Correlation and Regression
What you must understand
Correlation describes the strength and direction of a linear relationship between two variables. On a scatter diagram, look for positive (both increase), negative (one increases as the other falls) or no correlation. The product-moment correlation coefficient r lies between −1 and 1; values near ±1 indicate strong linear association, values near 0 weak or no linear association.
Regression goes further: the least-squares regression line of y on x has equation y = a + bx, where b = Sxy ÷ Sxx and a = ȳ − bx̄. Use it to predict y for a given x. Understand that regression describes association, not causation, and that predictions outside the data range (extrapolation) are unreliable.
Know the meaning of explanatory (independent) and response (dependent) variables, and the coefficient of determination r², the proportion of variation in y explained by x.
Traps that appear in questions
- Confusing correlation with causation: ice-cream sales correlate with drowning, but one does not cause the other.
- Swapping x and y: the regression line of y on x is not the same as x on y.
- Reading r = 0 as "no relationship" - there may be a strong curved relationship.
- Predicting far beyond the observed range of x.
- Forgetting to label axes and give the equation in context.
Worked example
For data with x̄ = 5, ȳ = 20, Sxy = 30 and Sxx = 12, find the regression line of y on x.
Answer: b = 30 ÷ 12 = 2.5; a = 20 − 2.5 × 5 = 7.5. So y = 7.5 + 2.5x. When x = 6, predicted y = 7.5 + 15 = 22.5. State clearly that this is a prediction, not a guaranteed value.
Rule of thumb: a strong r means points lie close to a straight line - it says nothing about which variable causes the other.