Correlation AnalysisPurpose of Correlation AnalysisCorrelation CoefficientSample Correlation CoefficientChapter 10Correlation AnalysisJ.C. WangGoal and ObjectiveTo learn about the correlation coefficient as measure ofstrength of the linear association between two variables.OutlineCorrelation AnalysisPurpose of Correlation AnalysisCorrelation CoefficientSample Correlation Coefficientframe title hereThe correlation measures the strength of the linear relationshipbetween numerical variables, for example, the height of menand their shoe size or height and weight. In these situations thegoal is not to use one variable to predict another but to showthe strength of the linear relationship that exists between thetwo numerical variables.Correlation CoefficientdefinitionThe strength of linear association between two numericalvariables in a population is determined by the correlationcoefficient, ρ, whose range is −1 to +1.Linear Relationship as Measured by ρstrong ←− weak no weak −→ strong− − − 0 + + +−1 −.65 −.35 0 .35 .65 +1The sign (positive or negative) of the correlation= the sign of the slope of straight line.Graphical Examplesr = sample correlation●●●●r == 14 8 12 16y●●●●r == −1●● ●●r == 0.874 8 12 16y2 3 4 5 6 7 8x●● ●●r == −0.872 3 4 5 6 7 8xGraphical Examplescontinued● ● ● ●r == 04 8 12 16y●● ●●●r == 02 3 4 5 6 7 8x●●●●●●●●●●●●●●●●●●●●●●●●r == 04 8 12 16y2 3 4 5 6 7 8xSample Correlation CoefficientSince our interest is the regression analysis, the samplecorrelation coefficient (r) is derived from the coefficient ofdetermination (R2, to be discussed in Chapter 11).R2=regressionSumOfSquarestotalSumOfSquares=SSRSSTThe correlation coefficient r = ±√R2where r takes the sign ofthe slope.Sample Correlation CoefficientcontinuedCorrelation coefficient can also be calculated byr =P(xi− x)(yj− y)p(xi− x)2q(yj− y)2Exampler = 0.9526Datax −2 2 5 −1 6y 0 3 10 1 15STAT → EDIT L1and L2STAT → TESTS ↓ LinRegTTestiClicker Question 10.1iClicker Question