Introduction to Statistics inPsychologyPSY 201Professor Greg FrancisLecture 11correlationIs there a relationship between IQand problem solving ability?CORRELATIONsuppose you get r ≈ 0.Does that mean there is no relationbetween the data sets?many aspects of the data may affectthe value of r• Linearity of data.• Homogeneity of group.• Size of group.• Restricted range.2LINEARITYr is partly an index of how well astraight line fits the data setr =0.903400 450 500 550 600 650 700Quantitative SAT Score405060Final Examination Score3NONLINEARITYwhen data points do n’ t fall along asingle line (nonlinear data)r =0.05400 450 500 550 600 650 700Quantitative SAT Score30405060Final Examination Score4NONLINEARITYthere are lots of types of nonlinearitiescurvilinear relationshipr =0.131400 450 500 550 600 650 700Anxiety Measure3040506070Final Examination Score5NONLINEARITYIt can get complicatedr = −0.200 5 10 15X020406080100Yr = −0.910 5 10 15X-50005001000Y6BOTTOM LINEPearson r is an index of a linearrelationship between variablesif another (nonlinear) relationshipexists, r might not notice itPearson r measures only simplerelationships between variablesif r is small, you might want to plot ascattergram to look at the data tonotice if other relationships exist7HOMOGENEITYsuppose you get r ≈ 0, and you cannotdetect any type of nonlinearrelationshipDoes this mean there is no relationshipbetween the variables?Not necessarily, it may be that thedata does not have enough variation initCorrelation measures how variable Xchanges with variable Yif one doesn’t change much, therewon’t be a strong correlation8HOMOGENEITYconsider the covariance formularxy=sxysxsywhere, covariance issxy=Σ(X − X)(Y − Y )n − 1if there is little change in Y from Y ,sxyis going to be small because +/−variations in X − X will be weightedby small values of Y − Ysimilarly, syis going to be small, so wedivide a small number by a smallnumber9HOMOGENEITYintutivelyr =degree to which X and Y vary togetherdegree to which X and Y vary separatelyif one of those variables (or both) isnot varying much at all, r will be smallyou need enough variability acrossboth sets of scores to adequatelymeasure correlation10HOMOGENEITYthe effects of homogeneity can besubtlerelationship between SAT scores andFinal exam grader =0.92400 450 500 550 600 650 700Quantitative SAT Scores38404244Final Exam Grade11HOMOGENEITYsuppose we looked at the relationshipamong only the best students(those with final exam scores above 44)r =0.62400 450 500 550 600 650 700Quantitative SAT Scores38404244Final Exam Grade12HOMOGENEITYor worst students(those with final exam scores below 40)r =0.62400 450 500 550 600 650 700Quantitative SAT Scores38404244Final Exam Gradecorrelation drops!13SIGNIFICANCEif you have r ≈ 0, it may be becausethere is not enough variation in yourdata sete.g.IQ and problem solving is probablyunrelated among a group of geniusesIQ and problem solving is probablyunrelated among a group of idiotsIQ and problem solving is probablystrongly related among a mix ofgeniuses, idiots, and normals14SIZE OF GROUPsuppose you have only two data pointsyou can always draw a straight lineconnecting themwhich implies perfect correlationr = −1.022 24 26 28 30 32 34X204060Y(correlation doesn’t tell us anythinguseful!)15SIZE OF GROUPif you have enough data points forcorrelation to be meaningful (> 2),and you have enough variation in thedata, thensize of group is not important indetermining the value of rwe will see later that it is important indetermining the accur acy of therelationship (hypothesis testing)16RESTRICTED RANGEif you sample data from a limitedrange you may not be able to trust thecorrelation values in generale.g., suppose you want to studyrelationships between IQ and creativityif you sample college students you willprobably get IQ’s between 110 and 140perhaps you find a strong correlation,e.g.r =0.78110 115 120 125 130 135 140IQ253035Creativity measure17RESTRICTED RANGEif you sample from the generalpopulation (not just college students)you would get a larger range of IQsyou may find a much weakercorrelation, e.g.r =0.1280 90 100 110 120 130 140IQ2025303540Creativity measure18RESTRICTED RANGEof course, it could be that you fail tofind a large r over a restricted range,but a larger range finds a large r (thisis slightly different from the issue ofhomogeneity)in generala correlation measure applies only tothe range of values used to compute ityou cannot extend the correlationvalue to other ranges19INTERPRETATION OF rif we calculate a value of rHow do we know what it means?How do we compare r values fordifferent data sets?Rule of thumb|r| Interpretation0.9 to 1.0 Very high correlation0.7 to 0.9 High correlation0.5 to 0.7 Moderate correlation0.3 to 0.5 Low correlation0.0 to 0.3 Little if any correlation20SCALE OF rvalues of r are ordinal measures ofcorrelation• higher r values indicate larger corre-lation• equal spacings of r values may notindicate equal spacings of correlationthus, r =0.90 is not twice ascorrelated as r =0.45the difference in correlation betweenr =0.90 and r =0.75 is not the sameas the difference in correlation betweenr =0.60 and r =0.45.21CONCLUSIONSPearson rsizeinterpretation22NEXT TIMEcoefficient of determinationcausationSpearman’s rhoDoes TV make you