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Pearson-Product Moment Correlation Coefficient (r) COVxy A measure of the relation between x and y, but is not standardized To standardize , we divide the covariance by the size of the standard deviations. Given that the maximum value of the covariance is plus or minus the product of the variance of x and the variance of y, it follows that the limits on the correlation coefficient are + or – 1.0 r COVxy sx s y Example: X (x x) Y 3 4 9 4 4 1 10 1 5 0 12 1 6 1 11 0 7 4 13 4 xx 5 sx 158 . COVxy 2.25 ( y y) x y 11 sy 158 . r 2.25 (158 . )(158 . ) r 0.9 COVxy r sx s y Compute the regression coefficient, but using standardized scores. X’ Y’ -1.27 -1.27 -.066 -.66 0 .66 .66 0 1.27 1.27 b= Why? Adjusted r (1 r 2 )( N 1) radj 1 ( N 2) From our example: radj 1 = .75 (1.81)(5 1) (5 2) r 2 = that proportion of the variance in y that is shared (accounted for) by x. Sometimes called the “coefficient of determination.” Thus, r = .9 and r2 = .81 Or x accounts for 81% of the variance in y. 2 r 2 R = .4, thus r R = .2, thus = .04 or 4% = .016 or 16% If on r is g times as large as a second r, then the proportion of the variance associated with the first r will be g(squared) times as great as that associated with the second. r 2 can also be misleading Factors Affecting r Range Restrictions Outliers Heterogeneous Subsamples Whole-Part correlations. This is were the score for variable x contributes to the score of variable y. Produces a + bias in r. Again, Correlation does not imply causality. Variables may be accidentally related, or both may be related to a third variable, or the may influence each other. Which is more informative, the slope of the regression line or the correlation coefficient?