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Topic 5 of 5 · Form 6 · Upper 6

Correlation and regression

Show the relationship between two variables: positive, negative or no correlation; linear or non-linear.

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Scatter diagrams

Show the relationship between two variables: positive, negative or no correlation; linear or non-linear.

Product moment correlation coefficient (PMCC)

r = Sxy ÷ √(Sxx Syy), where Sxx = Σx² − (Σx)²/n, Syy = Σy² − (Σy)²/n, Sxy = Σxy − ΣxΣy/n.

  • r ranges from −1 to 1; values near ±1 show strong linear correlation.
  • Correlation does not imply causation.

Spearman's rank correlation coefficient

rs = 1 − 6Σd²/(n(n² − 1)), used for ranked data or non-linear but monotonic relationships.

Worked example: For 5 learners, the sum of squared rank differences is Σd² = 4. Find rs.

rs = 1 − (6 × 4)/(5 × 24) = 1 − 24/120 = 0.8 (strong positive agreement)

Least squares regression

Regression line of y on x: y = a + bx, where b = Sxy/Sxx and a = ȳ − b x̄. Use it to predict y from x.

Worked example: Sxy = 120, Sxx = 60, x̄ = 4, ȳ = 15. Find the regression line.

b = 120/60 = 2; a = 15 − 2 × 4 = 7 → y = 7 + 2x

Reliability of predictions

Interpolation (within the data range) is more reliable than extrapolation (outside it).

Exam tip: State which variable is explanatory (x) and which is response (y) before finding a regression line.

Check yourself

  1. What does r = −0.95 indicate?
  2. Using y = 7 + 2x, predict y when x = 5.
  3. Why is extrapolation unreliable?
Show answers
  1. Strong negative linear correlation.
  2. 17
  3. The relationship may not continue outside the range of the data.
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