Correlation and Regression
What the exam tests
Questions test the direction and strength of linear association, Karl Pearson’s coefficient, Spearman’s rank correlation, and the two regression lines. The exam expects interpretation of (r), use of means and standard deviations, and prediction from a regression equation.
Core rules
- Pearson’s coefficient lies between (-1) and (1). Its sign shows direction; its magnitude shows strength.
- (r=\frac{\sum xy}{n\sigma_x\sigma_y}) when (x,y) are deviations from their means.
- Spearman’s rank correlation: (r_s=1-\frac{6\sum D^2}{n(n^2-1)}), where (D) is the difference of ranks.
- Regression of (y) on (x): (y-\bar y=b_{yx}(x-\bar x)), with (b_{yx}=r\frac{\sigma_y}{\sigma_x}).
- Regression of (x) on (y): (x-\bar x=b_{xy}(y-\bar y)), with (b_{xy}=r\frac{\sigma_x}{\sigma_y}). Also (r^2=b_{yx}b_{xy}).
Worked example
Given (r=0.8), (\sigma_x=2), (\sigma_y=5), (\bar x=10), (\bar y=20), find the regression line of (y) on (x). (b_{yx}=0.8\times\frac52=2). Thus (y-20=2(x-10)), so (y=2x). At (x=12), the predicted value is (y=24).
Traps
Both regression coefficients have the sign of (r). Do not interchange (\sigma_x) and (\sigma_y). The two regression lines are generally different and intersect at ((\bar x,\bar y)). Correlation does not prove causation. In rank correlation, assign tied ranks as averages and check the direction of (D).