Spearman's rank correlation coefficient

The formula

Rank each variable separately (1 for the smallest or for the best, but the same way for both). Find d, the difference in ranks for each item, then

r_s = 1 - 6Σd² / (n(n² - 1)).

It lies between -1 and 1: r_s = 1 means identical rankings, r_s = -1 exactly reversed rankings. Examiners report candidates forgetting to subtract from 1.

Ranks given by two judges

Example

For the judges: d = -1, 1, -1, 1, -1, -2, 2, 1, so Σd² = 14.0. r_s = 1 - 6 × 14.0/(8 × 63) = 1 - 84/504 = 0.8333: strong agreement between the judges.

When to use Spearman rather than the PMCC

Use Spearman's when:

• the data are given as ranks or are subjective judgements;
• the relationship looks non-linear but monotonic (always increasing or always decreasing);
• the data are not from a bivariate normal distribution (so the PMCC test is not valid).

The PMCC measures linear correlation of the actual values; Spearman's measures agreement of the ranks.

Tied ranks

Exam questions are not set with ties, but you should know the method: tied values share the mean of the ranks they would have occupied. Values 12, 15, 15, 18 get ranks 1, 2.5, 2.5, 4. With ties, the formula for r_s is only approximate; the exact method is to calculate the PMCC of the ranks.

Worked example

Find Spearman's rank correlation coefficient for x: 3.2, 4.1, 2.8, 5.0, 4.6, 3.9 and y: 18, 21, 15, 20, 24, 17.

1. Ranks of x: 2, 4, 1, 6, 5, 3
2. Ranks of y: 3, 5, 1, 4, 6, 2
3. Σd² = 8
4. r_s = 1 - 6 × 8/(6 × 35)

Answer: r_s = 0.7714

Check yourself

1. Using the two judges' rankings, find Σd².
   Show answer

   14. d = -1, 1, -1, 1, -1, -2, 2, 1; squares total 14.

2. Using the two judges' rankings, find Spearman's rank correlation coefficient. Give your answer to 3 significant figures.
   Show answer

   0.833. 1 - 84/504 = 0.8333.

3. For 10 pairs of ranks, Σd² = 42. Find r_s. Give your answer to 3 significant figures.
   Show answer

   0.745. 1 - 252/990 = 0.7455.

4. Find r_s for x: 3.2, 4.1, 2.8, 5.0, 4.6, 3.9 and y: 18, 21, 15, 20, 24, 17. Give your answer to 3 significant figures.
   Show answer

   0.771. Ranks give Σd² = 8, so r_s = 1 - 48/210 = 0.7714.

5. The values 12, 15, 15, 18 are ranked from smallest. What rank is given to each 15?
   Show answer

   2.5. The two 15s would occupy ranks 2 and 3, so each gets (2 + 3)/2 = 2.5.

6. Five items are ranked by two judges in exactly opposite orders. Find Σd².
   Show answer

   40. d = -4, -2, 0, 2, 4, so Σd² = 16 + 4 + 0 + 4 + 16 = 40, giving r_s = 1 - 240/120 = -1.

7. A scatter diagram shows y increasing with x but along a curve, not a straight line. Which coefficient is more suitable?
   • The PMCC, because it uses the actual values
   • The PMCC, because the data are not ranks
   • Neither, because there is no correlation
   • Spearman's rank, because the relationship is monotonic but not linear
   Show answer

   Spearman's rank, because the relationship is monotonic but not linear. Spearman's measures any always-increasing relationship; the PMCC only measures linear association.

8. What does r_s = -1 tell you?
   • There is no relationship between the rankings
   • The rankings are identical
   • The two rankings are in exactly the reverse order
   • The data lie on a straight line with negative gradient
   Show answer

   The two rankings are in exactly the reverse order. Perfect disagreement: the highest on one is lowest on the other.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel IAL Statistics S3 (WST03) mark scheme, June 2024
• Pearson Edexcel Mathematical Formulae and Statistical Tables

Last updated 8 November 2026. Spotted a mistake? Tell us.