Correlation Correlation Coefficient n n n aka Pearson

Correlation

Correlation Coefficient n n n 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 N 1 Ht. In. 60 Wt. Lbs. 102 2 62 120 3 63 130 4 65 150 5 65 120 6 68 145 7 69 175 8 70 170 9 72 185 10 74 210 When one variable increases, the other also increases. Example of a Positive Correlation

Negative Correlation N Study # Time Error Minutes 1 2 3 4 5 6 7 8 9 10 90 100 130 150 180 200 220 300 350 400 25 28 20 20 15 12 13 10 8 6 When one variable increases, the other decreases. Example of a Negative Correlation

Zero Correlation N SATV 1 450 Toe Size 1. 7 2 480 1. 8 3 500 1. 6 4 510 1. 8 5 520 1. 9 6 550 1. 7 7 600 1. 6 8 630 1. 7 9 650 1. 9 10 700 1. 7 No relation*: when one variable increases, the other variable neither increases nor decreases. Example of a Zero Correlation (*Strictly speaking, no linear relation. )

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) Cars sold $ 10 1000 2 15 1500 3 20 2000 4 25 2500 5 30 3000 6 35 3500 7 40 4000 8 45 4500 N 1 Notice the straight line. When r=+1 or -1, all the points will fall on a line.

Example Correlations Variable X Variable Y Salary Taxes paid Shyness Price of car # people greeted at party Prestige of car Price of quartz watch Income of sales people Anxiety Accuracy of time kept Number of cars sold Memory 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.

-1. 58 = (60 -66. 8)/4. 31 -1. 51=(102 -150. 7)/32. 2 Computational Example N 1 2 3 4 5 6 7 8 9 10 Mean S [N] (SD[N-1]) Ht 60 62 63 65 65 68 69 70 72 74 66. 8 4. 31 Wt 102 120 130 150 120 145 170 185 210 150. 7 32. 20 (4. 54) (33. 95) Zht Zwt Zh*Zw -1. 58 -1. 51 2. 39 -1. 11 -0. 95 1. 06 -0. 88 -0. 64 0. 57 -0. 42 -0. 02 0. 01 -0. 42 -0. 95 0. 40 0. 28 -0. 18 -0. 05 0. 51 0. 75 0. 39 0. 74 0. 60 0. 45 1. 21 1. 06 1. 29 1. 67 1. 84 3. 08 0 1 0. 96

Some examples X Y correlation Extroversion Job Performance of Managers . 10 Corporate social responsibility Corporate financial performance . 20 Job performance –supervisor report – peer report . 30 High School SAT College GPA (year 1) . 60 Weight-self report Weight-measured. 90

Review n n n 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 Which figure shows the most positive correlation? A, B, C, D? D

Test Questions n Which of the four choices below shows the correlation with the strongest association between X and Y? a. -. 50 b. . 0 c. . 25 d. 1. 02