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Pearson Correlation Coefficeint Symbol: r r can range from -1.0 to +1.0 Sign (+/-) indicates direction Value indicates strength Measures a linear relationship only (a) Direction of relationship between X, Y Positive (+r) = As X goes up, Y goes up Negative (-r) = As X goes up, Y goes down (b) Strength of a relationship between X, Y Closer to ( 1.0, stronger Closer to 0, weaker when r = 0 ( X,Y relationship not defined by a straight line   Pearson Correlation Coefficient -1.0 0 +1.0  Closer to 0 = weaker Closer to (1.0 = stronger r close to (1.0 very rare in social science r ( ( .30 considered important r ( 0, no linear relationship between X & Y    What does r represent? r = degree to which X & Y vary together degree to which X & Y vary separately r = covariance of X & Y variance of X & Y Definitional Formula for Pearson r: r = EMBED Equation.3 SP = Sum of Products SS = Sum of Squared Deviations SP = ((X- EMBED Equation.3 )(Y- EMBED Equation.3 ) SSX=((X- EMBED Equation.3 )2 SSy=((Y- EMBED Equation.3 )2 Variance interpretation of r : r 2 = % of variance in Y explained by its linear relationship with X (and vice versa) r 2 = Coefficient of determination % of shared variance between X & Y % of variance in Y predicted by X III. Factors that affect the size of r r ( 0 could mean many things: No relationship at all between X & Y Non-linear relationship between X & Y Restricted range on X and/or Y Outlier may be causing problem Non-linear relationships Curvilinear relationship Restricted range Low variability on X and/or Y Outliers Extreme value on X and/or Y Examples of how restricted range can distort a correlation   Example of how an outlier can distort a correlation  IV. Correlation vs. Causality: Correlation tells you two variables are related Does NOT tell you why!! Do not draw causal inferences from a correlation X ( Y  Y ( X  example: r = -.30 #friends, depression r = +.40 hours studying, grades Causal inferences require an experiment V. Other Correlation Coefficients Pearson r used when X & Y are at least interval level Many types of correlation coefficients for other data Spearman ( ordinal (rank) data Point-biserial ( nominal X, interval/ratio Y Phi ( nominal X & Y     PAGE 7 Y (Score) X (Hours Studying) No Linear Relationship Perfect Positive Relationship Perfect Negative Relationship X Y Z Third variable problem Figure 16-3 (p. 524) Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature. Figure 16-5 (p. 525) Examples of different values for linear correlations: (a) shows a strong positive relationship, approx +.90; (b) shows a relatively weak negative correlation, approx .40; (c) shows a perfect negative correlation, correlation = 1.0; (d) shows no linear trend, correlation = 0.0. r = .90 r = -.40 r = -1.0 r = .00 (a) In this example, the full range of X and Y values shows a strong, positive correlation, but the restricted range of scores produces a correlation near zero. (b) An example in which the full range of X and Y values shows a correlation near zero, but the scores in the restricted range produce a strong, positive correlation. 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