f e q u a t i o n (8 9 ) i s m u l t i p l i e d b y X, th e r e s p e c t i v e s i d e s a r e summed a n d m u l t i p l i e d b y N i t i s s e e n t h a t V x s sx y ’ Tsx • <9 2 > I t f o l l o w s fro m e q u a t i o n s (9 1 ) a n d ( 9 2 ) t h a t 2 2 2 2 s , = s + T s - 2 t b D* y x xy (9 3 ) o r m ore s im p ly g2 SD ‘ " Sy “ x The t o t a l v a r i a n c e o f t h e t w o - v a r i a b l e (m e n ta l a g e a n d c h r o n o l o g i c a l a g e ) e s t i m a t e d a c h ie v e m e n t s c o r e s b a s e d on t h e r e g r e s s i o n e q u a t i o n i s t h e c o r r e s p o n d i n g 4 - - 4 4 + 4 sc + b i b 2 s x o - ( 9 4 a > w h e re b ^ a n d bg a r e r e g r e s s i o n c o e f f i c i e n t s . T h u s , t h e 2 t o t a l v a r i a n c e ( s j) i) t*1® d i f f e r e n c e b e tw e e n a c t u a l 70 a c h ie v e m e n t s c o r e s a n d e s t i m a t e d a c h ie v e m e n t s c o r e s i s 2 2 e x p r e s s e d a s a f u n c t i o n o f s , s , a n d s • A jT Ajf The e r r o r (e ) o f t h e t r u e I n t e l l i g e n c e t e s t s c o r e a n d th e o b s e r v e d I n t e l l i g e n c e t e s t s c o r e i s g iv e n i n e q u a t i o n ( 1 ) . The e r r o r ( e ) o f th e o b s e r v e d d i f f e r e n c e v i n a c h ie v e m e n t a n d e s t i m a t e d a c h ie v e m e n t t e s t s c o r e s a n d t h e d i f f e r e n c e i n th e a c h ie v e m e n t a n d e s t i m a t e d a c h i e v e m ent b a s e d o n t r u e s c o r e s c a n b e e x p r e s s e d a s e y = Ya - Y - T (X* + X) - (y 5 ) £ L w h e re d ' i s d e f i n e d a s th e d i f f e r e n c e b e tw e e n a n a i n d i v i d u a l ' s e s t i m a t e d a c h ie v e m e n t t e s t s c o r e ( b a s e d on t r u e i n t e l l i g e n c e t e s t s c o r e s ) a n d h i s t r u e a c h ie v e m e n t s c o r e s . E q u a tio n (9 b ) c a n be w r i t t e n Y = d ' + TX* + e . (9b) a a y w ' From e q u a t i o n ( 1 ) , (9b), a n d t h e p r o d u c t o f th e two a n d b y t h e same p r o c e d u r e u s e d i n th e f i r s t d e r i v a t i o n , o f t h i s c h a p t e r i t i s s e e n t h a t 2 p 2 s ■ s „ + s , (97) x x * e ' ' 71 4 = s d - + + s e + 2 T sd ' x . < O 0 ) J a n d s x y = s 2 , x . + T s ^ . (9 9 ) 2 S o l v i n g f o r i t i s s e e n t h a t 2 s 2 s 2 2 2 ®yv 2 e x sd ' * 4 ' H P ~ 4 " — 5 — • (1 0 °) 8x y a x The t r u e v a r i a n c e o f th e t v o - v a r i a b l e ( m e n ta l a g e a n d c h r o n o l o g i c a l a g e ) e s t i m a t e d a c h ie v e m e n t s c o r e s b a s e d o n t h e r e g r e s s i o n e q u a t i o n i s t h e c o r r e s p o n d i n g 4> = 4 4 + 4 4 + b i b 2 sx c - 4 - b i 4 • < iooa> e y x S in c e t h e e r r o r v a r i a n c e i s e q u a l t o th e t o t a l v a r i a n c e m in u s t h e t r u e v a r i a n c e i t f o l l o w s t h a t 2 2 s s 2 2 2 2 e x S ■ 6 , “ S , , ■ S + ------- ------— e a , D* d ' y ^ The r e l i a b i l i t y w hen i n t e r p r e t e d a s t h e t r u e v a r i a n c e d i v i d e d b y t h e t o t a l v a r i a n c e i s g i v e n a s y 2 ~e 4 J s y s r d , a , = ------------2-----------g----------- * ---- . ( i o i ) s 2 - ! » y s 2 x A n o th e r e x p r e s s i o n f o r t h e r e l i a b i l i t y d e r i v e d b y t h e m e th o d o f m u l t i p l e c o r r e l a t i o n c a n b e g i v e n a s r d , d , = ~ t ( 1 0 2 ) 1 - r xy 2 s s x y e w h e re r d , x = — 3 , (1 0 3 ) x d ' 2 2 2 s - ts - T S -s y x y e e a n d r , = --------------- — - ---------------------- * . (1 0 4 ) y d 1 The r e l i a b i l i t y o f th e s c o r e s b a s e d on t h e tw o - v a r i a b l e s ( m e n ta l a g e a n d c h r o n o l o g i c a l a g e ) r e g r e s s i o n e q u a t i o n i s th e c o r r e s p o n d i n g 2 2 s e + b i s e ^ y l y l s p 2 "" ^ 2" ’ 2" * (1 0 4 a ) b I < + b I s c + 2b l b 2 s xc 73 V I I , DERIVATION O P THE RELIABILITY AND ERROR O F M EA SU R EM EN T FOR TH E DIFFERENCE BETW EEN ACHIEVEMENT TEST SCORES AND INTELLIGENCE TEST SCORES The d i f f e r e n c e b e tw e e n a n a c h ie v e m e n t t e s t s c o r e a n d a n i n t e l l i g e n c e t e s t s c o r e c a n b e e x p r e s s e d a s Da = ya ' V < 1 0 5 > From G u l l l k s o n ( 7 : 1 4 ) t h e r e l i a b i l i t y o f th e d i f f e r e n c e b e tw e e n two t e s t s c o r e s i s g i v e n a s th e t r u e d i f f e r e n c e v a r i a n c e d i v i d e d b y th e t o t a l d i f f e r e n c e v a r i a n c e . E x p r e s s i n g t h i s r e l a t i o n s h i p a s a n e q u a t i o n i t f o l l o w s t h a t 2 r DD * “ t ' (1 0 6 ) BD 2 2 w h e re s & i s g i v e n i n e q u a t i o n (1 6 ) a n d s ^ a s th e f a m i l i a r 2 2 2 s* - 2 s + s = s f . (1 0 7 ) x x y y D v ' A ls o fro m G u l l l k s o n ( 7 ;3 5 2 ) i t f o l l o w s t h a t s e = + s e ' (1 0 8 > D ex y 74 To q u o te fro m McNemar: When one r e g a r d s G ( s Y - X) a s a n o b t a i n e d g a i n s c o r e a n d r a i s e s t h e q u e s t i o n a s t o t h e u n d e r l y i n g t r u e g a i n G t, one r e a d i l y t h i n k s o f t h e r e g r e s s i o n e s t i m a t e o f a t r u e s c o r e fro m a n o b s e r v e d o r f a l l i b l e s c o r e , o r so c a l l e d r e g r e s s i o n s c o r e . I n , • . g r o s s s c o r e u n i t s , t h e e q u a t i o n w o u ld be w r i t t e n . • . . S u b s t i t u t i o n o f l e t t e r s h a s b e e n made i n t h i s f o r m u l a . M s t a n d s f o r th e d i f f e r e n c e (G) when u s i n g th e e q u a t i o n a f t e r McNemar. I f d i f f e r e n c e i s s u b s t i t u t e d f o r g a i n i n t h e McNemar s t a t e m e n t a n d t h e u s u a l a s s u m p t io n s a b o u t e x p e c ta n c y a r e made t h e n t h i s a d a p t a t i o n o f M cN em ar's e q u a t i o n may b e u s e d a s a n e x p e c ta n c y e q u a t i o n . I n e q u a t i o n (1 0 9 ) ( 9 7 :5 0 ) M * tt-x ) + (X - X) - (X - X ). (1 0 9 ) r, mm (1 1 0 ) w h e re a l l t h e te r m s h a v e b e e n p r e v i o u s l y d e f i n e d . 75 SU M M A R Y T h is c h a p t e r w as c o n c e r n e d w i t h t h e d e r i v a t i o n o f t h e v a r i o u s e x p e c ta n c y f o r m u l a s a n d a d a p t a t i o n o f e d u c a t i o n a l a n d p s y c h o l o g i c a l m e a su re m e n t e q u a t i o n s a n d t h e o r i e s t o th e c o n c e p t o f e x p e c t a n c y . A sum m ary o f t h e d e r i v e d e q u a t i o n s a n d t h e i r c o r r e s p o n d i n g r e l i a b i l i t i e s f o l l o w ; (The o r d e r o f p r e s e n t a t i o n o f t h e s e e q u a t i o n s i s n o t t h e o r d e r i n w h ic h t h e y w e re I n t r o d u c e d i n t h e c h a p t e r •) 1 . The A r i t h m e t i c D i f f e r e n c e e q u a t i o n a n d r e l i a b i l i t y a r e Da * Ya - Xa a n d ( 1 0 5 ) 2 PDD = “ I * (1 0 6 ) SD 2 . The H orn F o rm u la a n d r e l i a b i l i t y a r e Ha ■ Ya “ ^ a “ nCa a n d <6 7 ) 2 2 2 m s + s e x y (8 3 ) r ■ 1 -----2'--------5 2 -------------- — ------------------------------ s + m s + n s - 2ms - 2ms + 2mns y x c x y yc xc 76 3 , The p r e d i c t e d a c h ie v e m e n t b a s e d on t h e u s u a l t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n d i f f e r e n c e a n d i t s r e l i a b i l i t y a r e D' = YQ - TX + T X - Y a n d (8 8 ) a a a ' ' 2 2 s s 2 2 2 e x S - S - S r.-------- y x y e y V d - -----------------3 ----------------------- • <1 0 1 ) Ey J S X 4 . The e x p e c ta n c y d i f f e r e n c e f o r m u la a d a p t e d fro m MeNemar a n d r e l i a b i l i t y a r e M s r ram ( Y "X ) + <Y ~X ) " r mm ( Y “X ) a n d ( 1 0 9 ) 2 2 r s + r s - 2 r s s x x x y y y x y x y / t , « \ mm = ------3 * { } s + s - 2 r s s x y xy x y 5 , The e x p e c ta n c y f o r m u la a d a p t e d fro m L o rd b y C a f f r e y a n d t h e p r e s e n t i n v e s t i g a t o r to t h e e s t i m a t e d t r u e d i f f e r e n c e ( u s i n g th e t w o - v a r i a b l e o f X a n d Y) a r e 77 K * 6 + b 6 x . y < x a - x > + b 6 y . x < Xa ‘ (3 1 ) a n d s b + s„ b 5 x B x .y By B y .x , oc:X r , . = ----------------------------------- — . (3 5 ) R 6 6 . The e x p e c t a n c y f o r m u l a a d a p t e d fro m L ord by C a f f r e y a n d t h e p r e s e n t I n v e s t i g a t o r to th e e s tim a te d t r u e d i f f e r e n c e u s i n g t h e t h r e e - v a r i a b l e s (X, Y, an d C) a r e 6l * 6 + b 6 x .y o < x a - x > + b B y .x c < Xa - x > + V . x y ^ <3 6 > a n d s b + s b + s b r - 5 x B x .y c 5 y B y .x c 5c B c.x y (6l) B' 6 1 “ 2 * SB 7. The r e l i a b i l i t y o f t h e s c o r e s b a s e d on th e r e g r e s s i o n e q u a t i o n » b y x . c X + b y c . x C + K (S#*) i s g i v e n a s 2 2 b s y x . c e x b s + b b s + b s* y x . c x y x . c yc ,x xc y c .x c (104a) 78 8 . The r e l i a b i l i t y o f t h e s c o r e s b a s e d o n t h e H orn F o rm u la Yh = mX + nC (6 3 ) i s g i v e n a s m s e r = 1 ----- 5T5------------- • (8 3 a ) H H m s ^ + n s c + 2mnsxc A lth o u g h t h e A n t i c i p a t e d A c h ie v e m e n t d i f f e r e n c e i s n o t g i v e n a s a f o r m u l a , i t i s m e n tio n e d h e r e b e c a u s e i t i s a c o m p a r a tiv e m eth o d o f e x p e c ta n c y w h ic h i s i n v e s t i g a t e d i n t h e p r e s e n t s t u d y . CHAPTER IV PROCEDURE I t i s a x i o m a t i c t h a t s c h o o l d i s t r i c t p o p u l a t i o n s v a r y w i t h r e s p e c t t o t h e i r d i s t r i b u t i o n o f c h r o n o l o g i c a l a g e s , m e n ta l a g e s , a n d a c h ie v e m e n t t e s t s r e s u l t s . I t f o l l o w s t h a t a n y r e s e a r c h d e s i g n , w h e re p r a c t i c a l a n d a p p l i e d o u tc o m e s o f e x p e c ta n c y a r e a n t i c i p a t e d , w o u ld i n c l u d e d i s t r i c t s h a v in g d i f f e r e n c e s i n th e a b o v e - m e n tio n e d c h a r a c t e r i s t i c s . T h u s , i f g e n e r a l i z a t i o n s c a n n o t b e made f ro m a s t u d y o f s u c h s a m p le s a s th e t h i r t e e n d i s t r i c t s u s e d , t h e v a l i d i t y o f g e n e r a l i z i n g t o a l l d i s t r i c t s w o u ld b e q u e s t i o n e d . The d e s i g n o f t h i s s t u d y , t h e n , i n v o l v e d a c o m p a r i s o n o f t h e v a r i o u s m e th o d s o f e x p e c ta n c y d e s c r i b e d i n C h a p te r I I w hen a p p l i e d t o t h i r t e e n d i s t r i c t s d i f f e r i n g w i t h r e s p e c t t o m ean c h r o n o l o g i c a l a g e , m e n ta l a g e , a n d a c h ie v e m e n t t e s t s c o r e s . A l l s t r a i g h t l i n e e x p e c t a n c i e s w e re c o m p a re d w i t h t h e b a s i c c r i t e r i a , t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ( a s m o d i f i e d i n C h a p te r I I I fro m L o rd ) a n d t h e r e g r e s s i o n e q u a t i o n s . 80 P o p u l a t i o n u s e d i n t h e s t u d y . I n o r d e r t o o b t a i n a r e p r e s e n t a t i o n o f v a r i o u s s i z e s o f s c h o o l p o p u l a t i o n s , i t w as d e c i d e d t h a t th e p o p u l a t i o n s o f s e v e r a l s c h o o l d i s t r i c t s w o u ld be u s e d . T h i r t e e n d i s t r i c t s r e p r e s e n t i n g d i f f e r e n t e n r o l l m e n t m a g n itu d e s w ere u t i l i z e d . D a ta w ere o b t a i n e d f o r a l l f o u r t h a n d s i x t h g ra d e p u p i l s I n r e a d i n g a n d a r i t h m e t i c , g i v i n g t w e n t y - s i x s e t s o f d a t a . O n ly t h o s e p u p i l s w e re u s e d f o r whom c h r o n o l o g i c a l a g e , m e n ta l t e s t , a n d a c h ie v e m e n t t e s t d a t a w e re a v a i l a b l e . The d i s t r i c t s u s e d i n t h e s t u d y w e re s e l e c t e d fro m th o s e f o r whom d a t a w ere a v a i l a b l e i n t h e L os A n g e le s C o u n ty S u p e r i n t e n d e n t o f S c h o o ls O f f i c e , D i v i s i o n o f R e s e a r c h a n d G u id a n c e . T h ese t h i r t e e n d i s t r i c t s w e re i d e n t i f i e d a s D i s t r i c t 1 , D i s t r i c t 2 , a n d so on to D i s t r i c t 1 3 , " T o ta l" r e p r e s e n t s t h e t o t a l o f a l l t h i r t e e n d i s t r i c t s a n d may be th o u g h t o f a s a d d i t i o n a l d i s t r i c t . A l l d a t a w e re p u n c h e d on IBM c a r d s . The c a l c u l a t i o n o f th e c o e f f i c i e n t s f o r th e e q u a t i o n s a n d th e a p p l i c a t i o n o f th e e q u a t i o n s to i n d i v i d u a l datum w e re c a l c u l a t e d on t h e IBM 602A m a c h in e • M e a s u rin g i n s t r u m e n t s u t i l i z e d a n d t h e n a t u r e o f th e g r a d e p la c e m e n t s c o r e s . A l l r e a d i n g a n d a r i t h m e t i c 81 s c o r e s w e re o b t a i n e d fro m t h e e le m e n t a r y b a t t e r y , fo rm AA o f th e C a l i f o r n i a A c h ie v e m e n t T e s t a n d a l l m e n ta l a g e s c o r e s w e re o b t a i n e d fro m t h e 1950 s h o r t - f o r m e l e m e n t a r y b a t t e r y o f t h e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y . E ac h a c h ie v e m e n t v a lu e i s e x p r e s s e d i n g r a d e l e v e l te r m s c a l l e d " g ra d e p l a c e m e n t s " , w h ere i n 4 .0 m eans " b e g in n in g o f t h e f o u r t h g r a d e " , 4 . 5 m eans " h a lf w a y t h r o u g h th e f o u r t h g r a d e , " e t c . M e n ta l a g e s c o r e s a n d c h r o n o l o g i c a l a g e s w e re c o n v e r t e d t o g r a d e p la c e m e n t v a l u e s b y m eans o f a p p r o p r i a t e a g e - g r a d e c o n v e r s i o n t a b l e s . M a tu re o f t h e p o p u l a t i o n s . A summary o f t h e m ean v a l u e s f o r c h r o n o l o g i c a l a g e g r a d e p la c e m e n t (CAGP), m e n ta l a g e g r a d e p la c e m e n t (MAGP), r e a d i n g g ra d e p la c e m e n t (RGP), a n d a r i t h m e t i c g r a d e p la c e m e n t (AGP), o f e a c h p o p u l a t i o n i s g i v e n i n T a b le I I f o r g ra d e 4 a n d T a b le I I I f o r g r a d e 6, An a n a l y s i s w as made o f th e d i s t r i c t s i n o r d e r t o a s c e r t a i n t h e s i m i l a r i t y o f th e G rade 4 p o p u l a t i o n w i t h r e s p e c t t o CAGP, MAGP, RGP, a n d AGP. T a b le IV su m m a riz e s t h e c o m p a r a tiv e r e s u l t s w h ic h w e re o b t a i n e d b y t h e m eth o d o f a n a l y s i s o f v a r i a n c e . A l l P r a t i o s w e re f o u n d t o be s i g n i f i c a n t a t t h e 1 p e r c e n t l e v e l o f c o n f i d e n c e * When t h e s e s t a t l s i c a l f i n d i n g s a r e i n t e r p r e t e d , i t i s c o n c lu d e d 82 TABLE I I SUMMARY OF MEAN VALUES FOR CHRONOLOGICAL AGE GRADE PLACEMENT (CAGP), MENTAL AGE GRADE PLACEMENT (MAGP), READING GRADE PLACEMENT (RGP) AND ARITHMETIC GRADE PLACEMENT (AGP) OF EACH GRADE 4 POPULATION D i s t r i c t N CAGP MAGP RGP AGP 1 477 4 .2 5 .1 4 .3 4 .4 2 101 4 ,2 4 .5 3 . 4 4 .3 3 179 4 .1 4 .7 4 .3 4 .3 4 36 4 .3 4 .5 3 .8 4 . 0 5 526 4 .3 4 .3 4 .0 4 .2 6 20 4 .2 3 . 9 3 .2 3 . 6 7 316 4 .2 5 . 0 4 .8 4 . 6 8 273 4 .3 4 .2 3 . 4 4 .1 9 781 4 .1 4 .8 4 .2 4 . 4 10 142 4 .2 4 .5 4 ,5 4 ,3 11 49 4 .2 4 . 9 4 .0 4 .3 12 167 4 .2 4 .7 4 . 6 4 .4 13 692 4 .2 4 .9 4 .7 4 .7 T o t a l 3759 4 .2 4 .7 4 .3 4 .3 83 TABLE I I I SUMMARY OF MEAN VALUES FOR CHRONOLOGICAL AGE GRADE PLACEMENT (CAGP), MENTAL AGE GRADE PLACEMENT (MAGP), READING GRADE PLACEMENT (RGP) , AND ARITHMETIC GRADE PLACEMENT (AGP) OF EACH GRADE 6 POPULATION D i s t r i c t N CAGP MAGP RGP AGP 1 363 6 .2 7 .3 6 .4 6 .0 2 67 6 .4 6 .3 5 .9 5 .8 3 247 6 .1 7 .0 6 .1 6 .0 4 29 6 .3 6 .7 5 .5 5 .7 5 508 6 .4 6 .1 5 .2 5 .5 6 12 6 .1 6 .7 5.4 5 . 6 7 266 6 .3 7 . 6 6 .9 6 .3 8 252 6 . 4 6*6 5 .1 5 .4 9 648 6 .2 7 .1 6 .0 5 .8 10 162 6 .4 6 .7 5 .8 5 .8 11 44 6 .3 6.1 5 .7 5 . 8 12 188 6 .3 7 .4 6 .5 6.2 13 330 6 .3 7 .0 6 .1 6 .0 Total 3116 6 .3 6 .9 6 .0 5 .8 84 TABLE IV SUMMARY OF F-RATIOS AS A RESULT OF ANALYSIS OF VARIANCE BETWEEN DISTRICTS AND WITHIN DISTRICTS WITH RESPECT TO CAGP, MAGP, RGP, AND AGP FOR GRADE 4 V a r i a b l e s F - R a t i o s CAGP 9 .5 4 * MAGP 1 8 .9 9 * RGP 4 1 .1 1 * AGP 3 0 .2 7 * * S i g n i f i c a n c e t e s t e d a t t h e 1% l e v e l o f c o n f i d e n c e 85 t h a t t h e r e a r e g r e a t e r d i f f e r e n c e s b e tw e e n d i s t r i c t s , w i t h r e s p e c t to CAGP, t h a n t h e r e a r e w i t h i n d i s t r i c t s . The same i n v e s t i g a t i o n a n d i n t e r p r e t a t i o n w e re made w i t h r e s p e c t t o MAGP, RGP, a n d AGP. T a b le V i n d i c a t e s t h a t fro m t h e P r a t i o s th e same c o n c l u s i o n s made f o r t h e G rade 4 p o p u l a t i o n s a l s o h o l d f o r t h e G rade 6 p o p u l a t i o n s . T h a t i s , t h e r e e x i s t g r e a t e r d i f f e r e n c e s b e tw e e n d i s t r i c t s w i t h r e s p e c t t o CAGP ( a s w e l l a s MAGP, RGP, a n d AGP) t h a n w i t h i n d i s t r i c t s . F o r th e p u r p o s e o f t h i s s t u d y i t i s i m p o r t a n t t h a t t h i s r e p r e s e n t a t i o n o f v a r i o u s l e v e l s o f a b i l i t y , a c h i e v e m e n t, a n d c h r o n o l o g i c a l a g e b e tw e e n s c h o o l d i s t r i c t s e x i s t s i n o r d e r to com pare a d e q u a t e l y t h e v a r i o u s t y p e s o f e x p e c t a n c i e s b e tw e e n d i f f e r e n t k i n d s o f p o p u l a t i o n s . E s t i m a t e d d i f f e r e n c e e q u a t i o n s a c c o r d i n g to d i s t r i c t s . M eans, v a r i a n c e s , a n d c o v a r i a n c e s o f CAGP, MAGP, RGP, a n d AGP w e re c a l c u l a t e d f o r e a c h d i s t r i c t p r e p a r a t o r y t o t h e s o l u t i o n o f th e e q u a t i o n s w h ic h w e re t o b e u s e d f o r c o m p a r a tiv e p u r p o s e s . The s t a n d a r d e r r o r s o f m e a su re m e n t a c h ie v e m e n t t e s t s w e re t a k e n fro m t h e m anual o f t h e C a l i f o r n i a A c h ie v e m e n t T e s t (1 3 0 ) a n d t h e s t a n d a r d e r r o r o f m e a su re m e n t o f th e a b i l i t y t e s t was 8 6 TABLE V SUMMARY OF F-RATIOS AS A RESULT OF ANALYSIS OF VARIANCE BETWEEN DISTRICTS AND WITHIN DISTRICTS WITH RESPECT TO CAGP, MAGP, RGP, AND AGP FOR GRADE 6 V a r i a b l e s F - R a t l o CAGP 8 .2 7 * MAGP 2 8 .9 5 * RGP 3 6 .8 3 * AGP 3 3 .7 9 * * S i g n i f i c a n c e t e s t e d a t t h e 1 $ l e v e l o f c o n f i d e n c e . 87 t a k e n fro m th e m an u al o f t h e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y . (1 3 2 ) From a p r o c e d u r e u s e d b y L o rd (9 2 ) i n v o l v i n g e q u a t i o n (1 8 ) o f C h a p te r I I I t h e s e r e l i a b i l i t i e s a n d o t h e r known s t a t i s t i c s t h e e r r o r s o f t h e r e l i a b i l i t i e s w e re r e a d i l y c a l c u l a t e d f o r e a c h d i s t r i c t . W ith t h i s i n f o r m a t i o n v a r i o u s d i f f e r e n c e e q u a t i o n s w e re c a l c u l a t e d . The e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s f o r e a c h d i s t r i c t , a n d f o r G ra d e s 4 a n d 6 a r e g iv e n i n C h a p te r V. V a r io u s e x p e c t a n c i e s a p p l i e d t o e a c h p u p i l . A f t e r c o m p u tin g th e v a r i o u s e x p e c ta n c y e q u a t i o n s f o r e a c h d i s t r i c t , t h e e q u a t i o n s w e re c o m p a red o n t h e b a s i s o f s l o p e s a n d i n t e r c e p t s . The i n t e r c e p t s w e re c o m p a red b y r e d u c i n g ( i f n e c e s s a r y ) a l l e q u a t i o n s t o t h e t w o - v a r i a b l e e q u a t i o n s i n te rm s o f X a n d Y ( o t h e r v a r i a b l e s b e i n g h e l d c o n s t a n t . ) C o m p a riso n s w e re made b y m eans o f t h e W alsh T e s t , a n a l y s i s o f v a r i a n c e , a n d th e t t e s t . The c r i t e r i o n e q u a t i o n u s e d f o r d i f f e r e n c e e q u a t i o n s w as th e tw o - v a r i a b l e (X ,Y ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . C o m p a riso n o f e x p e c ta n c y e q u a t i o n s . T h e re i s a n e x p l a n a t i o n i n th e f o l l o w i n g c h a p t e r r e g a r d i n g th e d i s t i n c t i o n b e tw e e n d i f f e r e n c e e q u a t i o n s a n d e x p e c ta n c y e q u a t i o n s . I n t h e n e x t s t a g e o f t h e s t u d y th e r e l a t i o n 88 s h i p s b e tw e e n v a r i o u s e x p e c t a n c i e s b a s e d o n r e g r e s s i o n e q u a t i o n s w e re a n a l y z e d . The e f f e c t s o f g r a d e a c h ie v e m e n t a r e a , c h r o n o l o g i c a l a g e , a n d d i s t r i c t d i f f e r e n c e s w e re i n v e s t i g a t e d . T h ese c o m p a r is o n s w e re a l s o made b y m eans o f s l o p e s a n d I n t e r c e p t v a l u e s . The c r i t e r i o n e q u a t i o n u s e d w i t h r e g a r d t o e x p e c t a n c y w as t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n . E x p e c t a n c i e s a s c a l c u l a t e d fro m r e g r e s s i o n e q u a t i o n s f o r d i f f e r e n t a b i l i t y g r o u p s , T h e re w e re t h r e e o t h e r s e t s o f e x p e c ta n c y e q u a t i o n s d e r i v e d fro m d a t a b a s e d on a d i v i s i o n o f a l l p u p i l s i n t o t h r e e a b i l i t y l e v e l s . T h ese d a t a w e re o b t a i n e d b y f i r s t c o m b in in g th e IBM c a r d s o f a l l p u p i l s o f a l l th e d i s t r i c t s , t h e n g r o u p i n g t h e c a r d s a c c o r d i n g t o a b i l i t y (IQ, 85 a n d b e lo w , IQ 86 to 115* a n d IQ l l o a n d a b o v e ) . T h u s, t h r e e e q u a t i o n s b a s e d u p o n l e v e l s o f a b i l i t y w e re o b t a i n e d . T h ese e q u a t i o n s w e re c o m p a red w i t h t h e r e g r e s s i o n e q u a t i o n b a s e d o n th e t o t a l p o p u l a t i o n . The A n t i c i p a t e d A c h ie v e m e n t. A lth o u g h th e A n t i c i p a t e d A c h ie v e m e n t i s n o t a s t r a i g h t - l i n e ty p e o f e x p e c t a n c y , i t s t h e o r e t i c a l r a t i o n a l e h a s s u c h h ig h m e r i t t h a t i t s v a l u e s w e re o b t a i n e d a n d c o m p a red w i t h th e 89 str a ig h t-lin e regression equations. The A nticipated Achievement was not d ir e c tly obtainable from the te s ts used in th is study. The reason for th is i s that the A nticipated Achievement i s obtained d ir e c tly from the form W X Y Z ed ition of the C alifornia Achievement Tests rather than the form AA, which was the form used to obtain achievement data for the present study, A sim ilar discrepancy ex isted between the 195? and 1950 ed itio n s of the C alifornia Test of Mental M aturity, Conversion tab les y ield in g equivalent values between these two forms were supplied by the C alifornia Test Bureau, From these tables equivalent A nticipated Achievement values were obtained. The c r ite r io n of r e l i a b i l i t y . R e lia b ilit ie s fo r each of these equations were then computed, based upon the r e li a b i li t y formulas developed in Chapter I I , The r e l i a b i l i t i e s were in v estig a ted to in d icate evidence of s u ffic ie n tly high r e li a b i li t y to be acceptable as a comparatively con sisten t method of expectancy. Although o r ig in a lly the r e li a b i lit y c o e ffic ie n t of p u p il's scores (based on various expectancy equations) was assumed to be an important c r ite r io n , a clo ser a n a ly sis of the 90 i n d i c a t e d r e s u l t s l e d to a r e - e v a l u a t i o n o f I t s r e l a t i v e I m p o r t a n c e . A f u r t h e r d i s c u s s i o n o f t h i s i s f o u n d I n C h a p te r V. The c r i t e r i o n e q u a t i o n f o r e s t i m a t e d d i f f e r e n c e e q u a t i o n s . One a s p e c t o f th e r e s e a r c h d e s i g n I n v o l v e s th e c o m p a r is o n o f t h e v a r i o u s s t r a i g h t - l i n e e x p e c ta n c y e q u a t i o n . S in c e th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ( a f t e r L o rd ) h a s h i g h r a t i o n a l v a l i d i t y , t h i s e q u a t i o n w as s e l e c t e d a s t h e c r i t e r i o n e q u a t i o n o f th e b a s i s o f i t s f e a t u r e s : 1 . R e l i a b i l i t y o f t h e t e s t s u s e d i n t a k e n i n t o a c c o u n t . 