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How do Spearman's rank and the PMCC measure correlation, and how do they differ?

Calculating and interpreting Spearman's rank correlation coefficient; interpreting Pearson's product moment correlation coefficient; the distinction between rank correlation and product moment correlation.

A focused answer to Edexcel GCSE Statistics (Higher tier) on correlation coefficients, covering calculating and interpreting Spearman's rank correlation coefficient, interpreting Pearson's product moment correlation coefficient, and the distinction between rank correlation and linear product moment correlation.

Generated by Claude Opus 4.810 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this dot point is asking
  2. Spearman's rank correlation coefficient
  3. Interpreting Spearman's rank
  4. Interpreting the PMCC
  5. Distinguishing Spearman and PMCC

What this dot point is asking

Edexcel Higher tier codes 2e.05 to 2e.08 require you to calculate Spearman's rank correlation coefficient (the formula is on the formulae sheet), interpret both Spearman's rank and Pearson's product moment correlation coefficient (PMCC), and understand the distinction between them. The PMCC itself is not calculated at GCSE, only interpreted. Tied ranks are not tested, and a scientific calculator's functions are sufficient.

Spearman's rank correlation coefficient

You rank each variable separately (usually 11 for the highest or lowest, applied consistently), find the difference dd between the two ranks for each item, then apply the formula.

The steps are always the same: rank both variables, compute each dd, square and sum to get ∑d2\sum d^2, then substitute with nn.

Interpreting Spearman's rank

The sign and size of rsr_s tell the story:

  • rsr_s close to +1+1: strong agreement, the rankings rise together.
  • rsr_s close to −1-1: strong disagreement, one ranking reverses the other.
  • rsr_s near 00: little or no agreement.

Edexcel does not require a formal scale of "strong" versus "weak"; values closer to the limits indicate stronger correlation. Always interpret in context (for example "the two judges largely agree on the order of the cakes").

Interpreting the PMCC

At GCSE you interpret a given PMCC but do not calculate it. A PMCC of +1+1 means the points lie exactly on a rising straight line; 00 means no linear relationship (though a non-linear one could still exist).

Distinguishing Spearman and PMCC

The two coefficients answer different questions:

  • Spearman's rank measures whether the variables move together in order (monotonic agreement), so it works even when the relationship is curved.
  • The PMCC measures how close the points lie to a straight line (linear correlation only).

A revealing comparison: if there is a positive but non-linear relationship, both coefficients are positive, but Spearman's is greater than the PMCC, because the ranks agree well even though the points are not on a straight line. Recognising this difference is the core of the highest-mark questions in this topic.

Exam-style practice questions

Practice questions written in the style of Pearson Edexcel exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Edexcel 1ST0 20215 marksTwo judges rank 55 cakes. The differences in their ranks are d=1,−1,2,0,−2d = 1, -1, 2, 0, -2. Using rs=1−6∑d2n(n2−1)r_s = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}, calculate Spearman's rank correlation coefficient and interpret the result.
Show worked answer →

∑d2=12+(−1)2+22+02+(−2)2=1+1+4+0+4=10\sum d^2 = 1^2 + (-1)^2 + 2^2 + 0^2 + (-2)^2 = 1 + 1 + 4 + 0 + 4 = 10.

With n=5n = 5: n(n2−1)=5(25−1)=5×24=120n(n^2 - 1) = 5(25 - 1) = 5 \times 24 = 120.

rs=1−6×10120=1−60120=1−0.5=0.5r_s = 1 - \frac{6 \times 10}{120} = 1 - \frac{60}{120} = 1 - 0.5 = 0.5.

The value 0.50.5 is positive but only moderate, so the two judges show some agreement in their rankings, but the agreement is far from perfect.

Markers reward ∑d2=10\sum d^2 = 10, correct substitution, rs=0.5r_s = 0.5, and an interpretation of positive but moderate agreement.

Edexcel 1ST0 20223 marksFor a set of bivariate data the Spearman's rank correlation coefficient is 0.90.9 and the Pearson's product moment correlation coefficient (PMCC) is 0.60.6. (a) Interpret each value. (b) What does the difference between the two coefficients suggest about the relationship?
Show worked answer →

(a) rs=0.9r_s = 0.9 indicates strong positive rank correlation: as one variable's rank increases, so does the other's. PMCC =0.6= 0.6 indicates moderate positive linear correlation.

(b) Spearman's coefficient is higher than the PMCC, which suggests the relationship is positive but not linear: the variables increase together (high rank correlation), but the points do not lie close to a straight line (lower linear correlation), so the relationship is likely curved.

Markers reward interpreting each coefficient and recognising that a higher Spearman than PMCC indicates a positive but non-linear relationship.

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