ZIMSEC A Level · Statistics · Form 6
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).
Check yourself
- What does r = −0.95 indicate?
- Using y = 7 + 2x, predict y when x = 5.
- Why is extrapolation unreliable?
Show answers
- Strong negative linear correlation.
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- The relationship may not continue outside the range of the data.
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