2 . D e r i v a t i o n o f t h e e q u a t i o n fro m t r u e t e s t s c o r e s . 3 . T a k in g i n t o a c c o u n t th e e f f e c t s o f r e g r e s s i o n . 4 . E m p i r i c a l l y t e s t e d to y i e l d r e a l i s t i c r e s u l t s w hen c o m p a rin g two a c h ie v e m e n t t e s t s . The c r i t e r i o n e q u a t i o n f o r e x p e c ta n c y e q u a t i o n s . The s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n o f t h e fo rm bX + K was u s e d a s a c r i t e r i o n e q u a t i o n s im p ly b e c a u s e i t i s t h e s i m p l e s t o f a l l r e g r e s s i o n e q u a t i o n s . I t w as r e c o g n i z e d t h a t i f m a jo r d i f f e r e n c e s o c c u r e d b e tw e e n t h i s 91 c r i t e r i o n a n d t h e r e s u l t o f o t h e r r e g r e s s i o n e q u a t i o n s th e new r e g r e s s i o n e q u a t i o n m ig h t w e l l be i n v e s t i g a t e d a s a b a s i s f o r b e i n g t h e c r i t e r i o n e q u a t i o n . S lo p e s a n d I n t e r c e p t s u s e d a s a m eth o d o f c o m p a r is o n . Any s t r a i g h t - l i n e c a n b e r e p r e s e n t e d on a r e c t a n g u l a r c o o r d i n a t e s y s te m i f two c h a r a c t e r i s t i c s o f th e l i n e a r e know n. The two c h a r a c t e r i s t i c s o f th e l i n e a r e (1 ) i t s s l o p e a n d (2 ) i t s i n t e r c e p t . The s l o p e o f a l i n e may be d e f i n e d a s t h e a n g l e b e tw e e n th e l i n e a n d th e h o r i z o n t a l m e a s u re d c o u n t e r c l o c k w i s e fro m th e h o r i z o n t a l . The i n t e r c e p t o f a l i n e i s d e f i n e d a s t h e v e r t i c a l d i s t a n c e fro m a p o i n t o n th e l i n e t o t h e a b s c i s s a . T h u s, i n t e r c e p t s may b e m e a su re d a t a n y p o i n t a l o n g t h e a b s c i s s a a s i n d i c a t e d b y 1-^, 1 ^ , 1 ^ , a n d 1^ i n F i g u r e 1 , r e s p e c t i v e l y , f o r m e n ta l a g e s o f 2 . 1 , ^ . 1 , 6 . 1 , a n d 8 . 1 . SUMMARY The p u r p o s e o f t h i s c h a p t e r w as t o r e p o r t on th e v a r i o u s i te m s i n v o l v e d i n c o m p a rin g e x p e c t a n c i e s a n d e x p e c ta n c y e q u a t i o n s . F i r s t , a n a n a l y s i s was made o f th e g ro u p s i n v o l v e d i n th e s t u d y w i t h p a r t i c u l a r r e f e r e n c e to c h r o n o l o g i c a l Mental age grade placement FIGURE SCHEMATIC PfcESEjfTAT Xl f *2* 1 3[ A1T H MEHTAL AtffiS £ . 1 , O F T H E CEPTS 1, A M D 8 .1 93 a g e , a b i l i t y , m e n ta l a g e , a n d a c h i e v e m e n t . N e x t, th e p r o c e d u r e f o r o b t a i n i n g th e d i f f e r e n c e a n d e x p e c ta n c y e q u a t i o n s w as r e v e a l e d a l o n g w i t h t h e c o m p a r is o n s t o b e a n a l y z e d . F i n a l l y , th e c h a p t e r c o n c l u d e s w i t h a p r e s e n t a t i o n o f c r i t e r i a u s e d f o r c o m p a r in g e x p e c t a n c i e s . CHAPTER V EQUATIONS O P ESTIM ATED DIFFERENCES, THEIR SLOPES AND INTERCEPTS, AND THEIR RELIABILITIES The previous chapter described the ch a r a c te r istic s of the populations used in the study, the research design and procedure, and the c r it e r ia . The present chapter reveals the various d ifferen ce equations (between mental te s t scores and achievement te st scores) fo r the several d is t r ic t s involved in the study together with th eir corresponding slopes and in te r c e p ts. The a n alysis compares c h a r a c te r is tic s of various d is t r ic t s on the same equation, and the ch a r a c te r istic s of the difference equations on the same data. D ifferen ces of in terp reta tlo n between expectancy equations and d ifferen ce eq u ations. I t is reca lled that the four d ifferen ce equations are these: (1) the arithm etic d ifferen ce; (2) the two-variable estim ated true d ifferen ce equation; and (4-) the approximate estim ated true d ifferen ce equation. The three expectancy equations are these: (1) the sin g le variable regression equation; (2) the two-variable 95 regression equation; and (3) the Horn Formula. At the outset the in ten tio n was to compare the four d ifferen ce equations with each other and with the three expectancy equations. It would then be necessary for the expectancy equations to be expressed in terms of d iffe r e n c e s. This i s accomplished by subtracting the expectancy value from the actual obtained achievement score. For the purpose of th is study, th is d ifferen ce i s ca lled the exp ectan cy-d lfferen ce. Thus, the d ifferen ce formulas have to do with estim ating the d ifferen ce between the scores of the a b ilit y te s t and the achievement t e s t , whereas, the expectancy-dlfference equations estim ate the d ifferen ce between the expectancy value and the ach ieve ment te s t score. In the la t t e r case, the expectancy is not mental age d ir e c tly , but some regression function of mental a g e. In summary, the Im plication i s that mental age i s assumed to be the sole expectancy contributor in the estim ated d ifferen ce equations; whereas, adaptations of mental age by means of regression or in combination with other variables are used in expectancy d ifferen ce equation. When the equations of these two d ifferen ce 96 e q u a t i o n s a r e a p p l i e d t o p o p u l a t i o n a v e r a g e s , t h e a v e r a g e d i f f e r e n c e o f e s t i m a t e d d i f f e r e n c e e q u a t i o n s i s e q u a l t o t h e d i f f e r e n c e b e tw e e n t h e a v e r a g e m e n ta l a g e a n d th e a v e r a g e a c h ie v e m e n t; w h e re a s t h e l a t t e r a v e r a g e d i f f e r e n c e o f t h e e x p e c t a n c y - d i f f e r e n e e e q u a t i o n s i s th e d i f f e r e n c e b e tw e e n t h e e x p e c ta n c y s c o r e a n d t h e a c h ie v e m e n t s c o r e w h ic h , w hen a p p l i e d t o a r e g r e s s i o n e q u a t i o n , i s z e r o o r c l o s e t o z e r o . From a n a p p l i e d v i e w p o i n t , th e f o r m e r d i f f e r e n c e ( f o r a n y i n d i v i d u a l ) i s I n t e r p r e t e d a s th e d i s c r e p a n c y b e tw e e n a c h ie v e m e n t a n d I n t e l l i g e n c e s c o r e s , r e a l i z i n g t h e r e a r e c o n s i s t e n t a v e r a g e d i f f e r e n c e s ( t h e a v e r a g e m e n ta l a g e b e i n g h i g h e r i n a l l d i s t r i c t s I n v e s t i g a t e d i n t h i s s t u d y ) b e tw e e n th e m e n ta l a g e s c o r e s a n d a c h ie v e m e n t s c o r e s . The l a t t e r d i f f e r e n c e i s i n t e r p r e t e d a s t h e d i s c r e p a n c y b e tw e e n t h e a c h ie v e m e n t s c o r e a n d t h e " b e s t " e x p e c t e d v a lu e i n w h ic h r e g r e s s i o n e f f e c t s h a v e b e e n t a k e n i n t o a c c o u n t . S in c e t h e s e two p r o c e d u r e s do n o t y i e l d r e s u l t s w h ic h a r e w h o l l y c o m p a t i b l e , t h e r e s u l t s fro m t h e s e two p r o c e d u r e s a r e n o t d i r e c t l y c o m p a red i n t h i s s t u d y . The e s t i m a t e d ( t r u e ) d i f f e r e n c e s a r e c o m p a re d i n t h i s c h a p t e r . 97 The stra ig h t lin e expectancies rather than the d ifferen ces are compared in Chapter VI. C riterion estim ated d ifferen ce equation. Of the four stra ig h t lin e d ifferen ce equations In vestigated in th is study, only one was used as a c r ite r io n equation. As in dicated in Chapter IV, th is c r ite r io n equation is the two-variable estim ated true d ifferen ce equation (m odification of Lord). In th is chapter, than, the slop es, In tercep ts, and r e l i a b i l i t i e s of the other three equations are compared with th is c r ite r io n equation. C o m p a riso n o f t h e s l o p e s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w i t h i n a n d b e tw e e n s e t s o f e q u a t i o n s . I t w as a p p a r e n t t h a t i f t h e e s t i m a t e d t r u e d i f f e r e n c e i n th e t w o - v a r i a b l e (X ,Y ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , i s h e l d c o n s t a n t , a s i n g l e e q u a t i o n ( a s a f u n c t i o n o f X a n d Y) w o u ld t h a n b e o b t a i n e d f o r e a c h d i s t r i c t . T h u s , s l o p e s w e re c o m p u te d f o r e a c h d i s t r i c t a n d t h e e q u a t i o n s p l o t t e d o n r e c t a n g u l a r c o o r d i n a t e g r a p h p a p e r , a s I l l u s t r a t e d i n F i g u r e s 2 , 3 , 4 , a n d From t h i s , i t i s s e e n t h a t t h e r e a r e f o u r s e t s o f s l o p e s ; G rad e 4 R e a d in g , G rad e 4 A r i t h m e t i c , G ra d e 6 R e a d in g , a n d G rad e 6 A r i t h m e t i c . A sum m ary o f t h e s e s l o p e s I s g i v e n I n T a b le V I . ding g r a d e placement » * M ental age grade placem ent 4 * 6.0 5 . 0 6,0 Mental age grad© placement bstJcmk 'm L T w m ; DISTRECTS ■ER8NCE EQUATIO; FOR GRACE 6 RE I 6 * 0 7 . 0 Mental age grad* placement FIGURE 5 . LINES REPRESENTING THE ESTIMATED TRUE RATIONS FOR THE i I xnu» DISTRICTS FOE GRADE 6 ARITHMETIC | W HEN B' t O 102 TABLE VI SUMMARY OF THE SLOPES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE ESTIMATED TRUE DIFFERENCE EQUATION FOR GRADE 4 AND 6, READING AND ARITHMETIC D i s t r i c t G ra d e 4 R e a d in g G rad e 4 A r i t h m e t i c G rade 6 R e a d in g G rade 6 A r i t h m e t i c 1 .9 4 1 .2 3 .9 5 1 .1 b 2 1 .0 1 1 .2 6 1 .0 1 1 .4 7 3 .97 1 .1 4 1 .0 1 1 .3 0 4 1 .1 1 .9 7 .9 4 1 .3 9 5 1 .0 3 1 .3 9 1 .0 1 1 .3 3 b 1 .0 9 1 .1 2 o o • -A 1 .1 4 7 .9 2 l . l u 1 .0 0 1 .1 6 O 1 .0 9 1 .4 7 • 9b 1 .1 1 9 1 .0 9 1 .2 3 1 .0 1 1 .1 6 10 .9 9 .9 9 1 .0 1 1 .2 2 n 1 .0 0 1 .1 9 1 .0 1 1 .1 5 12 1 .0 9 1 .2 9 1 .0 4 1 .3 2 13 j - 1 . o o 1 .2 b 1 .0 1 1 .2 5 T o t a l .9 9 1 .2 0 1 .0 0 1 .1 9 103 I t w as h y p o t h e s i z e d , H^, t h a t t h e m ean s l o p e ( o f th e 13 d i s t r i c t s ) , w o u ld n o t b e d i f f e r e n t fro m th e s l o p e o f th e e q u a t i o n b a s e d on th e t o t a l p o p u l a t i o n . No a ssu m p t i o n s a b o u t th e d i s t r i b u t i o n s o f th e s l o p e s w ere m ade. T h e r e f o r e , t h e v e ry p o w e r f u l (14:84) n o n - p a r a m e t r l c s t a t i s t i c , th e W alsh t e s t , w as u s e d t o i n v e s t i g a t e t h e h y p o t h e s i s , • T h is h y p o t h e s i s , H^, w as a c c e p t e d a s t e n a b l e f o r e a c h o f t h e f o u r 3 e t s o f s l o p e s , a s i n d i c a t e d b y th e r e s u l t s su m m arized i n T a b le V I I . A s i m i l a r c o n c l u s i o n was r e a c h e d by th e u s e o f th e t t e s t (w here th e a s s u m p tio n o f n o rm al d i s t r i b u t i o n o f t h e s l o p e s h a d to be m ad e ) . The s l o p e s w e re t h e n i n v e s t i g a t e d b y m eans o f t h e F rie d m a n tw o-w ay a n a l y s i s o f v a r i a n c e (14) t o d e te r m in e w h e th e r o r n o t g r e a t e r d i f f e r e n c e s e x i s t e d b e tw e e n th e f o u r s e t s t h a n w i t h i n s e t s . The t e s t ( c h i - s q u a r e e q u a l to 31.46 w h ic h i s s i g n i f i c a n t o f t h a t 1 p e r c e n t l e v e l f o r 3 d f ) r e v e a l e d t h a t t h e n u l l h y p o t h e s i s i s r e j e c t e d , i n d i c a t i n g a g r e a t e r v a r i a t i o n e x i s t s b e tw e e n s e t s o f s l o p e s t h a n among s l o p e s w i t h i n s e t s o r t h a t t h e s l o p e i s d e p e n d e n t on th e a c h ie v e m e n t s u b t e s t a n d g r a d e . A ssu m in g a n o rm a l d i s t r i b u t i o n o f s l o p e s a n F t e s t y i e l d e d 104 TABLE V II A SUMMARY COMPARISON OF MEAN SLOPE OF THE ESTIMATED TRUE DIFFERENCE EQUATIONS OF EACH DISTRICT WITH THE TOTAL SLOPE OF ALL DISTRICTS FOR GRADE 4 AND 6 , READING AND ARITHMETIC, WHEN ESTIMATED TRUE DIFFERENCES ARE HELD CONSTANT AS COMPUTED BY THE WALSH TEST G rade A c h ie v e m e n t A re a S lo p e o f t h e T o t a l P o p u l a t i o n ( ± . S . D . ) S i g n o f W a lsh M ax. V a lu e S ig n o f W alsh M in. V alu e C o n c lu s io n * 4 R e a d in g . 9 8 8 ± .0 6 0 + - n o t s i g . 4 A r i t h m e t i c 1 .2 0 6 ± .1 3 4 + - n o t s i g . b R e a d in g 1 .Q0 0 ± .028 + - n o t s i g . 6 A r i t h m e t i c 1 .1 S 7 ± .1 1 4 + - n o t s i g . * T e s t w as c o n d u c te d a t t h e fyo l e v e l o f s i g n i f i c a n c e . 105 s i m i l a r r e s u l t s . The B a r t l e t t ' s t e s t f o r h o m o g e n e ity o f v a r i a n c e s gave e v id e n c e l e a d i n g t o th e r e j e c t i o n o f th e n u l l h y p o t h e s i s ( c h i - s q u a r e e q u a l t o 14.53 w h ic h i s s i g n i f i c a n t a t t h e 1 p e r c e n t l e v e l f o r 3 df). s u g g e s t i n g t h a t some o f t h e v a r i a t i o n b e tw e e n s e t s may b e a t t r i b u t a b l e t o u n e q u a l v a r i a n c e s b e tw e e n t h e s e t s . A f u r t h e r a n a l y s i s w as made t o i n v e s t i g a t e th e p o s s i b i l i t y o f u s i n g a s i n g l e r e g r e s s i o n l i n e f o r r e a d i n g a n d a r i t h m e t i c . A c o m p an io n a n a l y s i s i n v e s t i g a t e d th e p o s s i b i l i t y o f u s i n g a s i n g l e r e g r e s s i o n l i n e f o r G rade 4 R e a d in g a n d G rade 6 R e a d in g (a s i m i l a r I n v e s t i g a t i o n b e i n g made w i t h A r i t h m e t i c ) . T h is i n q u i r y , s t a t e d i n h y p o t h e s i s f o r m , i s g iv e n a s f o l l o w s ; H2 : T h e re a r e d i f f e r e n c e s b e tw e e n m ean s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s b a s e d on v a r i o u s d i s t r i c t s f o r R e a d in g a n d A r i t h m e t i c (a s s u m in g g r a d e c o n s t a n t ) . H^: T h e re a r e d i f f e r e n c e s b e tw e e n m ean s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s , b a s e d on v a r i o u s d i s t r i c t s f o r G rade 4 a n d G rade 6 R e a d in g a n d A r i t h m e t i c a n a l y z e d s e p a r a t e l y ) , Once a g a i n th e W alsh t e s t w as u s e d t o I n v e s t i g a t e t h e h y p o t h e s e s h2 a n d h y m eans o f t h e n u l l h y p o t h e s i s , 106 Hq . The f i n d i n g s w e re s u p p o r t e d b y t h e t t e s t , s h o u l d a s s u m p tio n s a b o u t a n o rm a l d i s t r i b u t i o n b e a c c e p t e d . The r e s u l t s o f t h e W alsh t e s t , su m m a riz e d i n T a b le V I I I , I n d i c a t e t h a t t h e N u ll h y p o t h e s i s , Hq, i s r e j e c t e d I n f a v o r o f t h e i n d i c a t e d h y p o t h e s e s b u t a c c e p t e d I n t h e c a s e o f t h e h y p o t h e s i s , H ^. T h u s, th e s l o p e s o f t h e e q u a t i o n s a r e d i f f e r e n t f o r d i f f e r e n t a c h ie v e m e n t t e s t s b u t n o t n e c e s s a r i l y f o r d i f f e r e n t g r a d e s . C o m p a riso n b e tw e e n t h e s l o p e s o f t h e t w o - v a r i a b l e (X ,Y ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d t h e t h r e e - v a r i a b l e (Xj Yj C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . The t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s f o r G rad e 4 a n d G ra d e 6 , R e a d in g a n d A r i t h m e t i c , f o r v a r i o u s d i s t r i c t s a r e r e p r e s e n t e d I n T a b le s X I I I , XIV, XV, a n d X V I. A p e r u s a l o f t h e t w o - v a r i a b l e e q u a t i o n s o f T a b le s IX , X, X I , a n d X II a n d t h e c o r r e s p o n d i n g t h r e e - v a r i a b l e e q u a t i o n s o f T a b le X I I I , XIV, XV, a n d XVI r e v e a l s i m i l a r i t i e s o f c o e f f i c i e n t s . A c o m p a r a tiv e l i s t o f th e s l o p e s o f t h e s e two b a s i c e q u a t i o n s I s g i v e n I n T a b le X V II. T h ese s i m i l a r i t i e s l e a d to t h e h y p o t h e s i s , HQ, t h a t t h e r e a r e no d i f f e r e n c e s b e tw e e n t h e s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a s t h e r e s u l t o f i n c l u d i n g 107 TABLE V I I I A SUMMARY COMPARISON OF THE SLOPES OF ESTIMATED TRUE DIFFERENCE EQUATIONS BETWEEN VARIOUS COMBINATIONS OF ACHIEVEMENT (READING AND ARITHMETIC) AND GRADE (FOURTH AND SIXTH) BY MEANS OF THE WALSH TEST C o m p a ra tiv e s e t s fro m w h ic h s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a r e b a s e d S ig n o f W alsh Max, V a lu e S ig n o f W alsh M in. V alu e C o n c lu s io n G rad e 4 R e a d , a n d G rade 6 R e a d , + n o t s i g , * * G rade 4 A r i t h , a n d G rade 6 A r l t h * + - n o t s i g . * * G ra d e 4 R e a d , a n d G rad e 4 A r i t h . + + s i g . * G ra d e 6 R e a d , a n d G rade 6 A r i t h • + + s i g . * * T e s t w as c o n d u c te d a t th e 1 p e r c e n t l e v e l o f s i g n i f i c a n c e . ** T e s t w as c o n d u c te d a t th e 5 p e r c e n t l e v e l o f s i g n i f i c a n c e • 108 TABLE IX ESTIMATED TRUE DIFFERENCE (5 1 ) EQUATIONS BASED ON GRADE 4 READING (Y) AND MENTAL AGE (X) DATA FOR THE DISTRICTS D i s t r i c t N T w o - v a r ia b le e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 477 5' = -.77X .82Y + .10 2 101 6* -.87X + .8 6 y - .11 3 179 b* = -.86X + .89Y - .19 4 36 b' s -•67X + .60Y + .06 5 526 b 1 = -,77X + .75Y - .98 6 20 b1 = ”.59X + .56Y - .18 7 316 6* = -.77X + • 8 4 y + .61 8 273 b ' r -.75X + .69Y + .03 9 781 b 1 - - .88x + .8 4 y + .08 10 142 b ' - -.88x + .92Y - .19 11 49 b' - -.79X + .79Y + .82 12 167 b ' - -.82X + .78y + .13 13 692 b ' s -.76X + .7 6 y - .06 T o t a l 3759 b* - -.82X + .8 3 y _ .11 109 TABLE X ESTIMATED TRUE DIFFERENCE (6 1 ) EQUATIONS BASED O N GRADE 4 ARITHMETIC (Y ) AND MENTAL AGE (X ) DATA FOR THE DISTRICTS D i s t r i c t N T w o - v a r i a b l e e s t i m a t e d , t r u e d i f f e r e n c e e q u a t i o n 1 477 6 ' = - .75X + • 6 l y + .4 9 2 101 6' ■ - • 85X + .6 6 y - .22 3 179 6* = - •86x + *76y - .62 4 36 & ' = - • 64 x + .62Y - .2 6 5 326 6 1 m - .79X + .5?Y - .11 6 20 6 ' s - .76X + • 6 8 y - .7 9 7 316 6 1 - - .7 6 x + .65Y + .4 1 8 273 6 ' - - .79X + .3 4 y + .0 1 9 761 6 ' s - • 83x + .6 7 y - .3 9 10 142 6 ' s - • 87X + • 8 8 y - .0 6 11 49 5 ' s - .82X + .69Y - .55 12 167 6 ' s mm • 84x + .67y - .32 13 692 6 ' ■ - *8lX + •64y - .21 T o ta l 3759 6 ' s mm ♦82X + • 6 8 y - .4 4 110 TABLE XI ESTIMATED TRUE DIFFERENCE (6 ' ) EQUATIONS BASED ON GRADE 6 READING (Y) AND MENTAL A G E E (X) DATA FOR THE DISTRICTS D i s t r i c t N T w o - v a r ia b le e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 363 &' s - • 62X + • 65 y - .5 3 2 67 6 ' m - • 78X + .77Y - .08 3 247 5 1 - mm • 83x + .82Y - .1 2 4 29 5 ' - - .62X + • 66y + .3 2 5 508 5 1 = - .72X + • 71Y - .1 6 6 20 5 1 m m - *9^X + .9^Y - .07 7 266 6 ' - - .75X + .75Y - .1 8 8 252 6 ' a - •80X + •83 y + .5 1 9 781 5 ' s - .7 6 x + .75Y - .4 1 10 162 & 1 ■ - •85X + • 84 y + .9 0 11 44 6 ' = - .8 4 x + •83 y - .0 1 12 188 6 ' ■ - • 77X + .7 4 y - .01 13 330 &' = - .78X + .77Y - .1 0 T o t a l 3116 6 ' mm - .7 8 x + 00 c- • - .2 5 I l l TABLE X II ESTIMATED TRUE DIFFERENCE (5*) EQUATIONS BASED O N GRADE 6 ARITHMETIC (Y ) AND MENTAL AGE (X) DATA FOR THE DISTRICT D i s t r i c t N T w o - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 363 6 ' m - • 80X + • 69Y + .36 2 67 6 * mm - •82X + . 56Y + .41 3 247 6 ' 8 - *84x + .65Y - .99 4 29 6 * 8 - .76x + • 55Y - .09 5 508 6* 8 - .81X + • 61Y + .01 6 12 5 1 8 - • 92X + •8lY - .46 7 266 5 1 - • 86x + • 73y + .62 8 252 6 ' - - •84x + . 76y + .24 9 648 5 1 8 - • 84x + • 71Y + .5 6 10 162 6 ' m m - .89X + • 73Y - .17 11 44 6 1 s - & 00 • + .77Y - .37 12 188 6 ' s - X ^3 - 00 • + • 64y + .07 13 330 6 ' = - .85X + • 68y + .81 T o t a l 3116 6' s - .85X + • 71Y + .6 6 112 TABLE X III ESTIMATED TRUE DIFFERENCE (&') EQUATIONS BASED ON GRADE 4 READING (Y), MENTAL AGE (X), AND CHRONOLOGICAL AGE (C) DATA FOR THE DISTRICTS D i s t r i c t N T h r e e - v a r i a b l e E s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 477 &' B “ .77X + .82Y - • 07C + .39 2 101 5 1 - .87X + .85Y - • 07C + .02 3 179 5 1 - - .8bX + ,87Y - • 08c + .02 4 36 6 ' = - .66x + .57Y + • 22C - • 08 5 92 b 5 1 - .77X + *74y - • 07C + .03 6 20 5 ' = - • 59X + .45Y # * • 31C + .14 7 316 5 * = - *77X + . 84y - • 09C + • 03 8 273 6 1 = - .75X + • 68y _ • IOC + .04 9 781 5 ' - - .89X + .80y - • 09C + .0 5 10 142 U ' « ■ - .87X + .90Y - • 13C + .04 11 49 6 1 - - .80X + .7 3 y - • 09C + .0 2 12 167 6 ' = - •8lX + .7 8 y m m • 08C + .04 13 692 6 * = - .73X + .79Y - • 07C + .0 2 T o t a l 3759 6 ' B - X O J 00 • + .82Y - • 07C + .0 2 1 1 3 ' TABLE XIV ESTIMATED TRUE DIFFERENCE (6 ' ) EQUATIONS BASED ON GRADE 4 ARITHMETIC (Y), MENTAL AGE (X), AND CHRONOLOGICAL AGE (C) DATA FOR THE DISTRICTS D i s t r i c t N T h r e e - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 477 6 ' — 2 101 5 1 s. 3 179 6 ' - 4 36 6 ' ■ 5 526 6 ' s 6 20 6 ' B 7 316 6* = 8 273 5 ' = 9 731 6 1 B 10 142 6 ' = 11 49 6 1 - 12 167 5 1 m 13 692 6 ' = T o t a l 3759 6 ' 8 78X + • 61Y m * • O O + .0 8 87X + • 64y - .09C + .13 CO p\ X + • 75Y - .08C + .07 67X + • 62Y - • 04c + .0 2 - ^ 1 00 X + . 56y - *04c + .1 0 64x + • 68y + .20C - .1 0 77X + • 64y - .1 0 c + .09 77X + «54y - .06c + .1 2 85X + • 68y - . 08c + .1 0 88X + • S6y - .IOC + .04 8 OX + • 69Y + .02C + .0 2 83x + .68y + .o o c + .0 6 ,78X + • 6 3 y - .02C + .07 83X + • 68y - .05C + .0 8 114 TABLE XV ESTIMATED TRUE DIFFERENCE ( 6 1) EQUATIONS BASED ON GRADE 6 READING (Y ), MENTAL AGE (X ), AND CHRONOLOGICAL AGE (C) DATA FOR THE DISTRICTS D i s t r i c t N T h r e e - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s 1 363 6 ' s - .62X + .6 4 y - .0 2 0 - .0 3 2 67 b ' ■ - *77X + *77Y ~ . 06c + .02 3 247 6 ' a - •83x + .82Y - .QIC + .0 0 4 29 b ' a - .61X + • 6 lY - .22C + .1 3 5 508 b ' - - • 72X + .69Y - .21C + .1 2 6 12 b 1 s - .94X + .93Y - . 08c + .0 4 7 266 6 ' = - .75X + .74Y - •04c + .01 8 252 5 1 a - .BOX + .8 3 y - .07C + .0 3 9 6 48 6 ' r m m . BOX +• .7 5 Y - .03C + .0 2 10 162 b ' a - . 85X + . 84y - .06C + .0 2 11 44 b ' a m t .84X + .8 3 y - . 01c + .00 12 188 b ' = - .77X + .7 4 y - C M O . + O O . 13 330 b ' a - .7 8 x + *76y - ,06C + .0 3 T o t a l 3116 b 1 a mm • 77X + .7 8 y - .05C + .02 115 TABLE XVI ESTIMATED TRUE DIFFERENCE (6 1) EQUATIONS BASED ON GRADE 6 ARITHMETIC (Y ), MENTAL AGE (X ), AND CHRONOLOGICAL AGE (C) DATA FOR THE DISTRICTS D i s t r i c t N T h r e e - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n 1 363 6 ' ■ - .79X + .69Y + .07C - .01 2 67 6' ■ - .82X + • 59Y - .02C + .1 5 3 247 b ' s - • 85X + • 69Y - .01C + . 3 * 1 4 29 b ' a - .76x + .54y - .030 + .11 5 908 5 1 « • - .8lx + • 60y - .04c + .13 6 12 5* ■ - .91X + • 77Y - .18C + .17 7 266 5 1 a - .85X + .72y - .02C + .07 8 292 5 1 = - *84x + .79Y - .04c + .09 9 648 5' - - .84x + • 7lY - .0 2 0 + .0 0 10 102 6 r = - .89X + • 72Y - .040 + .11 11 44 b * a - .88x + • 77Y + .010 + .0 5 12 188 6 1 m - •84x + • 63y - .070 + .1 5 13 330 6 1 S - • 00 * + • 66y - .070 + .13 T o t a l 3116 6 1 a - .85X + .7 3 y - .0 2 0 + .0 6 1 1 6 TABLE XVII SUMMARY COMPARISON OF THE SLOPES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE (X,Y) AND THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATIONS FOR GRADE 4 AND GRADE 6 , READING AND ARITHMETIC G rade 4 G rade 4 G rad e 6 G rade 6 R e a d in g A r i t h . R e a d in g A r i t h . D i s t . (X ,Y )(X ,Y ,C ) (X ,Y )(X ,Y ,C ) (X ,Y )(X ,Y ,C ) (X ,Y )(X ,Y ,C ) 1 .9 4 .94 2 1 .0 1 1 .0 2 3 .97 .9 9 4 1 .1 1 1 .1 6 5 1 .0 3 1 .0 4 6 1 .0 5 1 .3 2 7 .92 .9 2 6 1 .0 9 1 .0 9 9 1 .0 5 1 .0 6 10 .9 3 .97 11 1 .0 0 1 .0 2 12 1 .0 5 1 .0 4 13 1 ,0 0 1 .0 0 T o t a l .9 9 1 .0 0 1 .2 3 1 .2 8 .9 5 1 .2 8 1 .3 5 1 .0 1 1 .1 4 1 .1 3 1 .0 1 .97 1 .0 8 .9 4 1 .3 9 1 .3 9 1 .0 1 1 .1 2 .9 4 1 .0 0 1 ,1 6 1 .2 0 1 .0 0 1 .4 7 1 .4 3 .96 1 .2 3 1 .2 5 1 .0 1 .9 9 1 ,0 2 1 .0 1 1 .1 9 1 .1 6 1 .0 1 1 .2 5 1 .2 2 1 .0 4 1 .2 6 1 .2 4 1 .0 1 1 .2 0 1 .2 2 1 .0 0 .97 1 .1 6 1 .1 5 1 .0 0 1 .4 7 1 .4 7 1 .0 1 1 .3 0 1 .3 2 1 .0 0 1 .3 9 1 .4 1 1 .0 4 1 .3 3 1 .3 5 1 .0 1 1 .1 4 1 .1 9 1 .0 1 1 .1 8 1 .1 8 .90 1 ,1 1 1 .1 2 1 .0 6 1 .1 8 1 .1 8 1 .0 1 1 .2 2 1 .2 3 1 .0 1 1 .1 5 1 .1 4 1 .0 4 1 .3 2 1 .3 3 1 .0 3 1 .2 5 1 .2 8 .9 9 1 .1 9 1 .1 6 117 c h r o n o l o g i c a l a g e . T h e W a l s h t e s t ( s u m m a r i z e d I n T a b l e X V I I I ) i n d i c a t e s t h a t t h e h y p o t h e s i s H q , i s a c c e p t e d a s t e n a b l e f o r G r a d e 4 R e a d i n g , G r a d e 4 A r i t h m e t i c , a n d G r a d e 6 A r i t h m e t i c a n d r e j e c t e d f o r G r a d e 6 R e a d i n g . I t i s i n t e r e s t i n g t o n o t e t h a t , a l t h o u g h , i n g e n e r a l , t h e s l o p e s o f t h e t w o - v a r i a b l e (X,Y) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a r e l a r g e r t h a n t h e s l o p e s o f t h e t h r e e - v a r i a b l e (X ,Y ,C ) e q u a t i o n s , t h e u s e f u l n e s s o f t h i s f i n d i n g i s o f s t a t i s i c a l s i g n i f i c a n c e r a t h e r t h a n p r a c t i c a l s i g n i f i c a n c e . T h a t i s , t h e d i f f e r e n c e b e t w e e n c o r r e s p o n d i n g s l o p e s ( f o r G r a d e 6 r e a d i n g ) o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , w h e n c h r o n o l o g i c a l a g e i s a d d e d a n d w h e n i t i s n o t a d d e d i s o f n o p r a c t i c a l s i g n i f i c a n c e s i n c e a d i f f e r e n c e o f . 0 0 7 i n s l o p e r e v e r s e s t h e d e c i s i o n o f t h e W a l s h t e s t . T h i s d i f f e r e n c e o f . 0 0 7 i n s l o p e ( w h i c h i s a l s o t h e m e d i a n o f t h e d i f f e r e n c e s ) i s e q u a l t o a n a n g u l a r d i f f e r e n c e o f 1 2 * ( o n e - f i f t h o f o n e d e g r e e ) . I n o r d e r f o r t h i s d i f f e r e n c e t o m a n i f e s t a c h a n g e o f o n e m o n t h o f a c h i e v e m e n t (Y) t h e m e n t a l a g e d i f f e r e n c e (X) w o u l d h a v e t o t h e 14 y e a r s , a d i f f e r e n c e w h i c h , w h e n a d d e d o r s u b t r a c t e d f r o m G r a d e 4 i s m e a n i n g l e s s a n d s t a t i s t i c a l l y i m p r o b a b l e . 118 TABLE XVIII A S U M M A R Y C O M P A R I S O N B E T W E E N T H E S L O P E S O F T H E T W O - V A R I A B L E A N D T H E T H R E E “ V A R I A B L E E S T I M A T E D T R U E D I F F E R E N C E E Q U A T I O N F O R G R A D E 4 R E A D I N G A N D A R I T H M E T I C A N D A R I T H M E T I C G R A D E 6 R E A D I N G C O M B I N A T I O N S A N D G r a d e o f a c h i e v e m e n t t e s t f r o m w h i c h t h e s l o p e o f t h e t w o - v a r i a b l e ( X , Y ) a n d t h e t h r e e - v a r i a b l e ( X , Y , C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a r e c o m p a r e d S i g n o f W a l s h M a x . V a l u e S i g n o f W a l s h M i n . V a l u e C o n c l u s i o n s * G r a d e 4 R e a d i n g + - s i g . G r a d e 4 A r i t h m e t i c + - n o t s i g . G r a d e 0 R e a d i n g + + n o t s i g . G r a d e 0 a r i t h m e t i c + - n o t s i g . * T e s t w a s c o n d u c t e d a t t h e 3 ^ l e v e l o f s i g n i f i c a n c e . 119 C o m p a r i s o n b e t w e e n t h e s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d t h e a p p r o x i m a t e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . S i n c e t h e c o e f f i c i e n t s o f X a n d Y a r e i d e n t i c a l I n t h e a p p r o x i m a t e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s , a s s e e n I n T a b l e s X I X , X X , X X I , a n d X X I I , t h e s l o p e s o f t h e e q u a t i o n s a r e t h e s a m e a n d a r e e q u a l t o u n i t y . C o r r e s p o n d i n g t o e a c h o f t h e f o u r s e t s o f e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a r e f o u r s e t s o f a p p r o x i m a t e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . D i f f e r e n c e s b e t w e e n t h e m e a n s o f t h e s e c o r r e s p o n d i n g p a i r s w e r e c o m p a r e d . T h e h y p o t h e s i s , H ^ , s u g g e s t i n g t h a t t h e s l o p e s o f t h e e s t i m a t e d a n d a p p r o x i m a t e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a r e n o t t h e s a m e a n d c a n n o t b e i n t e r c h a n g e d , w a s t e s t e d . U p o n i n t e r p r e t i n g t h e r e s u l t s o f t h e V / a l s h t e s t , w h o s e s u m m a r y i s g i v e n I n T a b l e X X I I I , i t i s c o n c l u d e d t h a t , i n t h e N u l l h y p o t h e s i s , w h i c h c o r r e s p o n d t o t h e c o n t r a r y h y p o t h e s i s , H ^ , i s r e j e c t e d f o r G r a d e 4 a n d 6 A r i t h m e t i c a n d t e n a b l e f o r G r a d e s 4 a n d 6 R e a d i n g . T h e r e f o r e , H ^ , i s a c c e p t e d a s t e n a b l e f o r A r i t h m e t i c a n d i n g e n e r a l , t h e s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ( a f t e r L o r d ) a n d t h e c o r r e s p o n d i n g a p p r o x i m a t e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ( a f t e r M c N e m a r ) a r e n o t 120 TABLE XIX APPROXIMATE ESTIMATED TRUE DIFFERENCE (D , ) EQUATION BASED O N GRADE 4 READING (Y) AND MENTAL AGE (X) DATA OF THE DISTRICTS D i s t r i c t N A p p ro x im a te E s t i m a t e d D i f f e r e n c e E q u a t io n 1 4 7 7 d m - mm - • S O X + • 8 0 Y - • 1 4 2 1 0 1 d m ' = - • 8 6 X + • 86 y - . 1 5 3 1 7 9 d m » = - • 8 7 X + • S 7 Y - . 0 5 4 3 b d m ' = - • 6 7 X + • 6 7 Y - • 2 2 5 5 2 6 d m ' ** - • 7 6 X + • 7 6 Y - ,0 6 6 20 d m * = - • 5 S X + • 5 8 y - . 4 2 7 3 1 6 d m ' = - • 8 2 X + • 8 2 Y - . 0 3 8 2 7 3 d m ' - • 7 3 X + • 73y - .2 0 9 7 8 1 d m * = - • 8 4 X + •84 y - .0 8 1 0 1 4 2 d m - = - • 9 0 X + • 9 0 Y - .0 0 11 49 d m * = - • 7 9 X + • 7 9 Y - . 1 9 1 2 1 6 7 d m ‘ B - • S O X + • 8 0 Y - .0 2 13 692 d m ' = - • 7 5 X + • 7 5 Y - .08 T o t a l 3759 1 s - • S 2 X + • 8 2 Y - ,0 8 121 TABLE XX APPROXIMATE ESTIMATED TRUE DIFFERENCE (Dm , ) EQUATION BASED ON GRADE 4 ARITHMETIC (Y ) AND MENTAL AGE (X) DATA O F THE DISTRICTS D i s t r i c t N A p p ro x im a te E s t i m a t e d D i f f e r e n c e E q u a t io n 1 477 Dm t M' B - .76X + .7 6 y — .17 2 101 DM | M1 B - • 86x + • 00 S' - .0 4 3 179 Dm , M' = - .84x + •84y - .06 4 36 dm' - - .65X + .65Y - .19 5 526 dm‘ s - • 8lX + . 81Y mm .0 1 6 20 dm ‘ = - .75X + .75Y - .0 6 7 316 D., | M' = - .75X + .75Y - .0 9 8 273 dm« = - .80X + .80Y - .0 1 9 781 dm * z - .83X + .83Y - .07 10 142 dm ‘ s - .87X + .87Y mm .0 2 11 49 dm' s - .82X + . 82Y mm .1 0 12 167 dm i mm - • 86x + .8 6 y - .0 5 13 69 2 dm * = - .82X + .82Y mm .0 4 T o t a l 3759 dm ‘ mm - .82X + . 82y - .0 6 122 TABLE XXI APPROXIMATE ESTIMATED TRUE DIFFERENCE (E^,) EQUATION BASED ON GRADE 6 READING (Y) AND MENTAL AGE (X) DATA O F THE DISTRICTS D i s t r i c t N A p p ro x im a te E s t i m a t e d D i f f e r e n c e E q u a t io n 1 363 dM » ■ - • 63X + •63 y - .31 2 67 dm' m - • 77X + • 77y m m .1 0 3 247 dm* - .82X + .82Y - .1 6 29 dm« ~ - .65X + .65Y - .42 5 508 dm’ - • 71X + • 71Y - .25 6 12 dm‘ = - • 9^X + .9^Y - ,0 8 7 266 dm * P * - • 74x + .74Y .1 8 8 252 dm* = - .82X + .82Y - .27 9 o46 dm' * * _ .75X + .75Y - .2 8 10 162 dm' * * - .85X + .85Y - .14 11 44 dm' s - *84x + .8 4 y - .07 12 188 dm' = •* .76x + .7 6 y - .2 1 13 330 dm' - - .77X + .7 7 y - .1 8 T o t a l 3116 dm' 8 - X C O # + 00 r — • - .2 1 123 TABLE XXII APPROXIMATE ESTIMATED TRUE DIFFERENCE (Dm<) EQUATION BASED O N GRADE 6 ARITHMETIC (Y ) AND MENTAL AGE (X) DATA O F THE DISTRICTS D i s t r i c t N A p p ro x im a te E s t i m a t e d D i f f e r e n c e E q u a t io n s 1 363 dm ' ■ - .8ox + •80Y - .26 2 67 dm - s - •88x + • 88y - .06 3 247 dm ' s - .87X + .87Y - .12 4 29 dm ' ■ - .83x .83Y - .18 5 608 dm ' M - $ G O • + •86y - .08 6 12 dm * m - .93X + .93y - .08 7 266 dm ' a - .86x + .86y - .19 8 262 dm» - - • 84x + .8 4 y - .19 9 648 dm ' = - .86x + C O • - .18 10 162 dm > s - • 91X + • 91Y - .09 11 44 d m « ■ - • 91X + .91Y - .03 12 188 dm > ■ - •88x + .88y - .19 13 330 dm ' s - .87X + .87Y - .13 T o t a l 3116 dm » B - • 91X + • 91Y - .10 1 2 4 TABLE X X III A SUMMARY COMPARISON BETWEEN THE SLOPES OF THE ESTIMATED TRUE DIFFERENCE EQUATION (AFTER LORD) AND THE APPROXIMATE ESTIMATED TRUE DIFFERENCE EQUATION (AFTER MCNEMAR) FOR GRADE 4 READING AND ARITHMETIC AND GRADE 6 READING AND ARITHMETIC G ra d e a n d A c h ie v e m e n t T e s t S i g n ■ h o C O fro m t h e S l o p e s o f t h e o f E s t i m a t e d T ru e D i f f e r e n c e th e th e a n d t h e a p p r o x im a te E s t i m a t e d W alsh W alsh t r u e D i f f e r e n c e E q u a t i o n s V a lu e V a lu e a r e c o m p a red M ax, M in , G rad e 4 R e a d in g + - s i g . G ra d e 4 A r i t h m e t i c + + s i g . G ra d e 6 R e a d in g + - s i g . G ra d e 6 A r i t h m e t i c + + s i g . * T e s t w as c o n d u c te d a t th e 1; % l e v e l o f s i g n i f i c a n c e 1 25 th e sa m e • C o m p ariso n b e tw e e n th e s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d t h e a l g e b r a i c d i f f e r e n c e e q u a t i o n s . The c o e f f i c i e n t s o f X a n d Y o f th e a l g e b r a i c d i f f e r e n c e e q u a t i o n (D» Y-X) a r e b o th u n i t y . A lth o u g h t h e s e c o e f f i c i e n t s a r e n o t th e same a s t h o s e o f th e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , t h e s l o p e s o f e a c h o f t h e s e e q u a t i o n s a r e I d e n t i c a l ( u n i t y ) . T h e r e f o r e , th e same c o n c l u s i o n s t h a t h o ld f o r th e a p p r o x i m ate e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w i t h r e g a r d to s l o p e a l s o a p p ly t o th e a l g e b r a i c d i f f e r e n c e e q u a t i o n s . T h a t i s , th e s l o p e s o f th e a l g e b r a i c d i f f e r e n c e e q u a t i o n s w i t h r e s p e c t t o a r i t h m e t i c a r e n o t th e same a s th o s e o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . Summary c o m p a ris o n o f s l o p e s f o r th e v a r i o u s s t r a i g h t - l i n e e x p e c ta n c y e q u a t i o n s . S t a t i s t i c a l c o m p a r i s o n s b e tw e e n th e v a r i o u s s t r a i g h t - l i n e e x p e c t a n c i e s e q u a t i o n s hav e b e e n r e p o r t e d . S t a t i s t i c a l I n v e s t i g a t i o n s a r e o b j e c t i v e , a s w e l l a s n e c e s s a r y , when th e r i g o r o u s a n a l y s i s o f d a t a i s u n d e r t a k e n . They a r e , h o w e v e r, o f t e n made I n d e p e n d e n t o f a n y r e f e r e n c e t o a m e a n in g f u l , c l e a r , o r p l c t o r i a l l y o b s e r v a b l e p r e s e n t a t i o n o f th e d a t a . 126 The s l o p e s o f t h e d i f f e r e n c e e q u a t i o n s u s e d i n t h i s s t u d y a r e p r e s e n t e d i n T a b l e s XXIV, XXV, XXVI, a n d XX V II, S in c e t h e u n i t o f m e a su re m e n t m o st common t o th e a n g le i s th e d e g r e e , t h e s l o p e s o f t h e l i n e s a r e e x p r e s s e d a s d e g r e e s fro m th e X - a x is (w here t h e a n g le i s e q u a l t o th e a r c t a n o f t h e s l o p e . ) C o m p a riso n b e tw e e n v a r i o u s I n t e r c e p t s f o r th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w hen 6 - 0 . S in c e b o t h th e s l o p e a n d ( a x i s ) i n t e r c e p t o f a l i n e m u st b e known f o r a c o m p le te k n o w le d g e o f t h e c h a r a c t e r i s t i c s o f t h a t l i n e , t h e n e x t I n v e s t i g a t i o n I n t h i s s t u d y d e a l t w i t h th e i n t e r c e p t s . I t i s o b v io u s t h e i n t e r c e p t s o f G rade 4 a n d G rade 6 a r e n o t t h e sam e, b u t , on t h e a v e r a g e , d i f f e r e n t by two y e a r s . C o n s e q u e n tl y , th e i n t e r c e p t s b e tw e e n g ra d e l e v e l s a r e n o t c o m p a re d . S e p a r a t e h y p o t h e s e s , c o m p a rin g r e a d i n g a n d a r i t h m e t i c , w e re made f o r G rad e 4 a n d G rade 6 . The f i r s t h y p o t h e s i s , H^, m a i n t a i n s t h a t t h e r e a r e d i f f e r e n c e s b e tw e e n t h e i n t e r c e p t s o f th e t w o - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s f o r r e a d i n g a n d a r i t h m e t i c f o r G rad e 4 t h a t c a n n o t be a t t r i b u t e d t o c h a n c e . F iv e d i f f e r e n t i n t e r c e p t s l e v e l s o f m e n ta l a g e w ere c o m p a re d . T h ese f i v e l e v e l s I n c l u d e , (1 ) " a t g r a d e " ( 4 .1 f o r G ra d e 4 a n d 6 .1 127 TABLE XXIV SUMMARY OF THE SLOPES BY DEGREES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE (6 * _ ) , THE x y THREE-VARIABLE (& 'x y c ) / THE APPROXIMATE (Dm, ) TRUE DIFFERENCES EQUATIONS AND THE ALGEBRAIC DIFFERENCE (D ), GRADE 4 READING D i s t r i c t S lo p e o i 5 1 xy S lo p e o f 6 * xyc S lo p e o f Dm I M' S lo p e D 1 43 °1 2 1 43 °12 * 49° 49 2 4 5 ° 2 0 ‘ 4 5 ° 4 0 1 49° 49 3 4 4 ° 0 8 1 4 4 ° 4 o 1 490 49 4 4 8 ° 0 9 ' 4 9 ° 1 1 1 49° 49 4 9 ° 4 9 ' 46® 081 490 49 6 4 b ° 3 0 1 92 ° 4 0 1 49° 49 7 42°31* 42 ° 3 1 ' 49° 49 8 47 ° 2 3 1 4 7 ° 4 6 1 490 49 9 46 °2 0 1 4 6 ° 4 4 ' 490 49 10 46°17 • 4 4 °Q 2' 490 49 11 4 5 ° 0 0 ' 4 9 ° 4 4 ' 490 49 12 4 6 ° 2 6 ‘ 4 b 00 9 1 49° 49 13 4 3 ° 0 0 1 4 9 ° 0 0 ' 490 49 T o t a l 4 4 ° 3 y ' 4 9 ° 0 0 1 490 49 128 TABLE XXV SUMMARY OF THE SLOPES BY DEGREES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE (6 * ) , THE THREE-VARIABLE (& 'x y c )> THE APPROXIMATE (DM, ) TRUE DIFFERENCES EQUATIONS, AND THE ALGEBRAIC DIFFERENCE (D), G R A D E 4 ARITHM ETIC D i s t r i c t S lo p e o f 6 ' xy S lo p e o f 6 1 xyc S lo p e o f D M I S lo p e o f D 1 5 0 °5 3 ' 51 °5 8 ' 450 45° 2 3 2 °1 0 ' p3 °4 0 1 450 450 3 4 8 °3 2 ‘ 45055, 45° 450 4 47 °1 3 1 470131 450 45° 8 3 4 *1 1 ' 5 4 0 2 0 ' 450 450 6 48°1 1 » 43°1 6 « 450 450 7 49°2 8 ' 50°16' 450 450 8 83°3 9 ' 5 4 0 5 7 , 450 ^9° 9 3 1 °0 3 1 3 1 °2 0 ' 45° 450 10 44°4Q' 4 5 °3 9 ' 450 450 11 49055, 49013, 450 450 12 51 °2 5 ' 50°4 0 ' 45° 450 13 5 I04 1 1 51 °0 4 1 45° 450 T o t a l 3 0 °2 0 ' 50°4 0 ' 45° 450 129 TABLE XXVI SUMMARY OF THE SLOPES BY DEGREES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE (6 ' ) , THE w THREE-VARIABLE (& 'x y c )> THE APPROXIMATE (Dm, ) TRUE DIFFERENCES EQUATIONS, AND THE ALGEBRAIC DIFFERENCE (D ), GRADE 6 READING D i s t r i c t S lo p e o f 5 ' xy S lo p e o f 6 ' xyc S lo p e o f dm « Slope D 1 4 3 ° 3 9 ‘ 44°05» 45° 45 2 4 5 ° 2 2 ' 4 5 ° 0 0 ' 45° 45 3 4 5 ° 2 1 1 4 5 ° 2 1 1 45° 45 4 4 3 ° 1 3 ' 4 5 ° 0 0 ' 45° 45 5 4 5 ° 2 4 ‘ 4 6 ° 1 3 1 45° 45 6 4 5 ° 0 0 ' 4 5 ° 1 8 1 45° 45 7 4 5 ° 0 0 1 4 5 ° 2 3 1 45° 45 a 4 3 °^7 i 43 °57 1 45° 45 9 4 5 ° 2 3 ‘ 4 6 ° 5 1 1 45° 45 10 4 5 o2 0 ' 4 5 ° 2 0 ‘ 45° 45 11 4 5 °2 Q ' 4 5 ° 2 0 1 45° 45 12 4 6 ° 0 8 1 4 6 ° 0 8 ' 45° 45 13 4 5 °2 2 ' 4 5 ° 4 5 ‘ 45° 45 T o t a l 4 5 ° 0 0 ' 4 4 ° 3 8 ' 45° 45 1 3 0 TABLE XXVII SUMMARY OF THE SLOPES BY DEGREES OF THE REGRESSION LINES BASED UPON THE TWO-VARIABLE (&' ) , THE xy THREE-VARIABLE (& 'x y c )# THE APPROXIMATE (Dm ,) TRUE DIFFERENCES EQUATIONS, AND THE ALGEBRAIC DIFFERENCE (D ), GRADE 0 ARITHMETIC D i s t r i c t S lo p e o f 6 1 x y S lo p e o f 6 ' x y c S lo p e o f DM' Slope D 1 4 9 ° 1 3 1 4 8 ° 5 2 1 45° 49 2 9 8 ° 4 0 ' 9 5 ° 4 0 1 49° ^9 3 9 2 °1 6 ' 92 ° 3 b 1 49° 49 4 9 4 o0 6 ' 9 4 ° 3 u ' 49° 49 9 9 3 ° 0 1 1 93°28« 45° 49 u 4 8 ° 3 d 1 4 9 ° 4 o * 49° 49 7 4 9 ° 4 o ' 4 9 ° 4 4 1 49° 49 b 4 7 ° 9 2 ' 4 b ° 1 4 1 490 49 9 49°48* 49°48* 49° 49 10 5 0 ° 3 d ' 9 1 ° 0 2 « 49° 49 11 4 9 ° 0 8 ' 4 8 ° 4 9 ‘ 45° 49 12 9 2 ° 4 2 1 9 3 ° 0 8 1 49° 49 13 9 1 0 2 0 ' 9 1 ° 9 1 ' 49° 49 T o t a l 8 0 ° o8 ' 4 9 ° 2 1 ' 49° 49 131 f o r G rade 6 , (2 ) o n e y e a r b e lo w g r a d e , ( 3 ) two y e a r s b e lo w g r a d e , (4 ) one y e a r a b o v e g r a d e , a n d (5 ) two y e a r s a b o v e g r a d e . The s e c o n d c o m p a n io n h y p o t h e s i s , H^, i s th e sa m e, e x c e p t t h a t i t a p p l i e s t o G rad e 6 r a t h e r t h a n t o G rad e 4 , The r e s u l t s o f th e t t e s t , a s a p p l i e d t o th e h y p o t h e s i s H^, a r e p r e s e n t e d i n T a b l e s X X V III, T hey i n d i c a t e t h a t a l t h o u g h two o f t h e f i v e i n t e r c e p t s c a n b e c o n s i d e r e d e q u a l , th e o t h e r t h r e e a r e s t a t l s l c a l l y d i f f e r e n t . The d i f f e r e n c e s a r e g r e a t e s t i n th e b e lo w g r a d e a n d a b o v e g r a d e r a n g e . H y p o th e s is H -, i s a c c e p t e d f o r a l l l e v e l s o f m e n ta l a g e ( e x c e p t f o r o n e ) s i n c e th e c o r r e s p o n d i n g N u ll h y p o t h e s i s , HQ, i s r e j e c t e d . I n g e n e r a l , t h e n , th e e v id e n c e I n d i c a t e s t h a t t h e same d i f f e r e n c e e q u a t i o n s c a n n o t a p p r o p r i a t e l y b e u s e d f o r r e a d i n g a n d a r i t h m e t i c . C o m p a riso n b e tw e e n th e I n t e r c e p t s o f t h e tw o - v a r i a b l e (X ,Y ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a n d t h e t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s when 5 i 0 , A lth o u g h i t h a s b e e n shown t h a t t h e s l o p e s o f th e t w o - v a r i a b l e (X,Y) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s a r e s i m i l a r t o th e c o r r e s p o n d i n g s l o p e s o f t h e t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s t h e i n t e r c e p t s w ere i n v e s t i g a t e d i n o r d e r t o d e te r m i n e 132 TABLE XX VIII SUMMARY COMPARISON OF (FIVE) MENTAL AGE INTERCEPTS FOR READING AND ARITHMETIC ESTIMATED TRUE DIFFERENCE EQUATIONS FOR VARIOUS DISTRICTS AND GRADES BY MEANS OF THE t TEST G rade M e n ta l a g e t - R a t i o l e v e l 4 2 .1 5 .8 9 * * 4 3 .1 4 . o l * * 4 4 .1 2 .0 0 4 5 .1 2 .0 6 4 6 .1 4 .5 1 * * 6 4 .1 8 .1 4 * * o 8 .1 8 . 