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Spearman's rank correlation test
Edexcel IAL Psychology · WPS02 Biological Psychology, Learning Theories and Development
Spearman's rank (rho, rs) is the test of relationship named for Topic C. It tells you whether a correlation is significant (spec 3.2.4 and List B).
When to use Spearman's rank
Choose Spearman's rank when:
- the hypothesis predicts a relationship (a correlation), and
- the data are at least ordinal (can be ranked).
Levels of measurement: nominal (categories, e.g. yes/no), ordinal (ranked, e.g. ratings out of 10), interval/ratio (equal units, e.g. hours). Spearman's works on ordinal data or above, so it is a non-parametric test.
Calculating rs
1. Rank each co-variable separately (lowest = 1). Tied scores share the average rank.
2. Find d = rank of A minus rank of B for each participant.
3. Square each d and add them: Σd².
4. Use rs = 1 - (6Σd²) / (N(N² - 1)), where N is the number of pairs.
The formula and critical value tables are given in the exam, and you can use a calculator.
Significance
Compare the observed rs (ignoring its sign) with the critical value for your N, level of significance and one- or two-tailed test. For Spearman's: the observed value must be equal to or greater than the critical value to be significant.
- Levels of significance: p ≤ .05 is the usual level (a 5% or lower probability the result is due to chance); p ≤ .10 is less strict and p ≤ .01 stricter.
- One-tailed for a directional hypothesis; two-tailed for non-directional.
- Type I error: rejecting the null hypothesis when it is true (false positive), more likely with a lenient level like p ≤ .10.
- Type II error: keeping the null hypothesis when it is false (false negative), more likely with a strict level like p ≤ .01.
| N | One-tailed p ≤ .05 | Two-tailed p ≤ .05 | Two-tailed p ≤ .01 |
|---|---|---|---|
| 8 | 0.643 | 0.738 | 0.881 |
| 10 | 0.564 | 0.648 | 0.794 |
| 12 | 0.503 | 0.587 | 0.727 |
| 15 | 0.446 | 0.521 | 0.654 |
Extract from Pearson's Spearman's rank critical values table
Worked data
Ten students: hours of screen time before bed (A) and hours of sleep (B).
| P | A | Rank A | B | Rank B | d | d² |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 8.2 | 9 | -7 | 49 |
| 2 | 2.5 | 5 | 7.1 | 6 | -1 | 1 |
| 3 | 0.5 | 1 | 8.6 | 10 | -9 | 81 |
| 4 | 4 | 8 | 6 | 2 | 6 | 36 |
| 5 | 3 | 6 | 7.4 | 7 | -1 | 1 |
| 6 | 1.5 | 3 | 7.9 | 8 | -5 | 25 |
| 7 | 5 | 10 | 5.8 | 1 | 9 | 81 |
| 8 | 2 | 4 | 7 | 5 | -1 | 1 |
| 9 | 3.5 | 7 | 6.9 | 4 | 3 | 9 |
| 10 | 4.5 | 9 | 6.3 | 3 | 6 | 36 |
| Σd² | 320 |
Worked example
Use the worked data (N = 10) to calculate rs and decide if it is significant for a two-tailed test at p ≤ .05.
- Σd² = 320.
- rs = 1 - (6 x 320) / (10 x 99) = 1 - 1920 / 990 = -0.939.
- Critical value (N = 10, two-tailed, p ≤ .05) = 0.648.
- 0.939 ≥ 0.648, so the result is significant: reject the null hypothesis.
Answer: rs = -0.939: a strong, significant negative correlation.
Check yourself
- N = 12 and Σd² = 80. Calculate rs to 3 decimal places.
Show answer
0.720. rs = 1 - 480 / (12 x 143) = 1 - 480 / 1716 = 0.720.
- For N = 12, rs = 0.720, two-tailed test, p ≤ .05 (critical value 0.587). What do you conclude?
- Significant: reject the null hypothesis
- Not significant: keep the null hypothesis
- Significant only at p ≤ .10
- The test cannot be used with N = 12
Show answer
Significant: reject the null hypothesis. 0.720 ≥ 0.587, so it is significant.
- N = 8 and Σd² = 120. Calculate rs to 3 decimal places.
Show answer
-0.429. rs = 1 - 720 / (8 x 63) = 1 - 720 / 504 = -0.429.
- N = 8, rs = -0.429, one-tailed test at p ≤ .05, critical value 0.643. Is the result significant?
- Yes, because the value is negative
- No, because 0.429 is less than 0.643
- Yes, because N is small
- No, because Spearman's needs nominal data
Show answer
No, because 0.429 is less than 0.643. Ignore the sign: 0.429 < 0.643, so not significant.
- A researcher predicts a relationship between ratings of aggression (out of 10) and hours of sleep. Which test should be used?
- Chi-squared
- