Correlation

Correlation Coefficient aka Pearson Product-Moment Correlation Coefficient. Correlation coefficient summarizes the relations b/t 2 variables, both direction and degree (closeness). Scattergram summary. Sample r; population  (Greek rho). Can take values from –1 thru 0 to +1. Sign tells direction (+ or -); magnitude or value tells closeness or degree.

Positive Correlation NHt. In. Wt. Lbs Example of a Positive Correlation When one variable increases, the other also increases.

Negative Correlation N Study Time Minutes # Error Example of a Negative Correlation When one variable increases, the other decreases.

Zero Correlation N SAT- V Toe Size Example of a Zero Correlation (*Strictly speaking, no linear relation.) No relation*: when one variable increases, the other variable neither increases nor decreases.

Correlation Sign & Magnitude The sign indicates the direction of the relationship. If positive, they increase together. If negative, when one goes up the other goes down. The absolute value tells the strength of the relationship. Values close to +1 and –1 indicate very strong relations. As r  0, the relationship is weaker. If r=0, no relation. If |r|=1, perfect relation.

Perfect Positive Correlation (r = 1) N Cars sold $ Notice the straight line. When r=+1 or -1, all the points will fall on a line.

Example Correlations Variable XVariable Y SalaryTaxes paid Shyness# people greeted at party Price of carPrestige of car Price of quartz watch Accuracy of time kept Income of sales people Number of cars sold AnxietyMemory exam Correlation

Computing the Correlation The definition: The correlation coefficient, r, is the average cross-product of z scores. z X is X in z-score form, z Y is Y in z-score form and we multiply the two. We add them all and divide by N to get the average.

Computational Example NHtWtZ ht Z wt Z h *Z w Mean S [N] (SD[N-1]) 4.31 (4.54) (33.95) = ( )/ =( )/32.2

Some examples XYcorrelation ExtroversionJob Performance of Managers.10 Corporate social responsibility Corporate financial performance.20 Job performance –supervisor report Job performance – peer report.30 High School SATCollege GPA (year 1).60 Weight-self reportWeight-measured.90

Review What are the maximum and minimum values of r? What does r summarize? What is the symbol for the population value of r? How does r show direction of relations? How does r show magnitude of relations?

Test Questions A B C D Which figure shows the most positive correlation? A, B, C, D?

Test Questions Which of the four choices below shows the correlation with the strongest association between X and Y? a b..0 c..25 d. 1.02