0I * * 0 6 ,1 ,09 6 7 .1 6 .7 4 * * 6 8,1 7.68 * * * S i g n i f i c a n t a t t h e 5# l e v e l o f c o n f i d e n c e f o r 13 d f . ** S i g n i f i c a n t a t t h e 1 % l e v e l o f c o n f i d e n c e f o r 13 d f . 133 s i m i l a r i t i e s o f a l l c h a r a c t e r i s t i c s b e tw e e n t h e s e l i n e s . The f o l l o w i n g h y p o t h e s i s , H? , i n d i c a t e d t h a t t h e r e a r e no d i f f e r e n c e s b e tw e e n th e I n t e r c e p t s o f v a r i o u s m e n ta l ag e o r d i n a t e s f o r G rade 4 R e a d in g when a v e r a g e c h r o n o l o g i c a l a g e I s a d d e d t o th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . S i m i l a r h y p o t h e s e s Hy, H^, a n d H^Q, w ere s u g g e s t e d , r e s p e c t i v e l y , f o r t ir a d e 4 A r i t h m e t i c ; G rade 6 R e a d in g ; a n d G rad e 6 A r i t h m e t i c . None o f th e t r a t i o s o f a n y o f t h e tw e n ty c o m p a r a tiv e I n t e r c e p t s was s i g n i f i c a n t , t h u s , a c c e p t i n g t h e n u l l h y p o t h e s i s Hq, a s w e l l a s H^, Hy, H^, a n d H1 0 « The d a t a i n d i c a t e t h a t th e e f f e c t s o f a v e r a g e c h r o n o l o g i c a l a g e s do n o t a p p r e c i a b l y c h an g e th e i n t e r c e p t s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . The summary r e s u l t s i n te rm s o f t r a t i o s a r e g i v e n i n T a b le XXIX, C o m p a riso n b e tw e e n th e i n t e r c e p t s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n a n d t h e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n w hen 0 . The s l o p e s o f t h e a p p r o x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s w e re fo u n d t o b e d i f f e r e n t fro m th e s l o p e s o f t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . A r e s u l t i n g h y p o t h e s i s , H j^ , s u g g e s t e d t h a t th e i n t e r c e p t s f o r G rade 4 R e a d in g w e re n o t t h e same f o r e a c h o f t h e s e e q u a t i o n . T h re e s i m i l a r 1 34 TABLE XXIX SUMMARY COMPARISON BETWEEN THE FIVE MENTAL AGE INTERCEPTS OF THE TWO-VARIABLE (X,Y) ESTIMATED TRUE DIFFERENCE EQUATIONS AND THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATIONS FOR GRADE 4 READING, GRADE 4 ARITHMETIC, GRADE 6 READING, AND GRADE 6 ARITHMETIC BY MEANS OF THE t TEST Grade Mental age t-ratio g r a d e p la c e m e n t ( d f - 1 3 ) O r d i n a t e 4 2 .1 1 .0 0 4 3 .1 0 4 4 .1 1 .0 0 4 5 .1 1 .1 6 4 6 .1 1 .1 2 4 2 .1 1 .0 0 4 3 .1 1 .0 0 4 4 .1 0 4 5 .1 1 .0 0 4 o . l 1 .0 0 o 4 .1 1 .0 0 6 5 .1 1 .0 0 6 6 .1 0 6 7 .1 1 .0 0 o 8 .1 1 .4 0 6 4 .1 0 d 5 .1 0 0 o . l 0 6 7 .1 0 u 8 .1 1 .0 0 135 h y p o t h e s e s , H1 2 , Hl y , a n d H -^ , f o r G rad e 4 A r i t h m e t i c , G rad e 6 R e a d in g , a n d G rade 6 A r i t h m e t i c , R e s p e c t i v e l y , w e re g iv e n * The n u l l h y p o t h e s e s , c o r r e s p o n d i n g t o th e h y p o t h e s e s , H -^ a n d w e re r e j e c t e d . The n u l l h y p o t h e s e s c o r r e s p o n d i n g t o th e h y p o t h e s e s a n d w e re fo u n d t o b e t e n a b l e . T h u s, th e h y p o t h e s e s H -q a n d H-^ w ere r e j e c t e d a n d th e h y p o t h e s e s a n d w ere r e g a r d e d a s t e n a b l e * T h e r e f o r e , a l t h o u g h n o t t r u e f o r th e c o n t e n t a r e a o f r e a d i n g i t may be c o n c lu d e d t h a t , i n g e n e r a l th e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n c a n n o t a d e q u a t e l y r e p l a c e th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , T a b le XXX p r e s e n t s a summary o f th e t r a t i o s . C o m p a riso n b e tw e e n th e I n t e r c e p t o f th e a r i t h m e t i c d i f f e r e n c e s a n d t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s when 5» 0 . A lth o u g h th e a r i t h m e t i c d i f f e r e n c e i s t h e m ost common ty p e o f d i f f e r e n c e , L o rd d i d n o t assum e i t t o b e a s a c c u r a t e a s th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . The r e s u l t i n g h y p o t h e s i s , m a i n t a i n s t h a t t h e d i f f e r e n c e s b e tw e e n th e i n t e r c e p t s f o r f i v e m e n ta l a g e v a l u e s f o r G rade 4 R e a d in g a r e d i f f e r e n t b y m ore t h a n c h a n c e . The h y p o t h e s e s H -^ a n d a r e e x c e p t f o r G rade 4 A r i t h m e t i c , G rade 6 R e a d in g , a n d G rade 6 A r i t h m e t i c . 136 TABLE XXX SUMMARY COMPARISON BETWEEN THE FIVE MENTAL AGE INTERCEPTS OF THE ESTIMATED TRUE DIFFERENCE EQUATIONS AND THE APPROXIMATE ESTIMATED TRUE DIFFERENCE EQUATIONS FOR GRADE 4 READING, GRADE 4 ARITHMETIC, GRADE 6 READING, AND GRADE 6 ARITHMETIC BY MEANS OF THE t TEST G rad e A c h ie v e m e n t M e n ta l a g e t - r a t i o A re a g r a d e p la c e m e n t O r d i n a t e 4 Reading 2 .1 1 .4 2 4 Reading 3 .1 .9 6 4 Reading 4 .1 .8 9 4 Reading 5 .1 .09 4 Reading 6 .1 . 2o 4 Arithmetic 2 .1 6 . 9 0 ** 4 Arithmetic 3 .1 0 • 6l * * 4 Arithmetic 4 .1 2 . 8 8 * 4 Arithmetic 5 .1 4 .8 5 * * 4 Arithmetic u . l 5 .8 9 ** u Reading 4 .1 0 0 Reading 5 .1 0 6 Reading u .1 0 o Reading 7 .1 0 6 Reading 6 .1 0 6 Arithmetic 4 .1 0 .7 1 * * 6 Arithmetic 5 .1 4 .8 4 * * 6 Arithmetic 6 .1 2 .6 8 * 6 Arithmetic 7 .1 8 . 2 8 ** 6 Arithmetic 8 .1 8 . 1 9 ** * S i g n i f i c a n t a t th e 5/6 l e v e l o f c o n f i d e n c e f o r 13 d f . ** S i g n i f i c a n t a t t h e Vp l e v e l o f c o n f i d e n c e f o r 13 d f . \ 137 T a b le XXXI g i v e s th e r e s u l t s o f t h e t t e s t s a s a p p l i e d t o t h e n u l l h y p o t h e s i s , Hq, c o r r e s p o n d i n g t o H -^ , % £ * H1 7 ' a n d Hl 8 . T h ese r e s u l t s i n d i c a t e t h a t f o u r o f t h e G rade 4 R e a d in g i n t e r c e p t s a r e s i g n i f i c a n t ; f o u r o f th e G rade 4 A r i t h m e t i c i n t e r c e p t s a r e s i g n i f i c a n t ; a l l o f th e G rade b R e a d in g i n t e r c e p t s a r e s i g n i f i c a n t ; a n d f o u r o f t h e G rade 6 A r i t h m e t i c i n t e r c e p t s a r e s i g n i f i c a n t . C o m p a riso n o f t h e r e l i a b i l i t i e s o f t h e e s t i m a t e d d i f f e r e n c e e q u a t i o n s . R e l i a b i l i t i e s o f t h e s c o r e s fro m t h e v a r i o u s d i f f e r e n c e e q u a t i o n s a r e i m p o r t a n t b e c a u s e : (1 ) t h e y g iv e a n i n d i c a t i o n o f t h e c o n s i s t e n c y o f s c o r e s w h ic h a r e d e r i v e d fro m th e e q u a t i o n s a n d (2 ) t h e y may be u s e d f o r c o m p a rin g th e m u l t i p l e c o r r e l a t i o n s o f one s e t o f 2 e q u a t i o n s w i t h a n o t h e r ( s i n c e r ^ = Rr ) . T a b le s XXXII, ' oo o .x y ' X X X III, XXXIV, a n d XXXV p r e s e n t th e r e l i a b i l i t i e s o f th e f o u r d i f f e r e n c e s f o r m u la s f o r e a c h d i s t r i c t a n d f o r G rade 4 R e a d in g , G rade 4 A r i t h m e t i c , G rade 6 R e a d in g , a n d G rade 6 A r i t h m e t i c . Prom t h e s e d a t a i t i s s e e n t h a t t h e a p p a r e n t r e l i a b i l i t i e s o f th e s c o r e s o f t h e v a r i o u s d i f f e r e n c e e q u a t i o n s show d i f f e r e n c e s w h ic h a r e n i l o r s l i g h t . A p p a r e n t l y t h e a d d i t i o n o f c h r o n o l o g i c a l a g e a s a v a r i a b l e e f f e c t s o n l y s l i g h t l y ( i f a n y ) th e r e l i a b i l i t i e s o f t h e 1 3 8 TABLE XXXI SUMMARY COMPARISON BETWEEN THE FIV E MENTAL AGE INTERCEPTS OF THE ESTIMATED TRUE DIFFERENCE EQUATION AND THE ARITHMETIC DIFFERENCE FOR GRADE 4 READING, GRADE 4 ARITHMETIC, GRADE 6 READING, AND GRADE 6 ARITHMETIC BY MEANS OF THE t TEST Grade Achievement Mental age t-r a tio Area grade placement Ordinate 4 Reading 2.1 2 .61* 4 Reading 3 .1 3.17 ** 4 Reading 4.1 2 .06* 4 Reading 6*1 2 . 31* 4 Reading 6.1 2 .03 4 Arithm etic 2.1 4 .8 9 ** 4 Arithmetic 3.1 3 . 30** 4 Arithmetic 4.1 1.57 4 Arithmetic 5*1 7 ,99** 4 Arithmetic b .l 1 0 . 95** 6 Reading 4.1 5.10** o Reading 5 .1 o.08** 6 Reading o .1 6.08** 0 Reading 7.1 b .83 ** o Reading 8.1 7 . 33** 6 Arithmetic 4 .1 3 .18** 6 Arithm etic 5 .1 1.16 6 Arithmetic 6.1 9 . 56** 6 Arithmetic 7.1 10.67 ** 6 Arithmetic 8.1 9 . 98** * S ig n ific a n t at the 8$ le v e l of confidence fo r 13 d f . ** S ig n ific a n t at the 1^ le v e l of confidence fo r 13 d f . 139 TABLE XXXII SUMMARY OF THE RELIABILITY OF SCORES BASED UPON THE TWO- VARIABLE (X,Y ) ESTIMATED TRUE DIFFERENCE EQUATION (5 THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATION (6 vir„ ) , THE APPROXIMATE ESTIMATED TRUE xyc DIFFERENCE EQUATION (5 ' ) AND THE ARITHMETIC DIFFERENCE (D ), FOR VARIOUS DISTRICT GRADE 6 READING strict Bx y 6xyc 5 1 D 1 .80 .81 .8 0 .8 0 2 .8 6 .8 b ,8 b .8 6 3 .87 .8 8 .87 .87 4 .67 .7 0 .67 .67 5 .7 6 .7 6 .7 6 .7 6 0 .9 8 .61 .5 8 .5 8 7 .82 .83 .82 .82 8 .73 .73 .73 .7 3 9 .8 4 .8 4 .84 .8 4 10 .9 0 .9 0 .9 0 .90 11 .7 9 .7 9 .7 9 .79 12 .8 0 .81 .8 0 .8 0 13 .7 5 .7 5 .7 5 .75 Total .8 2 .8 2 .82 .82 140 TABLE X X X III SUMMARY OF THE RELIABILITY OF SCORES BASED UPON THE TWO- VARIABLE (X,Y ) ESTIMATED TRUE DIFFERENCE EQUATION (5 ) , THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATION (6 ) , THE APPROXIMATE ESTIMATED TRUE xy c DIFFERENCE EQUATION (6 * ) AND THE ARITHMETIC DIFFERENCE (D ), FOR VARIOUS DISTRICT GRADE 4 ARITHMETIC D i s t r i c t 6 6 1 D x y x y c 1 .77 o C O • .76 .76 2 .87 .87 .8 6 . 8 6 3 .84 -4 - co • .84 .84 4 .65 . 6 8 .o5 .65 .8 2 .8 2 .8 1 .8 1 6 .75 .75 .75 .75 7 .75 .77 .75 .75 d .8 1 .8 0 . 8 0 .8 0 9 .83 .85 .83 .83 1 0 .87 .87 .87 .87 11 .8 2 .8 2 .8 2 .8 2 12 .87 .87 . 8 6 .8 6 13 .8 2 .8 2 .8 2 .8 2 ital . 8 2 C Q . .82 .8 2 141 TABLE XXXIV SUMMARY OF THE RELIABILITY OF SCORES BASED UPON THE TWO- VARIABLE (X,Y ) ESTIMATED TRUE DIFFERENCE EQUATION (B ) , THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATION THE APPROXIMATE ESTIMATED TRUE xyc DIFFERENCE EQUATION (6 1) AND THE ARITHMETIC DIFFERENCE (D ), FOR VARIOUS DISTRICT GRADE 6 READING D i s t r i c t 6x y 5x y c 6 ' D 1 • 63 .63 .63 *63 2 .77 .77 .77 .77 3 .8 2 .8 2 .8 2 .8 2 4 .67 .65 .65 5 .71 .71 .71 .71 6 .94 .94 .94 .94 7 .74 .74 .74 .74 8 .8 2 .83 .82 .8 2 9 .75 .76 .75 .75 10 .85 .85 .85 .85 11 .84 .84 .84 .84 12 .76 .76 .76 .76 13 .77 .77 .77 .77 Total .78 .78 .78 .78 142 TABLE XXXV SUMMARY OP THE RELIABILITY OP' SCORES BASED UPON THE TWO- VARIABLE (X,Y ) ESTIMATED TRUE DIFFERENCE EQUATION (6 ) , THE THREE-VARIABLE (X ,Y ,C ) ESTIMATED TRUE DIFFERENCE EQUATION THE APPROXIMATE ESTIMATED TRUE A j v DIFFERENCE EQUATION (6 1) AND THE ARITHMETIC DIFFERENCE (D ), FOR VARIOUS DISTRICT GRADE u ARITHMETIC D istric t 6 xy & xyc 6 1 D 1 .8 0 .81 .8 0 .8 0 2 00 • . 8 9 .8 8 .8 8 3 .93 .93 .87 .87 4 .85 .83 .83 5 .86 . 8 6 . 8 6 . 8 6 0 .93 .93 .93 .93 7 .87 .87 , 8 u . 8 6 8 .83 .84 .84 .84 y .8 0 . 8 6 • 8 u . 8 6 1 0 .91 .91 .91 .91 n .92 .92 .91 .91 12 .89 .8 9 .8 8 .8 8 13 .8 8 .8 9 .87 .87 Total .87 .87 .91 .91 143 s c o r e s o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . A l l o f th e r e l i a b i l i t i e s f o r G rad e 6 A r i t h m e t i c w e re f o u n d t o be .8 0 o r a b o v e . The r e l i a b i l i t y d a t a i n d i c a t e t h a t t h e r e i s a g e n e r a l te n d e n c y f o r e a c h d i s t r i c t o f G rade 6 t o h a v e a h i g h e r r e l i a b i l i t y f o r a r i t h m e t i c t h a n f o r r e a d i n g . T h e re w as fo u n d t o b e no su c h t r e n d f o r G rade 4 . SUMMARY S in c e a n y s t r a i g h t l i n e c a n b e i d e n t i f i e d b y i t s s l o p e a n d I n t e r c e p t , t h e e s t i m a t e d d i f f e r e n c e s e q u a t i o n s u s e d i n t h i s s t u d y w e re c o m p a red w i t h one a n o t h e r on th e b a s i s o f t h e s e c h a r a c t e r i s t i c s . R a th e r t h a n u s i n g a s i n g l e i n t e r c e p t a s a c o m p a r is o n , f i v e i n t e r c e p t s f o r v a r i o u s m e n ta l a g e s w e re I n v e s t i g a t e d . The r e s u l t s o f th e r e l i a b i l i t i e s o f th e s c o r e s b a s e d u p o n th e v a r i o u s d i f f e r e n c e e q u a t i o n s w e re p r e s e n t e d a n d a n a l y z e d . A t th e o u t s e t o f t h i s c h a p t e r th e d i f f e r e n c e b e tw e e n e s t i m a t e d d i f f e r e n c e e q u a t i o n s a n d e x p e c ta n c y ( p r e d i c t i o n t y p e ) e q u a t i o n s w e re d i s c u s s e d . The f o r m e r w e re p r e s e n t e d i n t h i s c h a p t e r a n d t h e l a t t e r p r e s e n t e d i n C h a p te r V I. CHAPTER VI EXPECTANCY EQUATIONS, THEIR SLOPES AND INTERCEPTS, AND THEIR RELIABILITIES The criterion equation used in this chapter was the single variable regression equation in which the achieve ment level is estimated from the mental age. Comparison of the slopes of the single variable regression equations between the four sets of populations. It was hypothesized, H^, that the slopes of the regression lines of the four sets of populations were different from grade to grade and achievement test to achievement test; that is, a single slope cannot adequately represent several grades or achievement tests. The single variable regress ion equations are presented in Tables XXXVI, XXXVII, XXXVIII, and XXXIX and Figures b, 7, d, and 9. Since the coefficients of Y are unity, the slopes of these equations are simply the (negative of the) regression coefficients. Since no assumption was made about the distribution of the slopes, the Friedman two-way analysis of variance statistical procedure was used. This test rejected the null hypothesis (which is the alternative hypothesis to 145 TABLE XXXVI SINGLE VARIABLE REGRESSION EQUATIONS ESTIMATING READING ACHIEVEMENT FOR GRADE 4 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y* = .71 X + . C D 2 Y* W .40 X + 1 .6 1 3 _ L , M .42 r +2.34 4 Y' m .61 X + 1 .1 1 5 Y' - .58 X + 1 .4 9 6 Y' s .60 X + .8 6 7 Y* ■ .59 X + 1 .8 5 8 Y» m .50 X + 1 .3 7 9 Y* s .31 X +2 .8 1 10 Y' = .25 X + 3 .3 5 11 Y* = .60 X +1 .0 9 12 Y' ■ .59 X +1 .8 1 13 Y' ■ .68 X + 1 .3 2 1 4 6 TABLE XXXVII SINGLE VARIABLE REGRESSION EQUATIONS ESTIMATING ARITHMETIC ACHIEVEMENT FOR GRADE 4 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y* m • CO *0 X + 2 .5 3 2 Y* m .22 X + 3 .2 8 3 Y* m .2 0 X + 3 .3 8 4 Y' m .3 0 X + 2 .6 4 5 Y' - .3 0 X + 2 .9 4 6 Y' s .3 0 X + 2 .4 6 7 Y* - .31 X + 3 .0 9 8 Y* .2 6 X + 3 .0 6 9 Y* = .2 5 X + 3 .1 9 10 Y* .2 6 X + 3 .1 7 11 Y* » .3 2 X + o C O . O J 12 Y* ■ .3 1 X + 2 .9 0 13 Y* ■ .3 1 X + 3 .2 0 T o t a l Y ' s .31 X + 2 .9 8 147 TABLE XXXVIII SINGLE VARIABLE REGRESSION EQUATIONS ESTIMATING READING ACHIEVEMENT FOR GRADE 6 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y* ■ .8 5 X + .2 3 2 Y 1 * .75 X + 1 .1 2 3 Y» ■ .65 X + 1 .5 7 4 Y ' B .93 X - .72 5 Y ' = .75 X + .6 4 6 Y* S .3 0 X + 3 .3 3 7 Y* B .73 X + 1 .5 6 8 Y* = .7 5 X + .12 9 Y ' = .81 X + .2 0 10 Y* S .64 X + 1 .4 5 11 Y* B .71 X + 1 .3 2 12 Y* B .69 X + 1 .5 ^ 13 Y ' - .7 0 X + 1 .2 5 Total Y* - .77 X + .6 1 1 48 TABLE XXXIX SINGLE VARIABLE REGRESSION EQUATIONS ESTIMATING ARITHMETIC ACHIEVEMENT FOR GRADE 6 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y* s .3 8 X + 3 .1 8 2 Y* - .2 9 X + 3 .9 6 3 Y» • .29 X + 4 .0 2 4 Y' ■ .41 X + 2 .9 3 5 Y» » .3 3 X + 3 .4 9 6 Y * - .2 1 X + 4 .1 4 7 Y' r .28 X + 4 .1 7 8 Y' ■ *35 X + 3 .0 8 9 Y* - .32 X + 3 .5 4 10 Y' » .22 X + 4 .2 7 11 Y* * .3 4 X + 3 .7 0 12 Y' » .29 X + 4 .0 1 13 Y1 ■ .31 X + 3 .7 9 T o t a l Y' = .33 X + 3 .5 7 Mental age grade placement FIGURE 6 LIKES REPRESENTING THE SINGLE VARIABLE REGRESSION EQUATIONS FOR THE DISTRICT8 FOR GRATE 4 READING 3 . 0 4* 0 5 » G Mental age grade placement FIGURE 7 : i DDS&S REPRESENTING THB SIBOLE VARIABLE REGRESSION EQUA.TIOHS FOR tHE DISTRICTS FOR GRAEE 4 ARITHMETIC r - - - T ' ........................... • " i Mental age grade placement FIGURE 8 LUCES REPRESENTING THE SINGLE VARIABLE REGRESSION EQUATIONS |*OR M B pISTRICTP FOR GRAEE 6 READING I d I o • H ■ P 4 ^.0 6*0 7.0 5.0 M e n tal a g e g ra d e p la c e m e n t PIQUBEi9 LINES REPRESENTING THE SINGLE VARIABLE REGRESSION EQUATIONS FOR THE DISTRICTS FOR GRADE 6 ARITHMETIC J i . 153 H-j^) a t th e one p e r c e n t l e v e l o f c o n f i d e n c e ( c h i - s q u a r e e q u a l t o 3 3 * 0 0 ) , T h u s, i s a n a c c e p t e d h y p o t h e s i s i n d i c a t i n g t h a t t h e r e a r e g r e a t e r d i f f e r e n c e s b e tw e e n g r a d e a n d a c h ie v e m e n t t h a n b e tw e e n d i f f e r e n t d i s t r i c t s w ith r e s p e c t t o s l o p e s o f t h e s i n g l e r e g r e s s i o n e q u a t i o n s . ' A summary o f th e s l o p e s ( e x p r e s s e d i n d e g r e e s ) , t o g e t h e r w i t h s l o p e s o f th e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n s a n d t h e H orn F o rm u la a r e p r e s e n t e d i n T a b le s XL, a n d X L I. C o m p ariso n o f t h e s l o p e s b e tw e e n th e tw o - v a r i a b l e (X .C ) r e g r e s s i o n e q u a t i o n a n d th e s i n g l e v a r i a b l e (X) r e g r e s s i o n e q u a t i o n . The s l o p e s o f th e t w o - v a r i a b l e e q u a t i o n (X,C) (w h ic h a r e r e p r e s e n t e d i n T a b le s X L II, X L I I I , XLIV, a n d XLV), a n d t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n c a n o n ly be co m p ared when th e a c h ie v e m e n t t e s t s c o r e (Y) a n d th e m e n ta l t e s t s c o r e (X) a r e th e d e p e n d e n t a n d i n d e p e n d e n t v a r i a b l e s , r e s p e c t i v e l y . T h u s, th e c h r o n o l o g i c a l a g e (C) w as h e l d c o n s t a n t . The h y p o t h e s i s , H2Q, a ssu m e s t h a t th e s l o p e s o f s i n g l e v a r i a b l e a n d t h e t w o - v a r i a b l e reg re ssd b n e q u a t i o n s b a s e d on G rade 4 R e a d in g d i f f e r o n ly b y c h a n c e . The h y p o t h e s i s w as t e s t e d f o r e a c h g ra d e a n d a c h ie v e m e n t c o m b in a tio n g i v i n g c o r r e s p o n d i n g h y p o t h e s e s , H2 2 , a n d 1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 5 4 TABLE XL SUMMARY OF THE SLOPES (IN DEGREES) OF THE GRADE 4 REGRESSION EQUATIONS S lo p e o f th e S lo p e o f t h e S lo p e o f t h e S i n g l e V a r i a b l e T w o -V a ria b le H orn F o rm u la R e g r e s s i o n E q u a . R e g r e s s i o n E q u a . R e a d in g A r i t h . R e a d in g A r i t h . 35°22* 2 0 ° 2 0 1 3 4 ° 5 4 ‘ 2 0 °0b 1 3 3 ° 4 l 1 21°55» 1 2 ° 3 6 1 22 ° 4 l ' 13 °0 8 1 3 3 ° 4 l 1 22 ° 3 9 1 1 1 ° 2 4 ' 2 3 ° l 8 1 l l ° 4 l 1 33 ° 4 l 1 3 1 ° l 8 ' 1 6 ° 3 3 1 3 4 ° 1 0 ' 1 6 ° 5 5 ' 33 ° 4 l 1 3 0 ° 1 2 ' 1 6 °5 0 1 290411 1 6 ° 3 8 ' 3 3 ° 4 l 1 3 1 ° 0 1 1 1 6 ° 3 7 ' 250421 1 8 ° 0 6 ‘ 3 3 ° 4 l ' 3 0 ° 1 2 1 17 ° 1 3 ' 3 0 ° 3 5 ' 17 ° 1 3 1 33 ° 4 l 1 26 °2 6 1 1 4 ° 2 7 ' 2 6 °5 1 1 1 4 ° 4 qi 33 ° 4 l ' 17 ° 0 1 ' l 4 ° l 8 1 17 ° 2 2 1 14°33» 33 ° 4 l * 1 4 ° 1 3 ' 14°20* 1 6 ° 2 0 1 1 5 °2 6 ' 33 ° 4 l * 3 0 ° 4 9 ' 1 7 o3 2 i 2 9 °0 1 ' 17 ° 0 0 1 33 ° 4 l 1 3 0 ° 4 l 1 170281 3 0 ° 4 9 ' 170341 33 ° 4 l 1 3 4 ° l 4 l 17 °12 1 33 ° 5 4 1 17 ° 0 5 1 33 ° 4 l ' 2 8 ° 5 1 1 1 6 65 9 1 290121 1 6 °5 2 ' 33 ° 4 l ' i 2 3 4 5 6 7 8 9 10 11 12 13 155 TABLE XLI SUMMARY OF THE SLOPES (IN DEGREES) OF THE GRADE 6 REGRESSION EQUATIONS S lo p e o f t h e S lo p e o f t h e S lo p e o f t h e s i n g l e v a r i a b l e two v a r i a b l e H orn F o rm u la R e g r e s s i o n E q u a . R e g r e s s i o n E q u a . R e a d in g A r i t h . R e a d in g A r i t h . 