CORRELATION. two variables may be related. SAT scores, GPA hours in therapy, self-esteem grade on homeworks, grade on exams

Size: px

Start display at page:

Download "CORRELATION. two variables may be related. SAT scores, GPA hours in therapy, self-esteem grade on homeworks, grade on exams"

Zoe Martin
5 years ago
Views:

1 Inrodcion o Saisics in sychology SY 1 rofessor Greg Francis Lecre 1 correlaion Did I damage my dagher s eyes? CORRELATION wo ariables may be relaed SAT scores, GA hors in herapy, self-eseem grade on homeworks, grade on exams nmber of risk facors, probabiliy of geing AIDS heigh, poins in baskeball... how do we show he relaionship? scaergrams SCATTERGRAMS plo ale of one ariable agains he ale of he oher ariable Semeser hors of mahemaics Academic abiliy 3 RELATIONSHIS Idenifying hese ypes of relaionships is one of he key isses in saisical analysis Consider a 1999 sdy ha repored a relaionship beween he se of nighlighs in a child s room and he endency of he child o need glasses My dagher slep wih a nighligh? COMLICATIONS Clearly here is a relaionship beween sing a nighligh and needing glasses Howeer, i s no clear wha he nare of he relaionship inoles I cold be ha he exra ligh somehow inflences he child s eyes and cases he need for glasses Or i cold be ha needing glasses will somehow co-occr wih he se of a nighligh (e.g., children who need glasses will wan a nigh ligh, or heir parens will wan a nighligh) Finding a relaionship is necessary for esablishing casaion, b i is no enogh OSITIVE CORRELATION Firs, we need o ndersand how o qanify he exisence of a relaionship. Increases in he ale of one ariable end o occr wih increases in he ale of he oher ariable SAT scores and exam scores Final Exam Score 6 4 Sden Sde Sden SAT Score 4 6

2 NEGATIVE CORRELATION Increases in he ale of one ariable end o occr wih decreases in he ale of he oher ariable emperare and nmber of people wih frosbie ERFECT CORRELATIONS perfec posiie correlaion NO CORRELATION no correlaion balance of larger and smaller ales Nmber of frosbie cases Degrees (Fahrenhei) perfec negaie correlaion CORRELATION COEFFICIENT qaniaie measre of correlaion bonded beween 1. & +1. correlaion coe cien of -1. indicaes perfec negaie correlaion correlaion coe cien of +1. indicaes perfec posiie correlaion correlaion coe cien of. indicaes no correlaion ales in beween gie ordinal measres of relaionship earson prodc-momen correlaion coe cien one correlaion coe cien for qaniaie daa (he mos imporan one) degree o which X and Y ary ogeher r = degree o which X and Y ary separaely 1. z-scores. Deiaion scores 3. Raw scores 4. Coariance seeral formlas all gie he same resl! z SCORES Two seps 1. Coner raw scores ino z scores. Find he mean of cross-prodcs

3 z SCORES wha does his calclaion do? sppose yo hae wo disribions ha hae a posiie correlaion hen a large ale of X will be aboe X and hae a posiie z x score and a corresponding Y will be aboe Y and hae a posiie z y score Ths he cross-prodc will be posiie also a small ale of X will be below X and hae a negaie z x score and he corresponding Y will be below Y and hae a negaie z y score Ths will again be posiie o find he aerage, sm all he prodcs (posiie nmbers) we diide by sill a posiie nmber! exacly he opposie is re for negaiely correlaed disribions hen a large ale of X will be aboe X and hae a posiie z x score and a corresponding Y will be below Y and hae a negaie z y score Ths will be negaie 1 while a small ale of X will be below X and hae a negaie z x score and he corresponding Y will be aboe Y and hae a posiie z y score Ths will again be negaie o find he aerage, sm all he prodcs (negaie nmbers) we diide by sill a negaie nmber! DEVIATION FORMULA i is awkward o coner o z scores we can ge he same nmber wih deiaion scores x = X y = Y X Y deiaion score formla r xy = xy s x y RAW SCORE FORMULA i is awkward o calclae deiaion scores r xy = raw score formla n XY X Y 4 n X ( X) 3 4 n Y ( Y )

4 COVARIANCE FORMULA coariance = s xy = (X X)(Y Y ) aerage cross-prodc of deiaion scores (similar o ariance) earson r rns o o be: r xy = s xy s x s y where s x and s y are he sandard deiaions of heir respecie disribions Final Exam Score 6 4 Sden X Y Sde Sden 8 sandard score formla = 1.67 =.9 X Y z xz y X =81 Y =77 z x =. z y =. z xz y = SAT Score 19 1 deiaion score formla xy r xy = p x y = 3. r =.93 (46.)(9.7) X Y x y xy X =81 Y =77 x =. y =. xy =3. x =46. and y =9.7 raw score formla n XY X Y rxy = r hn X ( X) ih n Y ( Y ) i (1)(448) (81)(77) q [(1)(4478) (81) ][(1)(4116) (77) ] =.93 X Y XY X =81 Y =77 XY =448 X =4478and Y = coariance formla r xy = s xy = s x s y (96.3)(1.11) =.93 where, s xy = xy = 3 =88.86 s x = s y = x = y = 46 = =1.11 4

5 CORRELATION r measres correlaion beween wo ariables no js any wo ariables 1. The wo ariables ms be paired obseraions.. Variables ms be qaniaie (ineral or raio scale). CONCLUSIONS correlaion scaergrams earson r formlas NEXT TIME facors a ecing r inerpreing r and Is here a link beween IQ and problem soling abiliy? 6 7

CORRELATION. two variables may be related. SAT scores, GPA hours in therapy, self-esteem grade on homeworks, grade on exams

CORRELATION. two variables may be related. SAT scores, GPA hours in therapy, self-esteem grade on homeworks, grade on exams Inrodcion o Saisics in sychology SY 1 rofessor Greg Francis Lecre 1 correlaion How changes in one ariable correspond o change in anoher ariable. wo ariables may be relaed SAT scores, GA hors in herapy,