Correlation and Regression
This domain tests the strength and direction of linear association and the use of a fitted line for prediction. You must distinguish correlation from causation, interpret r and r², and keep the explanatory and response variables the right way round.
Core rules
- Pearson correlation r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}; −1 ≤ r ≤ 1.
- r is unit-free and unaffected by changing units or swapping x and y.
- Coefficient of determination r² is the proportion of variation in y explained by the regression on x.
- Regression line: y = a + bx, where b = [nΣxy − ΣxΣy]/[nΣx² − (Σx)²] and a = ȳ − b x̄.
- b is the change in y per unit change in x; the line passes through (x̄, ȳ).
- Spearman rank correlation uses ranks and is suitable for ordinal data or non-linear monotonic relationships.
Worked example
For five pairs: Σx = 15, Σy = 25, Σxy = 88, Σx² = 55, Σy² = 145. b = (5×88 − 15×25)/(5×55 − 15²) = (440 − 375)/(275 − 225) = 65/50 = 1.3. a = 25/5 − 1.3×(15/5) = 5 − 3.9 = 1.1. So y = 1.1 + 1.3x. At x = 4, predicted y = 1.1 + 5.2 = 6.3. The slope means y rises by 1.3 units for each extra unit of x on average.
Traps
- Regressing x on y and using it to predict y.
- Claiming a strong r proves causation.
- Reporting r² as the correlation coefficient.
- Extrapolating beyond the observed x-range.
- Rounding the means before computing a and b.