40 **29* 2 1 *0 0 * 3 9 ° 5 5 ' 2 1 * 0 8 ' 3 0 u5O* 3 7 * 0 3 • 16*24* 3 7 * 0 9 ' 1 6 * 3 2 ' 20*50* 32*5^* 1 5 0 5 7 1 32*56* 1 5 * 5 4 ' 30*58* 42*50* 22*04* 3 9 *5 6 ' 20*48* 30*58* 36*56* 1 8 *2 6 ' 35*50* 17*46* 30*58* 16*47* 1 2 *0 5 * 2 0 *0 9 * 13 *41' 30*58* 36*09* 1 5 * 3 8 ' 3 5 * 1 9 ' 14*45* 30*58* 3 6 * 5 5 ' 19*29* 3 5 * 2 7 ' 1 8 * 3 5 ' 30*58* 39*06* 17*54* 3 9 c00* 17*48* 30*58* 3 2 * 4 5 ' 12*30* 3 2 * 4 4 ' 1 2 *2 9 * 3 0 * 5 8 ' 35*21* 18*37* 3 4 * 3 9 ' 18*12* 30*58* 34*21* 16*14* 34*13* 1 5 *4 4 * 30*58* 35°08* 17*18* 3 4 * 2 5 ' 1 6 * 2 9 ' 30*58* 3 7 * 4 6 ' 18*17 * 3 7 * 0 2 ' 1 7 * 4 5 ' 30*58* 1 5 6 TABLE X L II TWO-VARIABLE REGRESSION EQUATIONS ESTIMATING READING ACHIEVEMENT FOR GRADE 4 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y' B .70 X - .50C + 2.94 2 Y* M .42 X - .31C + 2 .8 2 3 Y' m .43 X - .56c + 4.58 4 Y* - .68 X + • 46c m m 1.15 5 Y' m .57 X - • 36c + 3.08 6 Y' s .48 X - 1.01C + 5.56 7 Y* s .59 X - . 56c + 4.22 8 Y' ■ .51 X - .29C + 2 .5 8 9 Y' = .31 X - • 30C + 3.98 10 Y' £ .29 X - 1.12C + 7.92 11 Y* B .55 X - .75C + 4.45 12 Y* = .6 0 X - ,38C + 3.39 13 Y' ■ .67 X - .360 + 2 .8 8 T o t a l Y' S S I •66 X - .44c + 3.50 157 TABLE X L III TW O “VARIABLE REGRESSION EQUATIONS ESTIMATING ARITHMETIC ACHIEVEMENT FOR GRACE 4 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 y = .37 X - .21 C + 3 .4 3 2 Y* s .23 X - .1 9 C + 4 .0 1 3 Y* .21 X - .21 C + 4 .2 1 4 Y» « .3 0 X - .07 C + 2 .9 0 5 Y' = .30 X - .13 C + 3 .4 9 6 Y' ■ .33 X + .0 8 C + 2 .0 1 7 Y* r .31 X - .24 C + 4 .1 0 8 Y* = .26 X “ .12 C + 3 .5 6 9 Y' mm .26 X - .21 c + C M O . 10 Y' s .28 X - .5 6 c + 5 .4 2 11 Y' ■ .31 X - .23 c 4 * 3 .8 2 12 Y ' s .3 2 X + .01 c + 2 .8 4 13 Y* B .3 1 X - .1 1 c + 3 .6 6 T o t a l Y* n .3 0 X _ .1 8 c + 3 .7 3 158 TABLE XLIV TWO-VARIABLE REGRESSION EQUATIONS ESTIMATING READING ACHIEVEMENT FOR GRADE 6 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y' s C O • X - .24 C + 1 .8 3 2 Y* = .76 X - .32 C + 3 .1 2 3 Y» e .65 X - .14 C + 2 .4 2 4 Y' J B .64 X - .85 C + 5 .2 9 5 Y ' r .72 X - .43 C + 3 .5 4 6 Y' = .37 X - .2 8 C + 4 .6 2 7 Y' s .71 X - .32 C + 3 .5 3 8 Y ' = .71 X - .59 C + 4 .1 2 9 Y ' w .81 X - .22 C + 1 .5 7 10 Y ' - .o4 X - .39 c + 3 .9 4 11 Y* - .69 X - .32 c + 3 .4 3 12 Y* s .68 X - .25 c + 3 .06 13 Y' ■ .69 X - .40 c + 3 .8 9 T o t a l Y* 2 .75 X - • 36 c + 2 .9 8 159 TABLE XLV TWO-VARIABLE REGRESSION EQUATIONS ESTIMATING ARITHMETIC ACHIEVEMENT FOR GRADE 6 FROM MENTAL AGE FOR THE DISTRICTS D i s t r i c t 1 Y' K .39 X - .02 C + 3 .0 4 2 Y' = .3 0 X - .07 C + 4 .3 6 3 Y» S .2 8 X - .09 C + 4 .5 7 4 Y* ■ .3 8 X - 00 C \J • C + 3 .8 7 5 Y1 m m .32 X - .19 c + 4 .7 5 6 Y1 s .24 X - .73 c + 8 .3 7 7 Y1 s .2 6 X - .23 c + 5 .7 5 8 Y' ■ .34 X - .20 c + 4 .8 7 9 Y' r .32 X - .12 c + 4 .2 7 10 Y' K .22 X - .15 c + 5 .2 3 11 Y' = .33 X - .1 4 c + 4 .6 0 12 Y' = .2 8 X - .25 c + 5.65 13 Y' r .30 X - .32 c + 5 .8 7 T o t a l Y* m .3 2 X - .19 c + 4 .8 0 1 60 H2 3 j f0I> Gratie ^ A l ,i t h n i e t i c # G rade 6 R e a d in g , a n d G rade 6 A r i t h m e t i c , r e s p e c t i v e l y . S in c e no a s s u m p tio n a b o u t th e d i s t r i b u t i o n o f t h e s l o p e s was m ade, th e W alsh t e s t was u s e d . The r e s u l t s o f t h e s e t e s t s r e v e a l e d t h a t t h e h y p o t h e s e s w e re a c c e p t e d a s t e n a b l e a n d t h a t th e s l o p e s o f th e e q u a t i o n s a r e in d e p e n d e n t o f th e i n c l u s i o n o f c h r o n o l o g i c a l a g e . The r e s u l t s o f th e W alsh t e s t a r e g i v e n i n T a b le XLVI. Comparison of the slopes between the two"variable (X,C) regression equation and the Horn Formula. Upon investigating the slopes of the two-variable (X,C) regression equation and the Horn Formula (whose equations are) Y 1 « .6 X + .4 C and Y' * .67 X + ,33 C for Grade 4 and Grade 6, respectively, there is evidence that the slopes of the Horn Formula tend to be the larger. Results of a Walsh test indicated that the difference between these slopes is significant at the one per cent level for Grade 4 Reading (the slope of the Horn Formula being the larger). The difference between the slopes of the average two-variable regression line and the Horn Formula was found to be 6° 4 ' ; while the difference between the slopes of the Horn Formula and the two-variable (X,C) regression 161 TABLE XLVI SUMMARY COMPARISON BETWEEN THE SLOPES OF THE SINGLE VARIABLE AND THE TWO-VARIABLE REGRESSION EQUATIONS FOR GRADE 4 READING AND ARITHMETIC AND GRADE 6 READING AND ARITHMETIC G rade A c h ie v e m e n t S ig n o f S ig n o f C o n c lu s io n * A re a W alsh W alsh M ax. M in . V a lu e V a lu e 4 R e a d in g + — n o t s i g . 4 A r i t h m e t i c + - n o t s i g . 6 R e a d in g + - n o t s i g . 6 A r i t h m e t i c + - n o t s i g . * T e s t w as c o n d u c te d a t th e 5$ l e v e l o f s i g n i f i c a n c e . 162 l i n e b a s e d u p o n th e t o t a l o f a l l t h e d i s t r i c t s w as 4° 2 9 1, S in c e e a c h s l o p e o f t h e G rade 4 A r i t h m e t i c tw o - v a r i a b l e r e g r e s s i o n l i n e w as s m a l l e r t h a n th e c o r r e s p o n d i n g s l o p e o f G rad e 4 R e a d in g a n d s i n c e t h e H orn F o rm u la re m a in s c o n s t a n t f o r t h e same p u p i l s , i t w as e v i d e n t t h a t th e d i f f e r e n c e s b e tw e e n t h e s e two s e t s o f s l o p e s w o u ld a l s o be s i g n i f i c a n t a t t h e one p e r c e n t l e v e l , a t l e a s t . The d i f f e r e n c e s b e tw e e n t h e H orn F o rm u la s l o p e a n d th e tw o - v a r i a b l e r e g r e s s i o n (X ,C ) l i n e s l o p e b a s e d on t h e a v e r a g e G rad e 4 A r i t h m e t i c s l o p e s w as 17° 5 1 ' j w h i l e th e d i f f e r e n c e b e tw e e n th e s l o p e s o f th e H orn F o rm u la a n d th e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n l i n e b a s e d on t h e t o t a l G rad e 4 A r i t h m e t i c s c o r e s w as 16° 5 0 '• By s i m i l a r r e a s o n i n g th e G rade 6 A r i t h m e t i c s l o p e s w e re s i g n i f i c a n t l y d i f f e r e n t , a t th e one p e r c e n t l e v e l , b e tw e e n th e H orn F o rm u la a n d th e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n l i n e . The a c t u a l a n g u l a r d i f f e r e n c e b e tw e e n H orn F o rm u la a n d m ean t w o - v a r i a b l e (X ,C) r e g r e s s i o n l i n e s a n d b e tw e e n H orn F o rm u la a n d t h e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n l i n e b a s e d on th e t o t a l d i s t r i c t s w e r e , 1 3 ° 8 1 a n d 1 2 ° 7 ' , r e s p e c t i v e l y . I n g e n e r a l , a l l b u t one o f t h e s l o p e s o f th e tw o - v a r i a b l e (X ,C ) r e g r e s s i o n l i n e s b a s e d on G rade 6 R e a d in g 163 w e re l a r g e r t h a n t h e s l o p e s o f t h e H orn F o rm u la s l o p e , by 4° 1 0 ' on th e a v e r a g e . The W alsh t e s t , h o w e v e r, d i d n o t i n d i c a t e a n y s i g n i f i c a n t d i f f e r e n c e s . The s l o p e o f th e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n l i n e b a s e d on th e t o t a l g ro u p w as l a r g e r t h a n t h e s l o p e o f th e H orn F o rm u la b y 6° 4 * . C o m p a riso n o f th e i n t e r c e p t s o f t h e tw o - v a r i a b l e r e g r e s s i o n e q u a t i o n s w i t h i n g r a d e l e v e l s . S in c e t h e i n t e r c e p t s o f r e g r e s s i o n e q u a t i o n s a r e n o t m ean t t o be e q u a l b e tw e e n g r a d e s , s u c h a d i r e c t c o m p a r is o n h a s l i t t l e m ea n in g f o r th e p u r p o s e s o f t h i s 3 t u d y . T h u s, s e p a r a t e h y p o t h e s e s w e re made f o r G rad e 4 a n d G rad e 6 . The f i r s t h y p o t h e s i s , H ^^, i n d i c a t e d t h a t t h e r e a r e d i f f e r e n c e s b e tw e e n th e I n t e r c e p t s o f t h e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n a t f i v e d i f f e r e n t l e v e l s o f m e n ta l ag e (tw o y e a r s b e lo w a v e r a g e MA, one y e a r b e lo w a v e r a g e MA, a t g r a d e w i t h r e s p e c t t o M A., one y e a r a b o v e a v e r a g e MA, a n d two y e a r s a b o v e a v e r a g e MA) b e tw e e n G rade 4 R e a d in g a n d G rade 4 A r i t h m e t i c . O p e r a t i o n a l l y s t a t e d , t h e same e q u a t i o n s c a n n o t be u s e d f o r G rade 4 R e a d in g a s f o r G rad e 6 A r i t h m e tic w hen c o m p a rin g i n t e r c e p t s . A c o m p a n io n h y p o t h e s i s H 2 ^ , I s s i m i l a r t o H 2 ^ , e x c e p t f o r G rade b i n s t e a d o f G rade 4 . 164 S in c e d i s t r i c t a v e r a g e t e s t s c o r e s a r e a ssu m e d t o be d i s t r i b u t e d n o r m a l l y , t h e e x p e c t a n c i e s w ere a l s o a ssu m e d t o b e n o r m a l ly d i s t r i b u t e d f o r t h e p u r p o s e o f t h i s s t u d y . T h e r e f o r e , a t t e s t w as u s e d t o i n v e s t i g a t e th e h y p o t h e s i s , H2 ^ , a n d H2 ^ . The t r a t i o s f o r G rade 4 a n d G rade 6 a r e su m m a riz e d i n T a b le X L V II. T h ese r a t i o s i n d i c a t e t h a t o u t o f a t o t a l o f e i g h t c o m p a r is o n s , s i x a r e s i g n i f i c a n t l y d i f f e r e n t fro m one a n o t h e r a t th e one p e r c e n t l e v e l o f c o n f i d e n c e . I n g e n e r a l , t h e n , i t may be c o n c lu d e d t h a t th e same e q u a t i o n s c a n n o t be u s e d f o r r e a d i n g a n d a r i t h m e t i c w i t h o u t i n t r o d u c i n g c o n s i d e r a b l e e r r o r . C o m p a riso n b e tw e e n th e I n t e r c e p t s o f th e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n a n d t h e tw o - v a r i a b l e r e g r e s s i o n e q u a t i o n . A lth o u g h th e s l o p e s o f th e r e g r e s s i o n e q u a t i o n s d i d n o t g iv e e v id e n c e o f b e i n g s e n s i t i v e t o c h r o n o l o g i c a l a g e ( f o r a n y g i v e n g r a d e ) th e I n t e r c e p t s w e re a l s o I n v e s t i g a t e d w i t h r e s p e c t to t h i s s e n s i t i v i t y . The h y p o t h e s i s , s u g g e s t s t h a t t h e i n t e r c e p t s o f th e r e g r e s s i o n e q u a t i o n s b a s e d on G rade 4 R e a d in g a r e t h e same w h e th e r c h r o n o l o g i c a l a g e i s I n t r o d u c e d o r n o t . I n t e r c e p t s c o r r e s p o n d i n g t o f i v e m e n ta l a g e s (one y e a r a p a r t ) w e re c o m p a re d . S i m i l a r h y p o t h e s e s , H2 7 , H2 c j, a n d H ^ , w ere i n v e s t i g a t e d f o r G rade 4 A r i t h m e t i c , G rade 6 R e a d in g a n d 1 65 TABLE XLVII SUMMARY OF t RATIOS COMPARING VARIOUS INTERCEPTS OF THE TWO-VARIABLE REGRESSION EQUATIONS BASED ON READING AND ARITHMETIC FOR GRADE 4 AND GRADE 6 G rade M e n ta l a g e I n t e r c e p t d f t 4 2 .1 13 7 .6 7 * * 4 3 .1 13 6 .1 2 * * 4 4 .1 13 4 .2 0 * * 4 5 .1 13 0 .9 7 4 6 .1 13 2 .0 6 6 4 .1 13 17 .7 0 * * o 5 .1 13 9 .9 3 * * 6 o . l 13 5 .7 1 * * 0 7 .1 13 3 . 9 0 ** 6 b . i 13 9 .3 5 * * * * S i g n i f i c a n t a t t h e one p e r c e n t l e v e l . 1 6 6 Grade 6 Arithmetic, respectively* The chronological age was taken as the average chronological age for each district, respectively. A l l t r a t i o s w h ic h c o m p a red t h e e f f e c t s o f a d d in g c h r o n o l o g i c a l a g e i n d i c a t e d t h a t t h e h y p o t h e s e s H ^ , H g?, a n d ( n u l l h y p o t h e s e s ) w e re fo u n d t o b e n o t s i g n i f i c a n t a t t h e 5 p e r c e n t l e v e l o f c o n f i d e n c e . A sum m ary o f th e t r a t i o s i s p r e s e n t e d i n T a b le X L V III. C o m p a riso n b e tw e e n t h e I n t e r c e p t s o f th e tw o - v a r l a b l e s ( X ,C ) r e g r e s s i o n e q u a t i o n a n d th e H orn F o r m u la . S in c e t h e t w o - v a r i a b l e (X ,C ) r e g r e s s i o n e q u a t i o n a n d th e H orn F o rm u la c a n b o t h b e p r e s e n t e d a s y s f (X ,C ), t h e s e two e q u a t i o n s w e re c o m p a re d . I n e a c h e q u a t i o n th e c h r o n o l o g i c a l a g e C, w as s e t e q u a l to th e mean c h r o n o l o g i c a l a g e o f th e d i s t r i c t , th u s l e a v i n g y » f (X). The h y p o t h e s i s , H ^ , w h ic h f o l l o w e d s t a t e d t h a t f o r v a r i o u s m e n t a l a g e s ( f i v e i n a l l , one y e a r a p a r t ) c h a n c e s a r e t h a t d i f f e r e n t i n t e r c e p t s w i l l r e s u l t i n d i f f e r e n t c o r r e s p o n d i n g i n t e r c e p t s d e p e n d in g o n w h e th e r t h e t w o - v a r i a b l e (X,C) r e g r e s s i o n e q u a t i o n o r t h e H orn F o rm u la i s u s e d f o r G rade 4 R e a d i n g . C om panion h y p o t h e s e s H ^ , H^2 > * * 3 3 * wer e s u b m i t t e d f o r G rad e 4 A r i t h m e t i c , G rad e 6 R e a d in g , a n d G rade 6 A r i t h m e t i c • 167 TABLE X LV III SUM M ARY OF t RATIOS COMPARING VARIOUS INTERCEPTS OF THE SINGLE VARIABLE REGRESSION EQUATION TO THE TWO- VARIABLE REGRESSION EQUATION WITH RESPECT TO GRADE 4 AND 6 , READING AND ARITHMETIC G r a d e A c h i e v e m e n t M e n t a l a g e t r a t i o A r e a l e v e l 4 R e a d i n g 2 .1 .27 4 R e a d i n g 3 .1 .5 6 4 R e a d i n g 4 .1 1 .0 0 4 R e a d i n g 6 .1 0 4 R e a d i n g o .1 0 4 A r i t h m e t i c 2 .1 1 .0 0 4 A r i t h m e t i c 3.1 1 .0 0 4 A r i t h m e t i c 4 .1 0 4 A r i t h m e t i c 6 .1 0 4 A r i t h m e t i c 0.1 1 .0 0 6 R e a d i n g 4.1 .90 6 R e a d i n g 6 .1 0 0 R e a d i n g 6.1 1 .0 0 o R e a d i n g 7.1 1 .0 0 6 R e a d i n g 6.1 • 56 6 A r i t h m e t i c 4 .1 1.47 6 A r i t h m e t i c 5 - 1 0 6 A r i t h m e t i c 6.1 0 6 A r i t h m e t i c 7.1 0 6 A r i t h m e t i c 8 .1 0 * S i g n i f i c a n t a t t h e l e v e l o f c o n f i d e n c e . * * S i g n i f i c a n t a t t h e l j o l e v e l o f c o n f i d e n c e . 1 68 S e v e n t e e n o f t h e t w e n t y c o m p a r i s o n s i n d i c a t e d t r a t i o s w h i c h w e r e s i g n i f i c a n t a t t h e f i v e p e r c e n t l e v e l o f c o n f i d e n c e , t h e r e b y a c c e p t i n g t h e h y p o t h e s e s a s g e n e r a l l y t e n a b l e . A s u m m a r y o f t h e s e r e s u l t s i s g i v e n i n T a b l e X L I X . C o m p a r i s o n o f t h e v a r i o u s I n t e r c e p t s b a s e d o n t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n b e t w e e n d i s t r i c t s . T h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n b a s e d u p o n t h e c h a r a c t e r i s t i c s o f t h e t o t a l d i s t r i c t s ( f o r a n y g i v e n a c h i e v e m e n t a r e a a n d g r a d e ) m a y s e r v e a s a k i n d o f " a v e r a g e " e q u a t i o n . T h e q u e s t i o n w a s t h e n r a i s e d r e g a r d i n g t h e a d e q u a c y o r r e p r e s e n t a t i v e n e s s o f t h i s e q u a t i o n w i t h r e g a r d t o e a c h o f t h e o t h e r s ( b a s e d u p o n t h e t h i r t e e n d i s t r i c t s ) . A n i n v e s t i g a t i o n c o m p a r i n g t h e i n t e r c e p t f u r t h e s t a b o v e t h i s " a v e r a g e e q u a t i o n " a n d t h e i n t e r c e p t f u r t h e s t b e l o w t h e s a m e " a v e r a g e e q u a t i o n " w a s m a d e . T h e f o r m e r i n t e r c e p t i s c a l l e d t h e u p p e r i n t e r c e p t a n d t h e l a t t e r i s c a l l e d t h e l o w e r i n t e r c e p t * T h e u p p e r a n d l o w e r i n t e r c e p t s w e r e c o m p u t e d f o r v a r i o u s m e n t a l a g e s ( f r o m t w o y e a r s b e l o w " a t g r a d e " m e n t a l a g e t o t w o y e a r s a b o v e ) . T h e I n v e s t i g a t i o n r e v e a l e d t h e f o l l o w i n g m a x i m a l d i f f e r e n c e s : 1 69 TABLE XLIX SUMMARY OP t RATIOS COMPARING VARIOUS INTERCEPTS OF THE TWO"VARIABLE REGRESSION EQUATION TO THE HORN FORMULA WITH RESPECT TO GRADE 4 AND 6 , READING AND ARITHMETIC G r a d e A c h i e v e m e n t M e n t a l a g e t R a t i o A r e a l e v e l 4 R e a d i n g 2 .1 1 .1 5 4 R e a d i n g 3 .1 2 .0 8 4 R e a d i n g 4 .1 3 .0 4 * * 4 R e a d i n g 5 .1 3 .4 7 * * 4 R e a d i n g b .1 3.7b** 4 A r i t h m e t i c 2 .1 9 . 1 b** 4 A r i t h m e t i c 3 .1 5.90** 4 A r i t h m e t i c 4.1 .41 4 A r i t h m e t i c 5 .1 5 .4 7* * 4 A r i t h m e t i c b . l 9 . 2 5 ** o R e a d i n g 4 .1 7 .73** 6 R e a d i n g 5 .1 8 .44** 6 R e a d i n g b . l 8 . 6 8** 6 R e a d i n g 7.1 8 .3 5* * b R e a d i n g b . l 7 . 0 5** 6 A r i t h m e t i c 4 .1 3.67 ** 6 A r i t h m e t i c 5 .1 3 .2 5 * * b A r i t h m e t i c b . l 2 1 . 9 0 ** 6 A r i t h m e t i c 7 .1 1 8 . 2 0 ** 6 A r i t h m e t i c 8 .1 30 .60** * S i g n i f i c a n t a t t h e 5 $ l e v e l o f c o n f i d e n c e . * * S i g n i f i c a n t a t t h e 1 $ l e v e l o f c o n f i d e n c e . 170 1 . F o r G r a d e 4 R e a d i n g t h e l a r g e s t d i s c r e p a n c y o f I n t e r c e p t a b o v e a n d b e l o w t h e e q u a t i o n b a s e d o n t h e t o t a l p o p u l a t i o n w a s o n e y e a r n o m o n t h s . I n a p r a c t i c a l s i t u a t i o n t h i s m e a n s t h a t a n i n d i v i d u a l e n t e r i n g a s c h o o l d i s t r i c t w o u l d h a v e a n e x p e c t a n c y o f a c e r t a i n l e v e l b a s e d u p o n t h e e q u a t i o n o f t h e t o t a l p o p u l a t i o n , w h e r e a s , h e c o u l d ( f o r a c e r t a i n d i s t r i c t a n d a p a r t i c u l a r m e n t a l a g e ) h a v e a n e x p e c t a n c y w h i c h w o u l d d i f f e r b y o n e y e a r . 2 . F o r G r a d e 4 A r i t h m e t i c t h e u p p e r d i f f e r e n c e I n e x p e c t a n c y i s o n l y t h r e e m o n t h s , w h e r e a s t h e l o w e r d i f f e r e n c e i s s i x m o n t h s . 3 . F o r G r a d e 6 R e a d i n g t h e d i f f e r e n c e s b e t w e e n t h e u p p e r I n t e r c e p t a n d t h e i n t e r c e p t o f t h e t o t a l p o p u l a t i o n a n d t h e l o w e r i n t e r c e p t a n d t h e i n t e r c e p t o f t h e t o t a l p o p u l a t i o n a r e 6 m o n t h s a n d o n e y e a r r e s p e c t i v e l y . 4 . F o r G r a d e u A r i t h m e t i c t h e u p p e r d i f f e r e n c e i n e x p e c t a n c y i s f i v e m o n t h s w h i l e t h e l o w e r d i f f e r e n c e i s f o u r m o n t h s . I t s h o u l d b e e m p h a s i z e d t h a t t h e s e d i f f e r e n c e s r e p r e s e n t t h e m o s t e x t r e m e c a s e s u t i l i z i n g t h e c o m b i n a t i o n o f d i s t r i c t a n d m e n t a l a g e t o a t t a i n t h e s e m a x i m u m d i f f e r e n c e s . Comparison of the reliabilities of the single and two-variable regression equations and the Horn Formula. T h e i m p o r t a n c e o f p r e s e n t i n g r e l i a b i l i t i e s o f s c o r e s a r e d i s c u s s e d i n t h e p r e v i o u s c h a p t e r . T h e s a m e d i s c u s s i o n a p p l i e s t o t h e p r e s e n t c h a p t e r . T h e r e l i a b i l i t i e s o f t h e s c o r e s b a s e d o n t h e 171 s e v e r a l e q u a t i o n s i n v e s t i g a t e d i n t h i s c h a p t e r a r e s h o w n i n T a b l e s L a n d LI f o r G r a d e 4- a n d G r a d e 6 , r e s p e c t i v e l y . T h e s e d a t a i n d i c a t e t h e f o l l o w i n g : 1 . T h e i n d i c a t e d r e l i a b i l i t i e s o f t h e s c o r e s o f t h e t w o - v a r i a b l e ( X , C ) r e g r e s s i o n e q u a t i o n a r e i n e a c h c a s e e q u a l t o o r l a r g e r t h a n t h e c o r r e s p o n d i n g i n d i c a t e d r e l i a b i l i t i e s o f t h e s c o r e s o f t h e s i n g l e v a r i a b l e ( X ) r e g r e s s i o n e q u a t i o n . 2 . G e n e r a l l y , t h e i n d i c a t e d r e l i a b i l i t i e s o f t h e s c o r e s b a s e d o n a r i t h m e t i c a r e l a r g e r t h a n t h e i n d i c a t e d r e l i a b i l i t i e s o f t h e s c o r e s b a s e d o n r e a d i n g . 3 . T h e i n d i c a t e d r e l i a b i l i t i e s o f t h e H o r n F o r m u l a e x c e e d a l l o t h e r I n d i c a t e d r e l i a b i l i t i e s b y a r a t h e r s i z a b l e m a r g i n . T h e s e r e s u l t s h a v e , i n p a r t a t l e a s t , r a t h e r l o g i c a l e x p l a n a t i o n s . S i n c e t h e r e l i a b i l i t y o f c h r o n o l o g i c a l a g e i s u n i t y , t h e i n c l u s i o n o f c h r o n o l o g i c a l a g e w o u l d t e n d t o I n c r e a s e t h e r e l i a b i l i t y . T h e r e l i a b i l i t y o f t h e s c o r e s o f t h e r e g r e s s i o n e q u a t i o n s i s a f u n c t i o n o f t h e r e l i a b i l i t y o f t h e t e s t . F r o m t h e t e s t m a n u a l t h e r e l i a b i l i t y c o e f f i c i e n t s o f t h e a r i t h m e t i c a n d r e a d i n g t e s t s w a s g i v e n a s . 9 5 a n d . 9 3 r e s p e c t i v e l y . T h u s , o n e m a y p a r t i a l l y a c c o u n t f o r t h e h i g h e r r e l i a b i l i t i e s f o r a r i t h m e t i c e x p e c t a n c y s c o r e s t h a n f o r r e a d i n g e x p e c t a n c y s c o r e s . C o m p a r e d t o t h e o t h e r r e l i a b i l i t i e s , t h e r e l i a b i l i t i e s o f t h e H o r n F o r m u l a a r e s o m e w h a t s p u r i o u s . T h e r e a s o n f o r 172 TABLE L SUMMARY OP THE RELIABILITY OP SCORES BASEL UPON THE SINGLE AND TWO-VARIABLE REGRESSION EQUATIONS FOR THE DISTRICT GRADE 4 READING AND ARITHMETIC D i s t . R e a d i n g S i n g l e T w o - V a r i a b l e v a r i a b l e ___________U) (X,C) 1 .78 .79 2 .0 6 .69 3 .53 • 60 4 .83 .8 6 5 o G O • i — 1 C O • 6 O 00 • .87 7 • c 0 0 .73 8 .79 .80 9 .59 10 .67 .68 n .78 .82 12 .8 2 .83 13 .82 .83 T o ta l .75 .7 8 A r i t h m e t i c S i n g l e T w o - H o r n V a r i a b l e V a r i a b l e F o r m u l a IX) (x.c)_____________ 82 .8 3 .9 5 75 .77 .9 6 58 .o 2 .9 5 86 .87 .9 5 82 .83 • 95 70 .72 .9 4 75 .77 • 9o 79 .8 0 • 96 75 .7 0 .9 5 .45 . 