Testing Spearman's rank correlation

Hypotheses

Use ρ_s for the population coefficient (examiners require a parameter, not r_s):

• H0: ρ_s = 0 (no correlation)
• H1: ρ_s > 0 (positive), ρ_s < 0 (negative) or ρ_s ≠ 0 (two-tailed).

The direction comes from the question ('agree' or 'higher ... higher' means positive).

Critical values

The table gives one-tailed critical values for n = 4 upwards at 5%, 2.5% and 1%. For a two-tailed test at 5%, use the 2.5% column. Reject H0 if |r_s| is at least the critical value (with the sign matching H1). Do not use the normal distribution; examiners report candidates wrongly using it to find the critical value.

Spearman's critical values, one-tailed (Pearson tables)

Example

Judges, n = 8, r_s = 0.8333. H0: ρ_s = 0, H1: ρ_s > 0 at 5%. Critical value 0.6429. 0.8333 > 0.6429, so reject H0: there is evidence of positive correlation between the judges' rankings, so the judges tend to agree.

The conclusion must use the context (ranks, judges, contestants) and the direction.

Worked example

A sample of 7 pairs gives r_s = -0.75. Test at the 5% level whether there is negative correlation.

1. H0: ρ_s = 0, H1: ρ_s < 0.
2. One-tailed 5% critical value for n = 7: -0.7143.
3. -0.75 < -0.7143: in the critical region.

Answer: Reject H0: there is evidence of negative correlation.

Check yourself

1. Find the critical value for a one-tailed test of ρ_s = 0 at the 5% level with n = 10. Give your answer to 4 decimal places.
   Show answer

   0.5636. From the table: 0.5636.

2. Find the critical value for a two-tailed test of ρ_s = 0 at the 5% level with n = 9. Give your answer to 4 decimal places.
   Show answer

   0.7000. Two-tailed 5% uses the 2.5% column: 0.7000.

3. Find the critical value for a one-tailed test of ρ_s = 0 at the 1% level with n = 7. Give your answer to 4 decimal places.
   Show answer

   0.8929. From the table: 0.8929.

4. Judges: n = 8, r_s = 0.8333, H1: ρ_s > 0 at 5%. Which conclusion earns the mark?
   • Reject H0: there is strong correlation
   • Reject H0: there is evidence of positive correlation between the two judges' rankings
   • Do not reject H0: the judges disagree
   • Reject H0: r_s is greater than 0
   Show answer

   Reject H0: there is evidence of positive correlation between the two judges' rankings. 0.8333 > 0.6429; the conclusion names the context and the direction.

5. n = 9 and r_s = 0.65. Is there evidence at the 5% level of correlation (two-tailed)?
   • Yes, because 0.65 > 0.6000
   • Yes, because 0.65 > 0.5
   • No, because r_s must equal 1
   • No, because 0.65 < 0.7000
   Show answer

   No, because 0.65 < 0.7000. Two-tailed 5% uses the 2.5% column, 0.7000, and 0.65 is less than this.

6. Which hypotheses are correct for testing whether two judges' rankings agree?
   • H0: r_s = 0, H1: r_s > 0
   • H0: ρ_s = 0, H1: ρ_s > 0
   • H0: ρ_s > 0, H1: ρ_s = 0
   • H0: ρ_s = 0, H1: ρ_s ≠ 0
   Show answer

   H0: ρ_s = 0, H1: ρ_s > 0. Agreement means positive correlation; hypotheses use the population parameter ρ_s.

7. A student uses the normal distribution to find a critical value for Spearman's test with n = 8. What is wrong?
   • The student should use χ²
   • Spearman's coefficient cannot be tested
   • Nothing: the normal distribution is required
   • The critical value must come from the Spearman table
   Show answer

   The critical value must come from the Spearman table. Examiners report this error: the critical value for Spearman's test comes from the table of critical values, not the normal distribution.

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) mark scheme, June 2024
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel Mathematical Formulae and Statistical Tables

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