Correlation and Regression
What the domain covers
This domain tests scatter diagrams, Karl Pearson's correlation coefficient, Spearman's rank correlation, regression lines and regression coefficients. You must distinguish association from prediction and know which variable is dependent.
Core rules
- Karl Pearson r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)²Σ(y − ȳ)²]; equivalently r = cov(x,y)/(σxσy). It lies between −1 and 1 and is unit-free.
- Spearman's rank correlation: R = 1 − [6Σd²]/[n(n² − 1)], with d = rank difference; adjust for ties.
- Regression of y on x: ŷ = a + bx, with b_yx = r(σy/σx). Regression of x on y: x̂ = c + dy, with b_xy = r(σx/σy).
- The two regression lines intersect at (x̄, ȳ); b_yx × b_xy = r².
- r measures linear association only; regression predicts one variable from another.
Worked example
Given r = 0.8, σx = 2 and σy = 5, the regression coefficient of y on x is b_yx = 0.8 × (5/2) = 2.0. The coefficient of x on y is b_xy = 0.8 × (2/5) = 0.32. Their product is 0.64 = r², a useful check. If x rises by 1 unit, predicted y rises by about 2 units.
Traps
Do not use the y-on-x line to predict x. A high r does not imply causation. r = 0 means no linear relation, not necessarily independence. Keep the ratio σy/σx the right way up; reversing it gives the other regression coefficient. Rank correlation needs ranks, not original values.