68 .9 6 ,80 .81 .9 5 ,83 .8 4 .97 ,82 .8 2 .9 5 >79 .8 0 .9 5 173 TABLE L I SUMMARY OF THE RELIABILITY OF SCORES BASED UPON THE SINGLE AND TWO-VARIABLE REGRESSION EQUATIONS FOR THE DISTRICT GRADE 6 READING AND ARITHMETIC D i s t . R e a d i n g S i n g l e T w o - V a r i a b l e v a r i a b l e (X) (X ,C ) 1 .8 4 00 * 2 .8 3 .8 4 3 .7 9 . 8 0 4 .8 5 C O C O • 5 .8 4 .8 5 6 . 2 1 .3 2 7 . 8 2 .83 8 .7 8 . 8 1 y .8 3 .83 1 0 .7 9 .8 1 n .8 3 -= ! ■ C O . 1 2 .83 .84 13 .8 2 .8 6 T o t a l .83 .84 A r i t h m e t i c S i n g l e T w o - H o r n V a r i a b l e V a r i a b l e F o r m u l a (X) _ ( x , c ) ______________ 83 .83 .95 84 .8 6 .95 83 .84 .95 87 .89 .94 85 •8b .95 80 .83 .95 79 .8 2 .94 81 .83 .95 83 .83 .95 79 .8 1 .95 85 .86 .95 85 .8 6 .95 84 .8 6 .95 .84 .85 .95 174 t h i s i s t h a t o n l y o n e e r r o r o f m e a s u r e m e n t ( t h a t o f t h e m e n t a l a g e ) i s i n c l u d e d i n t h e p r o c e d u r e f o r o b t a i n i n g t h e H o r n F o r m u l a r e l i a b i l i t y . S i n c e t h e o t h e r r e l i a b i l i t i e s i n v o l v e t w o e r r o r s o f m e a s u r e m e n t , t h e r e l i a b i l i t i e s o f t h e H o r n F o r m u l a w o u l d b e e x p e c t e d t o b e h i g h e r b u t s p u r i o u s l y h i g h e r . I t m a y b e c o n c l u d e d t h a t t h e s e o b t a i n e d r e l i a b i l i t i e s i n d i c a t e a r o u g h m e a s u r e o f c o n s i s t e n c y . P e r h a p s t h e m o s t a p p r o p r i a t e u s e o f t h e s e r e l i a b i l i t i e s i s f o r t h e i d e n t i f i c a t i o n o f t h o s e d i s t r i c t s h a v i n g l o w r e l i a b i l i t y v a l u e s . I t w o u l d b e n o t e d t h a t a l l r e l i a b i l i t i e s o f v a l u e s b a s e d o n t h e t o t a l p o p u l a t i o n s a r e . 7 5 o r g r e a t e r . I f t h e v a l u e o f . 7 5 w a s u s e d a s a m i n i m a l c r i t e r i o n , a l l t o t a l p o p u l a t i o n r e l i a b i l i t i e s w o u l d b e a c c e p t a b l e a s m e e t i n g t h e c r i t e r i o n a l t h o u g h s o m e d i s t r i c t r e l i a b i l i t i e s w o u l d f a i l t o m e e t t h i s c r i t e r i o n . A n a l y s i s o f t h e p r e d i c t i v e v a l u e o f t h e r e g r e s s i o n e q u a t i o n w i t h t h e I n c l u s i o n o f t h e e f f e c t s o f c h r o n o l o g i c a l a g e . S o m e e v i d e n c e w h i c h q u e s t i o n s t h e u s e f u l n e s s o f a d d i n g c h r o n o l o g i c a l a g e t o r e g r e s s i o n e q u a t i o n s h a s b e e n g i v e n . T h i s e v i d e n c e h a s b e e n I n t h e f o r m o f r e p o r t s f r o m t h e l i t e r a t u r e , n e g l i g i b l e d i f f e r e n c e s o f I n t e r c e p t s w h e n a v e r a g e d i s t r i c t c h r o n o l o g i c a l 175 a g e 1 b u s e d , a n d n e g l i g i b l e e f f e c t s o f t h e r e l i a b i l i t y o f s c o r e s w hen a v e r a g e d i s t r i c t c h r o n o l o g i c a l a g e i s u s e d . One o t h e r p h a s e o f t h i s i n v e s t i g a t i o n w as u n d e r t a k e n . T h is p h a s e co m p a red t h e c o r r e l a t i o n o f t h e a c h ie v e m e n t s c o r e s a n d th e m e n ta l a g e s c o r e s w i t h th e m u l t i p l e c o r r e l a t i o n o f a c h ie v e m e n t s c o r e s a n d t h e b e s t c o m b in a tio n o f m e n ta l a g e a n d c h r o n o l o g i c a l a g e . T h ese c o r r e l a t i o n s a r e p r e s e n t e d I n T a b le s L I I a n d L I I I . A lth o u g h a l l t h e m u l t i p l e c o r r e l a t i o n s a r e e q u a l t o , o r g r e a t e r t h a n t h e c o r r e s p o n d i n g t w o - v a r i a b l e c o r r e l a t i o n s , t h e d i f f e r e n c e s a r e n o t l a r g e , i n g e n e r a l * Some s m a l l e r d i s t r i c t s (u n d e r 1 0 0 ) I n d i c a t e m o d e ra te i n c r e a s e s . The d a t a w e re a n a l y z e d o n t h e b a s i s o f t h e c o r r e l a t i o n s b a s e d upon t o t a l p o p u l a t i o n s . The l a r g e s t I n c r e a s e i n c o r r e l a t i o n w as f o u n d t o be G rade 4 R e a d in g . The i n c r e a s e a s show n i n T a b le L I I i s fro m £ 9 t o . 6 2 . T h is I n c r e a s e a s a r e s u l t o f th e i n c l u s i o n o f c h r o n o l o g i c a l a g e I n d i c a t e s t h a t th e p e r c e n t r e d u c t i o n i n e r r o r s o f p r e d i c t i o n b y r e a s o n o f c o r r e l a t i o n i s i n c r e a s e d fro m 1 9 .2 6 to 2 1 ,5 4 . A n o th e r c o m p a r is o n o f t h e s e two c o r r e l a t i o n s w as made b y u t i l i z i n g th e s t a n d a r d d e v i a t i o n o f c o r r e l a t i o n 1 7 6 TABLE L I I SUMMARY OP THE CORRELATION COEFFICIENTS ( r ) BETWEEN xy MENTAL AGE (X) AND ACHIEVEMENT AGE (Y) AND MULTIPLE CORRELATION COEFFICIENTS (R„ ) BETWEEN ACHIEVEMENT y t A w AGE, MENTAL AGE, AND CHRONOLOGICAL AGE FOR GRADE 4 D i s t r i c t R e a d in g A r i t h m e t i c r R r R x y y .x c x y y .x c 1 .6 4 ,6 6 .6 2 .6 4 2 .4 9 .5 1 .5 0 .52 3 .4 1 .4 4 .37 .3 9 4 .77 .8 0 .7 4 .7 6 5 .6 9 .7 0 .6 2 • 62 6 .6 9 .81 .4 4 .51 7 .5 1 .5 5 .49 .52 8 .67 .55 • 56 9 .4 2 .4 4 .5 0 .5 1 10 .2 4 .4 8 .3 2 .42 11 .6 5 .7 0 .57 .5 9 12 .7 8 .7 4 .6 5 .6 6 13 .7 3 .7 4 .6 0 .61 T o t a l .5 9 .6 2 .5 6 .57 177 TABLE L I I I SUMMARY OP THE CORRELATION COEFFICIENTS ( r ) BETWEEN Ay MENTAL AGE (X) AND ACHIEVEMENT AGE (Y) AND MULTIPLE CORRELATION COEFFICIENTS (R,r ) BETWEEN ACHIEVEMENT y #ac AGE, MENTAL AGE, AND CHRONOLOGICAL AGE FOR GRADE 6 D i s t r i c t R e a d in g A r i t h m e t i c r x y y .x c r x y y .x c 1 .7 9 .7 9 .6 4 .6 4 2 .7 6 .7 8 .72 .7 3 3 .6 6 .67 .6 5 .6 5 4 .8 4 .8 9 .7 8 .8 2 5 .7 9 .81 .71 .7 3 6 .31 .57 .5 2 .6 4 7 .73 .7 4 .5 6 .5 9 8 .6 4 .67 .5 9 .6 2 9 .7 6 .77 .63 .63 10 .67 .6 9 .5 6 .5 8 11 .7 5 .7 8 .7 0 .72 12 .7 6 .77 .6 8 .7 1 13 .7 4 .7 5 .6 6 .6 9 T o t a l .7 6 .77 .67 .6 8 178 (which was found to be .0106). The correlation .62 l i e s w ithin the confidence lim its of plus or minus three standard deviations of the correla tio n .59* A ll other comparisons between the correla tio n s would then He w ell w ithin the confidence lim it s . Thus, the in clu sio n of chronological age does not appreciably aid p red ictio n . A nalysis of the sin g le variable regression equation based on three d iffer en t groups according to In te llig e n c e q u o tie n ts. Another assumption which was in v estig a ted was that the regression equations nay d iffe r in a con sisten t d irection according to a b ilit y groupings. The to ta l population for Grade 4 and Grade 6 were apportioned into three d istin c t groups. The below average a b ilit y group contained IQ's which were below 8 5 . The second group contained children in the average range, IQ's 8 6 to 118. The above average group included a l l those with IQ's higher than 116. For these three groups regression equations were calcu lated for Grade 4 Reading and Arithmetic as w ell as Grade 6 Reading and A rithm etic. The re su ltin g equations are presented in Table LIV. These equations were p lotted on coordinate paper as shown in Figures 10, 11, 12, and 13. 1 7 9 TABLE LIV SUMMARY OF THE SINGLE VARIABLE REGRESSION EQUATIONS BASED ON THREE DIFFERENT ABILITY GROUPS FOR GRADE 4 READING AND ARITHMETIC AND GRADE 6 READING AND ARITHMETIC Grade Achievement Ability Range Equation 4 Reading IQ: Below 85 Y '« .09X +2,96 4 Reading IQ: 85 to 115 Y’ - .47X +1.94 4 Reading IQ: Above 115 Y 1 * . 68x +1.01 4 Arithmetic IQ: Below 85 Y '- .19X +3.28 4 Arithmetic IQ: 85 to 115 Y '» .28X +3.10 4 Arithmetic IQ: Above 118 Y '« • 30X +3.04 6 Reading IQ: Below 85 Y 1 s • 2G X +3.03 6 Reading IQ: 8 5 to 115 Y* = .72X + .90 6 Reading IQ: Above 118 Y '« .75X + .88 6 Arithmetic IQ: Below 85 Y 's .05X +4.62 6 Arithmetic IQ: 8 5 to 118 Y*» • 30X +3.77 6 Arithmetic IQ: Above 118 Y 's • 34x +3,48 A n ticip a ted Achievem ent 6.0 2.8 3 .6 3 .2 Mental age grade plaoanent ElflUKE 1 0 LINES REPRESENTING THE ANTICIPATED ACHIEVEMENT AND REGRESSION EQUATIONS OF THREE DIFFERENT ABILITY GROUPS FOR GRADE ^ BEADING C O 1 _ A n ticip a ted Achievement 2,8 2,4 3 .6 4 .0 4 ,4 Mental age grade placem ent 6.0 3 .2 LINBS REPRESENTING THE ANTICIPATED ACHIEVEMENT REGRESSION EQUATIONS O F THREE DIFFERENT ABILITY GROUPS FOR GRADE 4 ARITHMETIC Reading grade placem ent 6.0 A n ticip a ted Achievem ent 5 . 2 Ift »b t e l o v 85 i i 7 .6 5 . 6 6 A 6.0 Mental age grade placem ent FIGURE 12 . ? LIKES REPRESENTING THE ANTICIPATED ACHIEVEMENT AND REGRESSION EQUATIONS < x > OF THREE DIFFERENT ABILITY GROUPS FOR GRADE 6 READING____ _ _ M Arithm etic grade placem ent A c h l e v e B > e n t 4 .8 5*6 6,8 6.0 7 .6 5*2 7 .2 M ental age grade placem ent FIGURE 13 LIKES KEPRESENTIHG THE ANTICIPATED ACHIEVEMENT AND REGRESSION EQUATIONS OF THREE DIFFERENT ABILITY GROUPS FOR ORAEB 6 ARITHMETIC H s 184 The t r e n d i n e a c h c a s e i s c o n s i s t e n t , a s th e a b i l i t y o f t h e g ro u p i n c r e a s e s , t h e s l o p e s o f t h e l i n e i n c r e a s e . I t i s a p p a r e n t t h a t t h e g r e a t e s t I n c r e a s e i n s l o p e i s b e tw e e n th e b elo w a v e r a g e g ro u p a n d t h e a v e r a g e g r o u p . D i f f e r e n c e s b e tw e e n t h e i n t e r c e p t s o f t h e s e e q u a t i o n s w e re c o m p a re d a n d a r e p r e s e n t e d i n T a b le s LV a n d L V I, The a b o v e a v e r a g e a b i l i t y g ro u p i n t e r c e p t s do n o t d e v i a t e fr o m t h e t o t a l g ro u p i n t e r c e p t s b y m ore th a n one m o n th w h e re t h e e q u a t i o n i s m o st a p p l i c a b l e . By t h e same t o k e n , t h e a v e r a g e a b i l i t y g ro u p i n t e r c e p t s h a v e no d i f f e r e n c e s t h e m a j o r i t y o f t h e t i m e , b u t may d i f f e r a s much a s t h r e e m o n th s . The b e lo w a v e r a g e g ro u p i n t e r c e p t s show l a r g e r d i s c r e p a n c i e s t h a n e i t h e r o f t h e o t h e r g r o u p s , t h e l a r g e s t d e v i a t i o n b e i n g s i x m o n th s . As a r e s u l t , t h e r e g r e s s i o n e q u a t i o n b a s e d u p o n t h e t o t a l p o p u l a t i o n c a n a p p a r e n t l y b e u s e d a s a p r e d i c t o r i n p l a c e o f t h o s e r e g r e s s i o n e q u a t i o n s b a s e d u p o n e i t h e r h i g h a b i l i t y a r e a v e r a g e a b i l i t y g r o u p s w i t h l i t t l e l o s s o f a c c u r a c y . E v id e n c e d o e s n o t s u b s t a n t i a t e th e same c o n c l u s i o n w i t h r e g a r d t o r e g r e s s i o n e q u a t i o n s b a s e d on b e lo w a v e r a g e g r o u p s . 185 TABLE LV SUMMARY OF THE DIFFERENCES, IN MONTHS, BETWEEN TOTAL INTERCEPTS AND THOSE FOR PUPILS OF THREE IQ CATEGORIES, FOR GRADE 4 READING AND ARITHMETIC, WITH FIVE MENTAL AGE GRADE PLACEMENT ORDINATES G ro u p , G rad e a n d M e n ta l a g e g r a d e p la c e m e n t I n t e r c e p t A c h ie v e m e n t 2 .1 3 .1 4 .1 5 .1 6 .1 IQ : B elow 85 R e a d in g , G rad e 4 3 - 2 -6 * -1 1 * -1 6 * IQ : 85 t o 115 R e a d in g , G rad e 4 1 * 0 0 -2 -3 * IQ : Above 115 R e a d in g , G rade 4 -4 * -3 * -1 0 1 IQ : Below 85 A r i t h . , G rade 4 1 0 - 1 * -2 * -3 * IQ : 85 t o 115 A r i t h . , G rade 4 1* 1 0 0 0* IQ : Above 115 A r i t h . , G rade 4 1 * 1* 1* 1 1 * T h is v a lu e i n d i c a t e s t h a t a l t h o u g h t h e r e g r e s s i o n e q u a t i o n s a r e b a s e d o n t h e a b o v e m e n ta l a g e g ra d e p la c e m e n t i t i s n o t a p p l i c a b l e a t t h i s l e v e l . 186 TABLE LVI SUMMARY OP THE DIFFERENCES, IN MONTHS, BETWEEN TOTAL INTERCEPTS AND THOSE FOR PUPILS OF THREE IQ, CATEGORIES, FOR GRADE 6 READING AND ARITHMETIC, WITH FIVE MENTAL AGE GRADE PLACEMENT ORDINATES G ro u p , G rade a n d A c h ie v e m e n t M e n ta l 2 . 1 a g e g ra d e 3 .1 p la c e m e n t i n t e r c e p t 4 .1 5 .1 6 .1 IQ : Below 85 R e a d in g , G rad e 6 0 - 6 -1 1 * -1 7 * -2 3 * IQ : 85 t o 115 R e a d in g , G rade 6 0* 0 0 -1 -2 * IQ : Above 115 R e a d in g , G rade 6 2* 1 * 2* 1 1 IQ : B elow 85 A r i t h , , G rade 6 -1 -4 -7 * -9 * -1 2 * IQ : 85 t o 115 A r i t h , , G rad e 6 1* 0 0 0 0* IQ : Above 115 A r i t h , , G rade 6 0 * - 1 * 0* 0 1 * T h is v a lu e I n d i c a t e s t h a t a l t h o u g h th e r e g r e s s i o n e q u a t i o n s a r e b a s e d o n th e a b o v e m e n ta l a g e g r a d e p la c e m e n t I t I s n o t a p p l i c a b l e a t t h i s l e v e l . 187 R e l a t i o n s h i p s b e tw e e n r e g r e s s i o n e q u a t i o n s a n d A n t i c i p a t e d A c h ie v e m e n t. The t e c h n i c a l r e p o r t on th e C a l i f o r n i a A c h ie v e m e n t T e s t I n d i c a t e s t h a t t h e A n t i c i p a t e d A c h ie v e m e n t s h o u l d n o t b e u s e d e i t h e r t o c o n f i r m o r c o n t r a d i c t r e s u l t s fro m o t h e r e x p e c ta n c y m e th o d s . As a r e s u l t o f t h i s c o u n s e l , f i n e d i s c r i m i n a t i o n s w e re n o t m ad e . H ow ever, g e n e r a l t r e n d s a n d r e l a t i v e l y g r o s s c o m p a r is o n s w e re made b e tw e e n A n t i c i p a t e d A c h ie v e m e n t a n d ( 1 ) th e r e g r e s s i o n e q u a t i o n b a s e d on th e t o t a l p o p u l a t i o n a n d (2 ) th e r e g r e s s i o n e q u a t i o n b a s e d on t h r e e d i f f e r e n t l e v e l s o f a b i l i t y . I t s h o u l d b e e m p h a s iz e d a t t h i s p o i n t t h a t t h e d a t a o b t a i n e d f o r t h i s s t u d y w e re b a s e d u p o n r e s u l t s o f t h e C a l i f o r n i a A c h ie v e m e n t T e s t Form AA a n d t h e C a l i f o r n i a T e s t o f M e n ta l M a t u r i t y s h o r t - f o r m 1951* The A n t i c i p a t e d A c h ie v e m e n t i s b a s e d upon a r e v i s i o n o f e a c h o f t h e s e t e s t s i n a d d i t i o n t o a r e s t a n d a r d i z a t i o n o f th e n o rm s . I n o r d e r to em p lo y t h e A n t i c i p a t e d A c h ie v e m e n t c o n c e p t t o th e I n s t r u m e n t s , u s e d i n t h i s s t u d y , c o n v e r s i o n t a b l e s f o r b o t h t h e a c h ie v e m e n t t e s t a n d th e a b i l i t y t e s t w e re u s e d t o o b t a i n s c o r e s c o m p a ra b le t o th e 1957 e d i t i o n s . From t h e s e c o n v e r s i o n s t a b l e s A n t i c i p a t e d A c h ie v e m e n ts w ere o b t a i n e d . H ow ever, a n o t h e r m o d i f i c a t i o n w as n e c e s s a r y . 188 The r e a s o n f o r t h i s w as t h e f a c t t h a t t h e A n t i c i p a t e d A c h ie v e m e n t y i e l d s two s c o r e s f o r r e a d i n g ( v o c a b u l a r y a n d c o m p r e h e n s io n ) a n d tw o s c o r e s f o r a r i t h m e t i c ( r e a s o n i n g a n d f u n d a m e n t a l s ) , w h i l e s i n g l e ( t o t a l ) s c o r e s f o r r e a d i n g a n d a r i t h m e t i c w e re i n v e s t i g a t e d i n t h i s s t u d y . The two A n t i c i p a t e d A c h ie v e m e n t s c o r e s f o r r e a d i n g w e re a v e r a g e d . The same p r o c e d u r e w as u s e d f o r a r i t h m e t i c . T h u s, a l t h o u g h some a l t e r a t i o n o f th e A n t i c i p a t e d A c h ie v e m e n t was m ade, a g e n e r a l i z e d c o m p a r is o n w as p o s s i b l e . B e c a u s e o f t h i s , c o m p a r is o n s w e re made o n t h e b a s i s o f d i r e c t i o n r a t h e r t h a n m a g n i t u d e . G r a p h i c a l p r e s e n t a t i o n s c o m p a rin g A n t i c i p a t e d A c h ie v e m e n t w i t h r e g r e s s i o n e q u a t i o n s b a s e d on a b i l i t y g r o u p s a r e show n i n F i g u r e s 1 0 , 1 1 , 1 2 , a n d 1 3 . F i g u r e 10 i n d i c a t e s t h a t f o r h i g h e r m e n ta l a g e s t h e r e g r e s s i o n e q u a t i o n b a s e d u p o n t h e t o t a l p o p u l a t i o n i s r o u g h l y p a r a l l e l t o a n d o n l y s l i g h t l y h i g h e r i n G rad e 4 R e a d in g E x p e c ta n c y t h a n t h e A n t i c i p a t e d A c h ie v e m e n t i n d i c a t e s ; w h e r e a s , f o r lo w e r m e n t a l a g e s t h e tw o l i n e s becom e i n c r e a s i n g l y d i v e r g e n t , th e r e g r e s s i o n l i n e y i e l d i n g t h e l o w e r e x p e c t a n c y . A lth o u g h t h e same com m ents w o u ld a p p l y , i n g e n e r a l , t o t h e r e l a t i o n s h i p b e tw e e n t h e 18 9 A n t i c i p a t e d A c h ie v e m e n t a n d t h e t h r e e r e g r e s s i o n e q u a t i o n s l i n e s b a s e d on d i f f e r e n t a b i l i t y g r o u p s , t h e s e l i n e s c o i n c i d e m ore c l o s e l y t o t h e c o n t o u r o f th e A n t i c i p a t e d A c h ie v e m e n t. F o r G rade 4 A r i t h m e t i c th e A n t i c i p a t e d A c h ie v e m e n t e x c e e d s , b u t g e n e r a l l y p a r a l l e l s , th e r e g r e s s i o n l i n e b a s e d on t h e t o t a l p o p u l a t i o n t h r o u g h o u t t h e m o st o f th e m e n t a l a g e ra n g e ( e x c e p t f o r low m e n ta l a g e s ) . F o r e x tr e m e l y low m e n ta l a g e s t h e d i v e r g e n c e o f t h e two l i n e s b eco m es g r e a t e r , th e A n t i c i p a t e d A c h ie v e m e n t i n d i c a t i n g a r e l a t i v e l y s m a l l e r e x p e c ta n c y v a l u e . I n g e n e r a l , th e t h r e e r e g r e s s i o n l i n e s b a s e d on a b i l i t y g r o u p in g (T a b le I I ) I n d i c a t e s a s i m i l a r c o m p a ris o n a l t h o u g h t h e s e l i n e s e n v e lo p e th e A n t i c i p a t e d A c h ie v e m e n t o v e r m o st o f th e m e n ta l a g e r a n g e . The G rade 6 R e a d in g A n t i c i p a t e d A c h ie v e m e n t l i n e a n d th e r e g r e s s i o n l i n e b a s e d on th e a v e r a g e a b i l i t y g ro u p i n d i c a t e r e m a r k a b ly c l o s e re s e m b l a n c e s f o r m e n ta l a g e s a b o v e th e " a t g r a d e " m e n ta l ag e l e v e l . The two l i n e s d l v e r a g e f o r m e n ta l a g e s lo w e r th a n t h o s e " a t g r a d e , " th e A n t i c i p a t e d A c h ie v e m e n t y i e l d i n g a h i g h e r e x p e c ta n c y v a l u e . When t h e s e two l i n e s a r e co m p a red f o r G rade 6 A r i t h m e t i c , t h e y i n d i c a t e v e r y s i m i l a r r e s u l t s w i t h a 190 s l i g h t te n d e n c y f o r t h e A n t i c i p a t e d A c h ie v e m e n t t o he h i g h e r f o r t h o s e m e n ta l a g e s w h ic h a r e i n t h e n e i g h b o r ho o d o f " a t g r a d e •" SUMMARY V a r io u s r e g r e s s i o n e q u a t i o n s w e re c o m p a red by m eans o f s l o p e a n d i n t e r c e p t . I t w as fo u n d t h a t c h r o n o l o g i c a l a g e h a d l i t t l e e f f e c t o n p r e d i c t i o n w hen c o m b in ed w i t h m e n ta l a g e . The r e g r e s s i o n e q u a t i o n s v a r i e d s i g n i f i c a n t l y b e tw e e n g r a d e a n d a c h ie v e m e n t a r e a . S in c e H orn F o rm u la i s u s e d a s a n e x p e c ta n c y in d e p e n d e n t o f a c h ie v e m e n t a r e a , i t to o w as f o u n d to be s i g n i f i c a n t l y d i f f e r e n t fro m t h e r e g r e s s i o n e q u a t i o n s . The i n c l u s i o n o f c h r o n o l o g i c a l ag e w as n o t f o u n d t o I n c r e a s e p r e d i c t i o n b y a n y a p p r e c i a b l e am ount i n m o st c a s e s . A lth o u g h r e g r e s s i o n l i n e s b a s e d o n th e t o t a l o f a l l t h e d i s t r i c t s f o r a n y g r a d e a n d a c h ie v e m e n t a r e a i s a good p r e d i c t o r f o r some d i s t r i c t s i t i s n o t a good p r e d i c t o r f o r o t h e r s . The r e g r e s s i o n e q u a t i o n s b a s e d u p o n a b i l i t y g r o u p i n g s v a r y p r o g r e s s i v e l y . When c o m p a red w i t h t h e A n t i c i p a t e d A c h ie v e m e n t t h e r e i s g e n e r a l a g re e m e n t i n b o t h s l o p e o f t h e l i n e s a n d m a g n itu d e o f e x p e c t a n c y . CHAPTER V II SUMMARY, CONCLUSIONS AND EDUCATIONAL IMPLICATIONS I . SUMMARY OF PROCEDURE The p u r p o s e o f th e p r e s e n t i n v e s t i g a t i o n w as to s t u d y t h e e f f e c t s o f g r a d e , a r e a o f a c h ie v e m e n t a n d v a r i a t i o n s i n s c h o o l d i s t r i c t s on v a r i o u s e x p e c ta n c y p r o c e d u r e s . A t th e o u t s e t o f t h e i n v e s t i g a t i o n , i t w as n e c e s s a r y to d e r i v e c e r t a i n e q u a t i o n s o f e s t i m a t e d d i f f e r e n c e s a n d e x p e c t a n c y , a n d e q u a t i o n s g i v i n g th e r e l i a b i l i t i e s o f t h e i r s c o r e s . Some o f t h e s e d e r i v a t i o n s i n v o l v e d m in o r c h a n g e s o f th e d e r i v a t i o n s o f r e l a t e d f o r m u la s w h ic h a r e p r e s e n t e d e l s e w h e r e . O th e r s w ere o r i g i n a l w i t h t h e p r e s e n t i n v e s t i g a t o r . M e n ta l a g e , c h r o n o l o g i c a l a g e , r e a d i n g g r a d e p l a c e m e n t , a n d m e n ta l a g e g ra d e p la c e m e n t fro m t h i r t e e n d i s t r i c t s w e re u s e d . Two g r a d e l e v e l s fro m e a c h d i s t r i c t w e re i n v e s t i g a t e d . T h ese w e re t h e f o u r t h g ra d e (N* 3 7 3 9 )a n d t h e s i x t h g ra d e (N* 3 1 1 6 ) . The n e c e s s a r y d a t a , o f a l l th e c o m b in a tio n s o f d i s t r i c t , a c h ie v e m e n t a r e a , a n d g r a d e s w e re a p p l i e d t o f o u r d i s t i n c t d i f f e r e n c e e q u a t i o n s . F o u r e s t i m a t e d d i f f e r e n c e e q u a t i o n s w e re u s e d ; ( 1 ) t h e t w o - v a r i a b l e (X,Y ) 192 e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , ( 2 ) th e t h r e e - v a r i a b l e (X ,Y ,C ) e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , (3 ) th e a p p r o x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , a n d (4 ) th e a r i t h m e t i c d i f f e r e n c e . T hese e q u a t i o n s w e re co m p a red by m eans o f t h e i r s l o p e s a n d e x p e c ta n c y i n t e r c e p t s , R e l i a b i l i t i e s o f t h e s c o r e s o f th e e q u a t i o n s w ere c o m p u ted a n d c o m p a re d . An e x p l a n a t i o n w as g i v e n i n d i c a t i n g t h e in c o m p a t i b i l i t y o f e q u a t i o n s e s t i m a t i n g t h e d i f f e r e n c e b e tw e e n t h e o b s e r v e d a c h ie v e m e n t a n d m e n ta l a g e a n d t h e d i f f e r e n c e b e tw e e n o b s e r v e d a c h ie v e m e n t a n d e x p e c ta n c y w h ic h i s n o t a s s t r a i g h t m e n ta l a g e , b u t some ( r e g r e s s i o n ) a d a p t a t i o n o f m e n ta l a g e . As a r e s u l t , e x p e c t a n c y e q u a t i o n s w e re i n v e s t i g a t e d , w h ere e x p e c ta n c y i s th e d e p e n d e n t v a r i a b l e r a t h e r t h a n th e d i f f e r e n c e s b e i n g th e d e p e n d e n t v a r i a b l e . T h re e o f t h e s e s t r a i g h t l i n e e x p e c ta n c y e q u a t i o n s w e re i n v e s t i g a t e d ; (1 ) t h e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n ( 2 ) th e t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n , a n d (3 ) th e H orn F o r m u la , R e l i a b i l i t i e s o f t h e s c o r e o f t h e s e e q u a t i o n s w e re a n a l y z e d . V a r i a t i o n s o f th e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n w i t h r e s p e c t t o m e n ta l a b i l i t y g r o u p i n g s w e re 193 I n v e s ti g a te d and compared w ith r e g r e s s io n e q u a tio n s based on t o t a l p o p u la tio n s . A n tic ip a te d Achievement was a n a ly ze d w ith r e s p e c t to the p re v io u s ly m entioned r e g r e s s io n e q u a tio n s . I I . FINDINGS A number of concepts were in v e s t i g a t e d and h ypotheses fo rm u late d i n t h i s s tu d y . The fo llo w in g s e c tio n s r e p r e s e n t a summary of th e se concepts and h y p o th e s e s . F in d in g s re g a rd in g the d if fe r e n c e e q u a tio n s as opposed to expectancy e q u a tio n s . Although d if fe re n c e e q u a tio n s may be c o n sid ere d expectancy e q u a tio n s , they do not y i e l d v a lu e s, d i r e c t l y , which one i s expected to a c h ie v e . For any expectancy based upon a r e g r e s s io n whose c h a r a c t e r i s t i c s a re d e riv e d from the p o p u la tio n b e in g I n v e s ti g a te d , the average expectancy value would equal the average achievem ent of t h a t p o p u la tio n . On the o th e r hand, the d if fe re n c e e q u a tio n s m a n ife ste d averages (based on m ental age av erag es and achievem ent av e ra g es) which were not z e r o . In f a c t , i n a l l d i s t r i c t s , the m ental age grade placem ents exceeded bo th o f the r e s p e c t i v e re a d in g grade placem ents and a r ith m e tic grade p la c e - 1 94 merits f o r Grade 4 and Grade 6 , This excess ranged from a very s l i g h t d if fe re n c e to a d if fe re n c e of one y ea r and f i v e months. T h erefo re, sin c e the average d if fe re n c e value of a l l d if f e re n c e eq u a tio n s f o r any d i s t r i c t i s equal to the d if fe r e n c e between the average sc o re s on the m ental and the average sc o re s on the achievem ent t e s t , i n t e r p r e t a t i o n s of d if f e r e n c e s e q u a tio n s must be made in l i g h t of th e se average d is c r e p a n c ie s . Thus, "Jo h n 's d if f e r e n c e in d ic a t e s th a t he i s s ix months below ex pectancy, b u t t h i s r e l a t i o n s h i p i s ty p ic a l of stu d e n ts i n our d i s t r i c t . " Whereas, i n r e g r e s s io n eq u a tio n s based on l o c a l p o p u la tio n s or i n the A n tic ip a te d Achievement ( i f the l o c a l p o p u la tio n was s im ila r to the norming p o p u la tio n ) i f John i s s ix months below expectancy, t h i s s i t u a t i o n would not be t y p i c a l . F in d in g s re g a rd in g the a p p l i c a b i l i t y of u s in g a s in g le e s tim a te d tru e d if f e re n c e e q i a t l o n f o r Grade 4 and Grade 6 Reading and A r ith m e tic . The a p p l i c a b i l i t y of u sin g a s in g le e s tim a te d tru e d if fe r e n c e e q u a tio n f o r Grade 4 and Grade b Reading and A rith m etic was i n v e s t i g a t ed by means of comparing the slo p e s and I n t e r c e p t s of th e se e q u a tio n s . The evidence in d ic a t e d , as a r e s u l t of a n a ly z in g the d a ta from the slo p e s and i n t e r c e p t s , th a t 1 9 5 th e same e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n c a n n o t a p p r o p r i a t e l y be u s e d f o r G rade 4 R e a d in g , G rade 4 A r i t h m e t i c , Grade 6 R e a d in g , an d G rade 6 A r i t h m e t i c w i t h o u t i n t r o d u c i n g a p p r e c i a b l e e r r o r ( f o r many c a s e s ) . F i n d i n g s r e g a r d i n g th e i n c l u s i o n o f c h r o n o l o g i c a l age to th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . When th e e f f e c t s o f i n c l u d i n g c h r o n o l o g i c a l a g e i n th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n w ere i n v e s t i g a t e d , i t was f o u n d t h a t n e g l i g i b l e d i f f e r e n c e s e x i s t e d i n th e i n t e r c e p t s a n d g e n e r a l l y good a g re e m e n t i n s l o p e e x i s t e d b e tw e e n th e e q u a t i o n s w i t h and w i t h o u t i n c l u s i o n o f c h r o n o l o g i c a l a g e . The s i m p l e r d i f f e r e n c e e q u a t i o n ( w ith o u t C) c an be u s e d w i t h e q u a l r e s u l t s ( i n m ost c a s e s ) . F i n d i n g s r e g a r d i n g th e u se o f th e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n i n p l a c e o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . The r e s u l t s fro m b o t h th e s lo p e and i n t e r c e p t c o m p a ris o n s r e v e a l t h a t , i n g e n e r a l , th e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n ( a f t e r McNemar) i s n o t a good s u b s t i t u t i o n f o r th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . 1 9 6 F i n d i n g s r e g a r d i n g th e c o m p a ris o n o f th e A r i t h m e t i c d l f f e r e n c e a n d th e e s t i m a t e d t r u e d i f f e r e n c e , The s l o p e s o f th e a r i t h m e t i c d i f f e r e n c e d i f f e r fro m th e s l o p e s of th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s . A lth o u g h n o t a l l i n t e r c e p t s (o f t h i s c o m p a ris o n ) d i f f e r s t a t i s t i c a l l y , m ost o f them d o . T h e r e f o r e , i t c an g e n e r a l l y be s a i d t h a t th e a r i t h m e t i c d i f f e r e n c e i s n o t a good s u b s t i t u t e f o r th e e s t i m a t e d t r u e d i f f e r e n c e . The e s t i m a t e d t r u e d i f f e r e n c e i s assum ed to th e more a c c u r a t e . F i n d i n g s r e g a r d i n g th e c o m p a ris o n o f th e r e l i a b i l i t i e s of e s t i m a t e d d i f f e r e n c e e q u a t i o n s . A lth o u g h th e r e l i a b i l i t i e s o f th e s c o r e s o f a n y d i f f e r e n c e e q u a t i o n v a ry o v e r a r e l a t i v e l y l a r g e i n t e r v a l t h e r e i s l i t t l e , i f a n y , v a r i a t i o n b e tw e e n th e v a r i o u s d i f f e r e n c e m ethods f o r a n y d i s t r i c t . T hus, th e m ethods u se d f o r o b t a i n i n g e s t i m a t e d d i f f e r e n c e s i n t h i s s tu d y c a n n o t be a c c e p t e d ( o r r e j e c t e d ) a s a s u p e r i o r (o r i n f e r i o r ) m ethod on th e b a s e s o f r e l i a b i l i t y a l o n e . However, r e l i a b i l i t y may s e r v e a s a c r i t e r i o n f o r e l i m i n a t i n g c e r t a i n d i s t r i c t e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s on th e b a s i s of l n c o n s l s t a n c y . 197 F i n d i n g s r e g a r d i n g th e u se o f a s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n t o be u s e d b e t v e e n g r a d e s a n d a c h ie v e m e n t a r e a s . An a n a l y s i s o f t h e d a t a o f t h e s l o p e s o f th e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n i n d i c a t e d t h a t a s i n g l e e q u a t i o n c a n n o t a d e q u a t e l y r e p r e s e n t Grade 4 R e a d in g , G rade 4 A r i t h m e t i c , G rade 6 R e a d in g a n d G rade 6 A r i t h m e t i c . T h is f i n d i n g was s u b s t a n t i a t e d by a c o r r e s p o n d i n g I n v e s t i g a t i o n o f t h e c o r r e s p o n d i n g I n t e r c e p t . F i n d i n g s r e g a r d i n g th e s i m i l a r i t y o f p r e d i c t i o n w i t h and w i t h o u t th e u se of c h r o n o l o g i c a l ag e i n th e r e g r e s s i o n e q u a t i o n s . As I n th e c a s e o f th e e s t i m a t e d d i f f e r e n c e e q u a t i o n , i t was f o u n d t h a t th e i n c l u s i o n of c h r o n o l o g i c a l a g e I n th e r e g r e s s i o n e q u a t i o n d i d n o t i n f l u e n c e th e s l o p e . The i n t e r c e p t s o f th e r e g r e s s i o n e q u a t i o n , a l s o , w ere n o t ch an g e d by v i r t u e o f i n c l u d i n g th e a v e r a g e c h r o n o l o g i c a l ag e th e d i s t r i c t t o a n i n d i v i d u a l ' s e x p e c t a n c y . The f a c t t h a t t h e a v e r a g e a g e o f th e p o p u l a t i o n was u s e d i n t h i s p a r t o f t h e i n v e s t i g a t i o n d i d n o t i n t r o d u c e a p p r e c i a b l e e r r o r t o th e i n d i v i d u a l ' s s c o r e s i n c e : 1 . The s t a n d a r d d e v i a t i o n of c h r o n o l o g i c a l a g e i s r e l a t i v e l y s m a l l , t h u s , a l l c h r o n o l o g i c a l a g e s t e n d t o be s i m i l a r f o r an y g i v e n g r a d e . 2 . The r e g r e s s i o n c o e f f i c i e n t s o f c h r o n o l o g i c a l 198 ag e a r e r e l a t i v e l y s m a l l , t h u s l e s s e n i n g th e e f f e c t o f a g e on p r e d i c t i o n , 3 . c o r r e l a t i o n b e tw e e n c h r o n o l o g i c a l ag e and a c h ie v e m e n t f o r a n y g iv e n g ra d e i s , i n g e n e r a l , c l o s e t o c h an c e r e l a t i o n s h i p (low n e g a t i v e ) , th u s i n d i c a t i n g a r e l a t i v e l y s m a ll r e l a t i o n s h i p of c h r o n o l o g i c a l age on a c h ie v e m e n t, 4 , The m u l t i p l e c o r r e l a t i o n I n d i c a t e s l i t t l e i n c r e a s e o f p r e d i c t i o n w i t h th e i n c l u s i o n of c h r o n o l o g i c a l a g e . F i n d i n g s r e g a r d i n g th e u se o f t h e Horn F o rm u la a s r e p r e s e n t a t i v e and a d e q u a te m easu re o f e x p e c t a n c y . T hat th e Horn F o rm u la i s a n a d e q u a te m ea su re o f e x p e c ta n c y f o r th e g ro u p fro m w h ic h t h e e q u a t i o n was d e r i v e d , i s n o t q u e s t i o n e d i n t h i s s t u d y , h o w ev er, th e e m p h a sis on c h r o n o l o g i c a l age i n d e t e r m i n i n g e x p e c ta n c y a n d th e u n i v e r s a l a p p l i c a b i l i t y o f th e Horn F o rm u la t o v a r i o u s a g e s a n d a c h ie v e m e n t a r e a s w ere r a i s e d a s i t e m s , w h ic h , when r e s p e c t i v e l y , w h o lly a c c e p t e d a s m ajo r c o n t r i b u t o r s a n d y i e l d i n g r e p r e s e n t a t i v e r e s u l t s , t h e i r v a l i d i t y was q u e s t i o n e d . The u se o f c h r o n o l o g i c a l age i n th e Horn F o rm u la i s q u e s t i o n e d on th e b a s i s o f th e f o l l o w i n g : 1 , The u s e f u l n e s s o f c h r o n o l o g i c a l age i n any r e g r e s s i o n e x p e c ta n c y e q u a t i o n h a s b e e n fo u n d to be r e l a t i v e l y s m a l l . 2 . H o r n 's a s s u m p t io n s r e g a r d i n g r e g r e s s i o n to w a rd th e mean m ig h t e q u a l l y be a p p l i e d t o th e mean a c h ie v e m e n t o f th e g ro u p o r th e a c t u a l g ra d e p l a c e m ent o f th e i n d i v i d u a l . 1 99 T h at th e r e g r e s s i o n e q u a t i o n s f o r v a r i o u s g rad e a c h ie v e m e n t a r e a s a r e d i f f e r e n t w ould im p ly t h a t th e u n i v e r s a l u s e of th e Horn F orm ula f o r d i f f e r e n t g r a d e s and a c h ie v e m e n t a r e a s i s a l s o q u e s t i o n e d . S in c e a r i t h m e t i c a c h ie v e m e n t h a s a d i s t r i b u t i o n w h ich i s t y p i c a l l y more r e s t r i c t e d t h a n th e r e a d i n g d i s t r i b u t i o n , i t i s f u r t h e r e v i d e n t t h a t th e same e x p e c ta n c y f o r m u la i s i n a d e q u a t e • F i n d i n g s r e g a r d i n g th e u se o f a s i n g l e r e g r e s s i o n e q u a t i o n a s a u n i v e r s a l e x p e c ta n c y e q u a t i o n f o r v a r i o u s d i s t r i c t s when g ra d e and a c h ie v e m e n t a r e a i s c o n s t a n t . T h at a l a r g e p o p u l a t i o n y i e l d s a more r e p r e s e n t a t i v e m odel, o t h e r t h i n g s b e in g e q u a l , m ig h t I n d i c a t e t h a t a l a r g e p o p u l a t i o n y i e l d s a r e g r e s s i o n e q u a t i o n w h ich w ould have b e t t e r p r e d i c t i v e v a l u e . P r e d i c t i o n , ho w ev er, i s i n p a r t a f u n c t i o n o f l o c a l c h a r a c t e r i s t i c s . The same d i s c u s s i o n m ig h t a p p ly t o e x p e c ta n c y . However, e x p e c ta n c y i s n o t n e c e s s a r i l y p r e d i c t i o n . E x p e c ta n c y may ta k e th e fo rm o f a n i n d i v i d u a l ' s e x p e c te d s c o r e w i t h a l a r g e ( p e r h a p s n a t i o n w i d e ) sam ple a s a r e f e r e n c e o r i t may t a k e th e fo rm , o f h i s e x p e c te d s c o r e w i t h a l o c a l (more l i m i t e d o r a d i s t r i c t ) p o p u l a t i o n a s a r e f e r e n c e . The r e l a t i v e im p o rta n c e o f t h e s e two a l t e r n a t i v e s d e p en ds upon o n e 's 200 t h e o r y of e x p e c t a n c y . I f one a c c e p t s th e p o i n t o f view t h a t e x p e c ta n c y s h o u ld n o t be c o n f i n e d t o l o c a l c h a r a c t e r i s t i c s , o u t s i d e c o m p a ris o n s a r e n o t t a k e n i n t o a c c o u n t . On t h e o t h e r h a n d , i f a ty p e o f e x p e c ta n c y w h ic h i s b a s e d on b r o a d e r d a t a i s u s e d , th e s t u d y i n d i c a t e s t h a t t h e r e may be w ide d i f f e r e n c e s b e tw e e n e x p e c t a n c i e s b a s e d on d i f f e r e n t d i s t r i c t s . F i n d i n g s r e g a r d i n g th e u s e o f r e g r e s s i o n e q u a t i o n s f o r e x p e c ta n c y when a b i l i t y g r o u p in g s a r e c o n s i d e r e d . When r e g r e s s i o n e x p e c ta n c y e q u a t i o n s a r e com puted on t h e b a s i s o f a b i l i t y (IQ) g r o u p in g s t h e s e e q u a t i o n s do n o t c o i n c i d e w i t h a s i n g l e r e g r e s s i o n e q u a t i o n b a s e d upon th e e n t i r e p o p u l a t i o n . The l a r g e s t d i s c r e p a n c y i s w i t h th e low a b i l i t y g r o u p . A l i n e o f b e s t f i t o f th e t h r e e a b i l i t y g r o u p in g r e g r e s s i o n e q u a t i o n s r e p r e s e n t s a c u rv e i n w h ich th e s lo p e I n c r e a s e s a s th e m e n ta l ag e I n c r e a s e . S in c e th e a v e r a g e a n d h i g h a b i l i t y g ro u p r e g r e s s i o n e q u a t i o n s a r e s i m i l a r , b u t a v e r a g e an d low much d i f f e r e n t fro m e a c h o t h e r , t h e r e I s i n d i c a t i o n t h a t two s t r a i g h t l i n e r e g r e s s i o n e q u a t i o n s (one b a s e d on th e below a v e r a g e g ro u p an d t h e o t h e r b a s e d on th e above a v e r a g e g ro u p ) may be p r o f i t a b l y u s e d a s i n d i c a t o r s o f e x p e c t a n c y . 201 F i n d i n g s r e g a r d i n g th e A n t i c i p a t e d A c h ie v e m e n t. In s p i t e o f c e r t a i n I n f l u e n c e s w h ic h c o n t r i b u t e d t o th e n o n c o m p a r a b i l i t y o f th e A n t i c i p a t e d A ch iev em en t w i t h o t h e r e x p e c t a n c i e s , th e A n t i c i p a t e d A chiev em en t n o t o n ly d u p l i c a t e d ( i n m ost a r e a s ) th e c o n t o u r s of th e r e g r e s s i o n e q u a t i o n s ( d e r i v e d a c c o r d i n g t o a b i l i t y ) b u t c o i n c i d e d w i t h th e a c t u a l e x p e c ta n c y v a l u e s , a s w e l l . T hus, th e A n t i c i p a t e d A c h ie v e m e n t, a l t h o u g h b a s e d on a s t a n d a r d i z e d n a t i o n w ide Bample, was i n harmony w i t h th e r e g r e s s i o n l i n e s w h ic h w ere b a s e d on th e sam ple o f t h i s s t u d y , th e l a r g e s t d i s c r e p a n c i e s b e in g i n th e a r e a o f low a b i l i t y . T h is c r o s s v a l i d a t i o n s u b s t a n t i a t e s th e A n t i c i p a t e d A ch iev em en t a s a n e x p e c ta n c y c o n g ru e n t t o th e f i n d i n g s o f t h i s s t u d y . The t h e o r y o f th e A n t i c i p a t e d A chievem ent i s sou n d , i n c o m p a ris o n w i t h o t h e r e x p e c ta n c y m e th o d s, i n t h a t i t i n v o l v e d a s im u lt a n e o u s d u a l s t a n d a r d i z a t i o n o f th e a c h ie v e m e n t t e s t a n d th e a b i l i t y t e s t . S in c e t h e s e t e s t s a r e r e l a t e d on a raw s c o r e b a s i s , a n d i n t u r n i n a g ra d e p la c e m e n t b a s i s , th e e x p e c ta n c y i t s e l f i s a ty p e of n o r m a tiv e e x p e c ta n c y . 202 I I I . CONCLUSIONS W ith r e g a r d t o e s t i m a t e d d i f f e r e n c e e q u a t i o n s , i t may "be c o n c lu d e d t h a t th e s im p le a r i t h m e t i c d i f f e r e n c e c a n n o t b e u s e d i n t e r c h a n g e a b l e w i t h th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . F o r i n d i v i d u a l s , d i f f e r e n t r e s u l t s w i l l o c c u r fro m t h e s e two m e th o d s . S in c e t h e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n t a k e s i n t o a c c o u n t t h e e f f e c t s o f r e g r e s s i o n an d th e r e l i a b i l i t y o f th e t e s t s i n v o l v e d ; w h e re a s t h e a r i t h m e t i c d i f f e r e n c e d o e s n o t ta k e t h e s e i n t o a c c o u n t , i t i s c o n c lu d e d t h a t th e f o r m e r p r o c e d u r e y i e l d s more a c c u r a t e r e s u l t s t h a n th e l a t t e r . I t I s a l s o c o n c lu d e d t h a t th e r e s u l t s o f t h e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n do n o t c o i n c i d e w i t h th o s e o f th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . These two e q u a t i o n s c a n n o t be u se d i n t e r c h a n g e a b ly w i t h o u t y i e l d i n g d i f f e r e n t r e s u l t s . The i n c l u s i o n o f c h r o n o l o g i c a l age i n th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n d oes n o t change a n I n d i v i d u a l ' s e x p e c ta n c y d i f f e r e n c e r e s u l t s th e way g ra d e l e v e l s a r e c u r r e n t l y com posed. T h u s, th e r e l a t i v e l y s im p le tw o - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n may b e u s e d i n p r e f e r e n c e t o t h e more c o m p l i c a te d t h r e e - v a r i a b l e 203 e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n w h ic h I n c l u d e s c h r o n o l o g i c a l a g e . I n g e n e r a l , th e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n whose c o e f f i c i e n t s a r e b a s e d upon a l a r g e r e p r e s e n t a t i v e p o p u l a t i o n may be u s e d w i t h r e l a t i v e l y s m a ll d i f f e r e n c e s b e tw e e n d i s t r i c t s . T h is s t a t e m e n t i s t r u e o n ly w i t h r e g a r d t o a c o n s t a n t a c h ie v e m e n t a r e a and a c o n s t a n t g r a d e . E s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n s , i n g e n e r a l , c a n n o t be i n t e r c h a n g e d b e tw e e n g ra d e an d a c h ie v e m e n t a r e a . The r e s u l t s o f e s t i m a t e d d i f f e r e n c e e q u a t i o n s m ust be I n t e r p r e t e d d i f f e r e n t l y fro m e x p e c ta n c y r e s u l t s . The same e x p e c ta n c y e q u a t i o n s c a n n o t be u s e d f o r r e a d i n g an d a r i t h m e t i c w h e t h e r t h e y be th e Horn F o rm u la o r s i n g l e o r t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n s . The Horn F o rm u la i s n o t a s t h e o r e t i c a l l y o r p r a c t i c a l l y u s e f u l a s r e g r e s s i o n e q u a t i o n s b a s e d upon a r e p r e s e n t a t i v e p o p u l a t i o n . The a c t u a l g ra d e l e v e l o f a n i n d i v i d u a l may be u s e d a s a v a lu e t o w h ic h t h e g ro u p r e g r e s s e s i n p r e f e r e n c e t o th e i n d i v i d u a l ' s a g e . The i n c l u s i o n o f c h r o n o l o g i c a l a g e i n a r e g r e s s i o n e q u a t i o n d o e s n o t c h an g e a n i n d i v i d u a l ' s e x p e c ta n c y r e s u l t s . T h ere t e n d s t o be more s i m i l a r i t y o f r e g r e s s i o n l i n e e x p e c ta n c y b e tw e e n v a r i o u s d i s t r i c t s f o r a r i t h m e t i c t h a n 2 0 4 f o r r e a d i n g . R e g r e s s i o n l i n e e x p e c ta n c y i s m ost a p p r o p r i a t e f o r l o c a l d i s t r i c t s , s i n c e t h e r e a r e som etim es w ide d i f f e r e n c e s b e tw e e n th e e x p e c t a n c i e s o f s t u d e n t s w i t h th e same m e n ta l age when a p p l i e d t o one d i s t r i c t a s o p p o se d t o t h a t o f a n o t h e r . The A n t i c i p a t e d A chievem ent a d e q u a t e l y r e p r e s e n t s r e g r e s s i o n e x p e c t a n c i e s th r o u g h o u t m ost o f th e ra n g e o f m e n ta l a g e . T h is e s p e c i a l l y t r u e i n th e " a v e r a g e " ran g e * The m ost p ro n o u n ce d d e v i a t i o n i s i n th e lo w e r m e n ta l age ra n g e s . IV. RECOMMENDATIONS The r e s u l t s o f t h i s s t u d y g iv e e v id e n c e w hich s u p p o r t s th e rec o m m en d atio n t h a t i f a d i s t r i c t i s c o n c e r n e d w i t h l o c a l c o m p a ris o n s th e s i n g l e v a r i a b l e r e g r e s s i o n e q u a t i o n i s recommended i n p r e f e r e n c e t o th e Horn F orm ula o r t h e more c o m p l i c a te d t w o - v a r i a b l e r e g r e s s i o n e q u a t i o n . I f d i f f e r e n c e s b e tw e e n m e n ta l a g e and a c h ie v e m e n t a r e d e s i r e d , i t i s recommended t h a t th e t w o - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n be u s e d i n p r e f e r e n c e t o th e a r i t h m e t i c d i f f e r e n c e , th e a p p ro x im a te e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n , o r t h e more c o m p lic a te d t h r e e - v a r i a b l e e s t i m a t e d t r u e d i f f e r e n c e e q u a t i o n . F o r b o th o f 205 th e above re c o m m e n d a tio n s, th e e q u a t i o n s s h o u ld be b a s e d upo n t h e g ra d e a n d a c h ie v e m e n t l e v e l t o w h ich t h e y w i l l be a p p l i e d . I f a n a t i o n a l ( o r b r o a d e r b a s e ) c o m p a ris o n o f e x p e c ta n c y i s d e s i r e d , i t i s recommended t o u se t h a t ty p e o f e x p e c ta n c y w h ic h was em ployed to o b t a i n th e A n t i c i p a t e d A c h ie v e m e n t; t h a t i s , e x p e c ta n c y o b t a i n e d b y a p r o c e s s o f d u a l s t a n d a r d i z a t i o n o f th e m e n ta l a b i l i t y t e s t a n d th e a c h ie v e m e n t t e s t . S in c e th e A n t i c i p a t e d A chievem en t o r o t h e r r e g r e s s i o n m etho ds do n o t ta k e i n t o a c c o u n t th e am ount o f v a r i a b i l i t y w h ich m ig h t be e x p e c te d fro m t h i s p r o c e d u r e , a n a d d i t i o n a l d im e n s io n i s i n o r d e r . T h is rec o m m e n d atio n i n v o l v e s th e u s e o f p e r c e n t i l e l i n e s o f c o n to u r s a ro u n d th e mean e x p e c ta n c y , e n a b l i n g some d e g re e o f m easure o f v a r i a b i l i t y t o be u s e d . T hus, a n i n d i v i d u a l " a t e x p e c ta n c y " w ould be a t th e 5 0 t h e x p e c ta n c y p e r c e n t i l e . Anyone "below e x p e c ta n c y " w ould be i d e n t i f i e d somewhere b e tw e e n th e f i r s t an d th e f o r t y - n i n t h p e r c e n t i l e ; w h ile anyone "above e x p e c ta n c y " w o u ld be i d e n t i f i e d somewhere b e tw e e n th e f i f t y - f i r s t a n d n i n e t y - n i n t h p e r c e n t i l e . A s c h e m a tic p r e s e n t a t i o n o f t h i s p r o c e d u r e i s p l c t o r i a l l y p r e s e n t e d i n F i g u r e 1 4 . T hus, th e A n t i c i p a t e d A c h ie v e m e n t, a l t h o u g h b a s e d on a s t a n d a r d i z e d n a tio n w id e sa m p le , was i n 206 harm ony w i t h th e r e g r e s s i o n l i n e s w h ich w ere b a s e d on th e sam ple o f t h i s s t u d y ; t h e l a r g e s t d i s c r e p a n c i e s b e i n g I n th e a r e a of low a b i l i t y . T h is c r o s s - v a l i d a t i o n s u b s t a n t i a t e s t h e A n t i c i p a t e d A chiev em en t a s a n e x p e c ta n c y c o n g r u e n t to th e f i n d i n g s of t h i s s t u d y . The t h e o r y o f th e A n t i c i p a t e d A ch iev em en t I s sound i n c o m p a ris o n w i t h o t h e r e x p e c ta n c y m ethods I n t h a t I t i n v o l v e d a s i m u lta n e o u s d u a l s t a n d a r d i z a t i o n o f th e a c h ie v e m e n t t e s t and th e a b i l i t y t e s t . S in c e th e s e t e s t s a r e r e l a t e d on a raw s c o r e b a s i s , an d i n t u r n on a g rad e p la c e m e n t b a s i s , th e e x p e c ta n c y I t s e l f i s a ty p e o f n o r m a tiv e e x p e c ta n c y . SCHEMATIC i FOR IHDIC. M e a ta l a g e j g ra d e p la c e m e n t FIGURE 14 OP A RE30MM ENDED FRi JffiVIATFIOKFR J R E KFECEANC? BIBLIOGRAPHY BIBLIOGRAPHY A . BOOKS D a v is , A l l i s o n , a n d K e n n e th E e l l s . p a v l s - E e l l s T e s t s of G e n e r a l I n t e l l i g e n c e o r P r o b le m - S o lv in g A b i l i t y M a n u a l. New Y ork; W orld Book Company, 19537 ----------- D ix o n , W i l f r i e d J . , a n d F r a n k J , M assey, J r . I n t r o d u c t i o n t o S t a t i s t i c a l A n a l y s i s . New Y ork: McGraw- H l l l Book Company, I n c . , 19 5 7 • 488 p p • E d w ard s, A l l e n L . E x p e r i m e n t a l D e s ig n i n p s y c h o l o g i c a l R e s e a r c h . New y o r k : R i n e h a r t an d Company, I n c . , 195^7 £46 p p . E e l l s , K e n n e th , A l l i s o n D a v is , R o b e r t J . H a v i g h u r s t , V i r g i l E . H e r r i c k a n d R a lp h W. T y l e r . I n t e l l i g en ce a n d C u l t u r a l D i f f e r e n c e s . C h i c a g o : The U n i v e r s i t y C h ica g o p r e s s , 1951 • 388 p p . G a r r e t t , H enry E . S t a t i s t i c s i n p s y c h o lo g y a n d E d u c a t i o n . New Y ork: Longm ans, G reen a n d Company, 1 9 5 3 . 4b0 p p . G u l l i k s e n , H a r o ld . T h eo ry o f M e n ta l T e s t s . New Y ork; J o h n W ile y a n d S o n s ,' In c . , 1 9 8 0 . 4'8o p p . G u i l f o r d , J . P . F u n d a m e n ta l S t a t i s t i c s i n p s y c h o lo g y a n d E du cation* ^ New Y ork; M cG raw -H ill Book Company, 1950* 833 p p . H orn, A l i c e M. Uneven D i s t r i b u t i o n o f E f f e c t s o f S p e c i f i c F a c t o r s . Los A n g e l e s : u n i v e r s i t y o f S o u t h e r n C a l i f o r n i a p r e s s , 1 9 4 7 . 107 p p , K e l l e y , Truman L . I n t e r p r e t a t i o n o f E d u c a t i o n a l M e a s u re m e n t. Y o n k e rs , New Y o r k : W orld Book Company, 1 9 2 7 . 210 1 0 . 11. 1 2 . 1 3 . 1 4 . 15. 1 6 . 1 7 . 1 6 . i y . 2 0 . 2 1 . M onroe, W a lte r S . E n c y c lo p e d ia o f E d u c a t i o n a l R e s e a r c h . New Y ork; The M a c M illia n Company, 1 9 5 0 . 1520 p p . M c C all, W illia m A . How t o M easure I n E d u c a t i o n . New Y ork; The M a c M illia n Company, 1 9 22 . 4 l 6 p p . O lso n , W i l l a r d C. C h i ld D e v e lo p m e n t. B o sto n : D. C. H e ath an d Company, 1 9 4 9 . 415 PP* P a t e r s o n , D onald G. P h y s iq u e a n d I n t e l l e c t . New Y ork; A p p l e t o n - C e n t u r y - C r o f t s , 1 9 3 0 . S i e g e l , S id n e y . N o n p a ra m e trlc S t a t i s t i c s f o r th e B e h a v i o r a l S c i e n c e s . New Y ork: M cG raw -H ill Book Company, I n c . , 1 9 5 6 . 312 p p . Spearm an, C h a r le s E . The A b i l i t i e s o f Man. New Y ork; The M a c M illia n Company, 1 9 2 7 . 415 p p . Thomson, G. H. F a c t o r i a l A n a l y s i s o f Human A b i l i t y . H oughton Company, 1 9 3 9 . 326 p p • T h u r s t o n e , L . L . C u r r e n t I s s u e s I n F a c t o r A n a l y s i s , p s y c h o l o g i c a l B u l l e t i n 27: 1 9 4 0 . 1 8 9 :2 3 6 p p . __________ , a n d T. G. T h u r s t o n e . F a c t o r i a l S t u d i e s o f I n t e l l i g e n c e . p s y c h o m e tr ic M onographs No. 2 , C h ic a g o : U n i v e r s i t y o f C h icag o p r e s s , 1 9 4 1 . W alk er, H elen M., and J o s e p h L e v . S t a t i s t i c a l I n f e r - e n c e . New Y ork; Henry H o lt an d Company, 195 3 . 510 p p . B. PERIODICALS Abrams, Elias N. "prediction of Intelligence from Certain Rorschach Factors," journal of Clinical psychology, vol. 11, No. 1 (January,"T955)» 61-83, A l l e n , M ild re d M. " R e l a t i o n s h i p B etw een K uhlm ann- A n d e rso n I n t e l l i g e n c e T e s t s a n d a c a d e m i c A c h ie v e m ent I n G rade I V , " The J o u r n a l o f E d u c a t i o n a l p s y c h o l o g y , V o l. 35 , No. 4 ( A p r i l , 1 9 4 4 ), 2 2 9 -2 3 9 . 211 2 2 . __________ . " R e l a t i o n s h i p B etw een th e I n d i c e s of I n t e l l i g e n c e D e riv e d fro m th e K uhlm ann-A nderson I n t e l l i g e n c e T e s t s f o r G rade 1 a n d th e same t e s t s f o r G rade I V , " The J o u r n a l o f E d u c a t i o n a l P s y c h o lo g y , " V o l . 36 , No. 4""~[April, 1 9 4 5 ) , 2 5 2 -2 5 5 2 3 . A l t u s , E . D. a n d G race M. Thompson. "The R o rsc h a c h a s a M easure o f I n t e l l i g e n c e , " J o u r n a l of C o n s u l t i n g p s y c h o lo g y , V o l. 1 3 , No. 5 ( O c to b e r , 1~949), m - m r . 2 4 . A l t u s , G race T. " R e l a t i o n s h i p s B etw een v e r b a l and N o n -V erb a l p a r t s o f th e CTMM a n d WISC," J o u r n a l o f C o n s u l t i n g p s y c h o lo g y , V o l. 1 9 , N o .2 ( A p r i l , 1 3 5 5 ) , 1 4 3 -1 4 4 “-------------- 2 5 . A r m ita g e , S te w a rd G. a n d p a u l D. G r e e n b e r g , D av id P e a r l , D avid G. B e rg e r a n d P a u l G. D a s to n . " p r e d i c t i n g I n t e l l i g e n c e fro m th e R o r s c h a c h ," J o u r n a l o f C o n s u l t i n g p s y c h o lo g y , V o l. 1 9 , No. 5 T o ct'o ^ er7 "1 9 5 '5 ), 321 ” 32^ --------- 2 u . Assum, A r t h u r L . a n d S id n e y J . L ev y . "a C o m p ara tiv e S tu d y o f th e Academic A b i l i t y a n d a c h ie v e m e n t o f Two G roups o f C o lle g e S t u d e n t s , " The J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 38 , No. 5 (May, T ^ 4 7 ) 3 0 7 -3 1 0 . 2 7 . B a e r , C lyde J . "The S c h o o l P r o g r e s s an d A d ju s tm e n t o f U nderage and O verage S t u d e n t s , " The J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 49, No. 1 ( F e b r u a r y , 195B ), 1 7 - 1 9 . 2 8 . B a r c h , Abram M. "The R e l a t i o n o f D e p a r tu re Time and R e t e n t i o n t o Academ ic A c h i e v e m e n t," The J o u r n a l E d u c a t i o n a l p s y c h o lo g y , V o l. 48, No. Z T o c t o b e r , 1 9 5 7 ) , 3 5 2 - 3 5 8 . 29* B a r r a t t , E r n e s t S . a n d D o ris L . B a u m g a rte n . "The R e l a t i o n s h i p o f th e WISC a n d S t a n f o r d - B i n e t t o S c h o o l A c h i e v e m e n t," j o u r n a l o f C o n s u l t i n g P s y c h o lo g y , v o l . 21, No. 2 ( A p r i l , 1 9 5 7 ) , 144 3 0 . B a y le y , N ancy. "Some I n c r e a s i n g p a r e n t C h ild S i m i l a r i t i e s D u rin g th e Growth o f C h i l d r e n , " J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 45 , 1945 212 3 1 . B e n d ig , A . W. “The v a l i d i t y o f Two Tem perament S c a l e s i n p r e d i c t i n g S t u d e n t A ch iev e m e n t i n i n t r o d u c t o r y p s y c h o l o g y ," J o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 50, No. 8 ( A p r i l , 1 95?) 5 7 1 -5 8 0 . 3 2 . B lechnew , j a n e l E . and H a ro ld D. C a r t e r , "R o rsc h ac h P e r s o n a l i t y F a c t o r a n d C o lle g e A c h ie v e m e n t," C a l i f o r n i a j o u r n a l o f E d u c a t i o n a l R e s e a r c h , v o l • 7 No. 2 (M arch, 1956)”T 2 -? 5 * 3 3 . B l l e s m e r , Emery P . "R ead in g A b i l i t i e s o f B r i g h t an d D u ll c h i l d r e n o f Com parable M ental a g e s , " The j o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 45, No. 6 , " ( o c t o b e r T ^ ^ T T ^ r ^ a r ; 3 4 . B o l to n , F lo y d B. "V alue o f S e v e r a l I n t e l l i g e n c e T e s t s f o r p r e d i c t i n g S c h o l a s t i c A c h ie v e m e n t, "j o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 61, No. 2 (O c to b e r , 1 9 4 7 7 ,' 1'31-T3B.---------------- 3 5 . Bond, Guy L . a n d T h e o rd o re W. C ly m er. " i n t e r r e l a t i o n s h i p o f th e SRA P rim a ry M en tal A b i l i t i e s , o t h e r M e n tal C h a r a c t e r i s t i c s , a n d R e a d in g A b i l i t y , " J o u r n a l o f E d u c a t i o n a l R e s e a r c h , v o l . 10510, N o ,2 ( O c to b e r , 1 9 5 5 )» 1 3 1 -1 3 F I 3 6 . B ro g d e n , H u r b e r t E . " I n c r e a s e d r’f f l c i e n c y o f S e l e c t i o n R e s u l t i n g fro m R e p la ce m en t o f a S i n g l e p r e d i c t o r w i t h S e v e r a l D i f f e r e n t i a l p r o d i c t o r s , " E d u c a t l o n - a l and p s y c h o l o g i c a l M e asu rem en t, V o l , 11, No. 2, “ (Summer, I 95I ) , 1 7 3 -1 9 5 , 3 7 . Brown, G i l b e r t L . "On The C o n sta n c y o f th e IQ ," j o u r n a l o f E d u c a t i o n a l R e s e a r c h , v o l . 44, No. 2 ( O c to b e r , 19507," 1 5 1 ^ 5 ^ 3 8 . Brown, M oroni H. a n d G. E l i z a b e t h B ry a n . "Sex D i f f e r e n c e s i n I n t e l l i g e n c e , " J o u r n a l o f C l i n i c a l P s y c h o lo g y , V o l. 1 1 , No. 3 ( J u l y , 1 9 5 5 ) / 3 ^ 3 -3 0 4 . 3 9 . __________• "The A l t i t u d e Q u o t i e n t a s a M easurem ent o f I n t e l l e c t u a l p o t e n t i a l , " j o u r n a l o f C l i n i c a l p s y c h o lo g y , V o l. 1 3 , No. 2 ( A p r i l , 1 9 5 ? ) , 1 3 6 -1 4 0 . 4 0 . __________. "Sex a s a v a r i a b l e I n I n t e l l i g e n c e T e s t P e r f o r m a n c e ," The J o u r n a l o f E d u c a t i o n a l P s y c h o lo g y , V o l. 48 , N o .5 ~ T M a y , 1 9 5 7 ) , 2 7 3 - 2 7 8 . 213 4 1 . Brown, W illia m F . a n d Wayne H. H o ltzm an , "A S tu d y A t t i t u d e s Q u e s t i o n n a i r e f o r P r e d i c t i n g Acadmeic S u c c e s s , " The J o u r n a l o f E d u c a t i o n a l p sy c h o l o g y , V o l. 46 , n o .2 ( F e b r u a r y , l95>FJT~75>-H4. 4 2 . B u r r a l l , L u c i l l e , " v a r i a b i l i t y i n A c h iev e m e n t o f P u p i l s a t th e F i f t h Grade L e v e l , " C a l i f o r n i a J o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 5 . No. 2 (M arch, 1 9 5 4 ) , 6 8 - 7 3 . 4 3 . C a f f r e y , J o h n . " E s tim a te d True G row th , " L o r d 's E q u a t i o n s A p p l i e d t o R e a d in g T e s t D a ta , C a l i f o r n i a J o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 7, No. 4 "[Septem ber,' 4 4 . C a r t e r , L o w e ll B u rn e y . "The E f f e c t o f E a r l y S c h o o l E n tr a n c e on t h e S c h o l a s t i c A c h iev e m e n t o f E le m e n ta r y S c h o o l C h i l d r e n i n t h e A u s t i n p u b l i c S c h o o l s , " J o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 5 0 , No. 2 ( O c to b e r , 1 9 5 6 ), 9 l - l 0 3 . 4 5 . C h a h b a z l, p a r i v z . " A n a ly s is o f C o r n e l l o r i e n t a t i o n I te m s on S tu d y H a b i t s an d T h e i r R e l a t i v e V alue i n P r e d i c t i o n o f C o lle g e A c h ie v e m e n t," J o u r n a l of E d u c a t i o n a l R e s e a r c h , V o l. 5 1 . No. 2 ( O c to b e r , 1957), 1 1 7 -1 5 7 :--------- 4 6 . C o h l e r , M i l to n J . " S c h o l a s t i c S t a t u s o f A c h i e v e r s an d N o n -A c h ie v e rs o f S u p e r i o r I n t e l l i g e n c e , " The J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 2 1 , No. 3 (NoVember, 1 9 4 1 ) , 6 0 3 -6 1 0 . 4 7 . C olem an, W illia m a n d Edward E . C u r e to n . " i n t e l l i g e n c e a n d A c h ie v e m e n t; The 'JANGLE FALLACY1 A g a i n ," V o l. 1 4 , No. 2 (Summer, 195*0 > 3 4 7 -3 5 1 . 4 8 . Colem an, W illia m a n d A nnie W. W ard, "a C om p ariso n o f D a v ls - E e lls and K uhlam ann-Finch S c o re s o f C h ild re n From High and Low Socio-E conom ic S t a t u s , " The J o u r n a l o f E d u c a tio n a l p sy c h o lo g y , V o l. 46," !N o. 8 (becem ber, 1 9 5 5 ) . 4b5-4b‘9". 4 9 . Corarey, Andrew L . "M en tal T e s t i n g a n d t h e L o g ic o f M e a s u r e m e n t," E d u c a t i o n a l a n d p s y c h o l o g i c a l M easu rem en t, V o l. l l , No. 3 (Autumn, I 95I ) , 3 2 3 -3 3 4 . 214 5 0 . C o o p e r, Jam es G. " P r e d i c t i n g S c h o o l A chievem ent f o r B i lin g u a l p u p i l s , " The J o u rn a l of E d u c a tio n a l p sy c h o lo g y , V o l. 49* No. 1 (F e b ru a ry , 195&) 3 1 - 3 6 . 5 1 . C o p p le , G eorge E . " E f f e c t i v e I n t e l l i g e n c e a s M easured by a n u n s t r u c t u r e d S e n te n c e -C o m p le tio n T e c h n iq u e ," J o u r n a l o f C o n s u l t i n g p s y c h o lo g y , V o l. 20 , No. 5 ( O c to b e r, 1 9 5 6 ), 35^- 3 6 0 , 5 2 . C o u r t i s , S .A . "The p r e d i c t i o n o f G ro w th ," j o u r : a l o f E d u c a t i o n a l R e s e a r c h , V o l. 26 (M arch, 1943) 4 8 1 -4 9 2 . 5 3 . C r o s s , P a t r i c i a K. an d Eugene L . G a i e r . "T echnique i n P r o b le m - S o lv in g a s a p r e d i c t o r o f E d u c a t i o n a l A c h ie v e m e n t," The j o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 4 6 , No. 4“ ( A p r l l , 1985) 1 9 3 -2 0 6 , 5 4 . D ie d e rlc h , P aul B. " p i t f a l l s i n th e Measurement o f G ains i n A ch iev em en t," The School R eview , ( F e b r u a r y , 1 9 5 6 ), 5 9 -6 3 . 55• D o p p e lt, Jero m e E . an d A le x a n d e r G. Wesman. "The D i f f e r e n t i a l A p t i t u d e T e s ts a s p r e d i c t o r s o f A chievem ent T e s t S c o r e s , " The J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 4 3 , No. 4 ~ 7 A p r il, 1 9 5 2 ), 210-2171 5 6 . _______ . " p ro g re s s i n th e M easurement o f M ental A b i l i t i e s , " E d u c a t i o n a l and p s y c h o l o g i c a l M easurem en t, V o l. 1 4 ,' No. 2 (Summer, 1 9 5 4 ), 2 6 1 -2 6 4 .------ 5 7 . D rew s, E l i z a b e t h Monroe and J o h n E . T ea h an . " p a r e n t a l A t t i t u d e s a n d Academ ic A c h ie v e m e n t." J o u r n a l o f C l i n i c a l p s y c h o lo g y , V o l. 1 3 , No, 4 (O c to b e r, 1 9 5 7 ), 3 2 8 -3 3 1 . 5 8 . D u r r e l l , D. D. " I n f lu e n c e o f R e a d in g on I n t e l l i g e n c e M e a s u r e s ," j o u r n a l of E d u c a t i o n a l p s y c h o lo g y , V o l. 34 , 1 9 4 3 . 1 6 5 . 59* E d m ls to n , R. W. "The R e l a t i o n s h i p o f p e r s i s t e n c e to A c h ie v e m e n t," The J o u r n a l o f E d u c a t i o n a l p sy c h o lo ^ - V o l. 4 0 , n o . 1 ( J a n u a r y , 19*4^), 4 7 - 5 1 . 215 6 0 , F e l n b u r g , H e n ry , "IQ C o r r e l a t e d w i t h EQ, " J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l, 27, No, 8 (November, 1 9 4 1 ), 6 1 7 -6 2 3 , 6 1 , F i e l d i n g , B e n ja m in and F r e d Brown, " p r e d i c t i o n o f I n t e l l i g e n c e fro m C e r t a i n R o rsc h a c h F a c t o r s , " J o u r n a l o f C l i n i c a l p s y c h o lo g y , V o l. 1 2 , N o ,2 ( A p r i l , I9 5 '6 7 ,1 9 6 - 1 9 7 :--------- 6 2 , F i f e r , G ordon. "Grade p la c e m e n t o f S e c o n d a ry Schoo l P u p i l s i n R e l a t i o n t o Age an d A b i l i t y , " "C a l i f o r n i a J o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 3, No. 1 ( J a n u a r y , 1 9 5 2 ), 3 1 -3 5 . 6 3 , F i n d l e y , W arren G. 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" F a c t o r s R e l a t i n g t o th e A cadem ic A c h i e v e m ent o f H ig h -S c h o o l S t u d e n t s , " The J o u r n a l o f E d u c a t i o n a l p s y c h o l o g y , V o l . 40, No. 2 ( F e b r u a r y , 1 9 4 9 ) , 6 5 - 7 8 . __________ , "What D e te rm in e s th e A cadem ic A c h ie v e m e n t o f H igh S c h o o l S t u d e n t s , " j o u r n a l o f E d u c a t i o n a l R e s e a r c h , V o l. 46 , No. 5 ( J a n u a r y , “T 9 5 3 ) , 6 2 1 - ^ 3 l • G ra y , S u s a n W ., "The R e l a t i o n of i n d i v i d u a l v a r i a b i l i t y t o I n t e l l i g e n c e , " The J o u r n a l o f E d u c a t i o n a l p s y c h o lo g y , V o l. 3 5 , No. 4 ( A p r i l , IT 4 4 T , 2 0 1 -5 1 0 . G re e n , C l i n t o n W a l l a c e . 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Asset Metadata

Creator Garlock, Jerry Craig (author)

Core Title An Empirical Study Comparing Various Methods Of Achievement Expectancy Bymeans Of Mental Ability

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Education

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag education, educational psychology,OAI-PMH Harvest

Language English

Advisor Meyers, Charles Edward (committee chair), Carnes, Earl F. (committee member), Guilford, Joy P. (committee member), Lefever, David Welty (committee member), Thorpe, Louis P. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-42815

Unique identifier UC11357825

Identifier 5904389.pdf (filename),usctheses-c18-42815 (legacy record id)

Legacy Identifier 5904389.pdf

Dmrecord 42815

Document Type Dissertation

Rights Garlock, Jerry Craig

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA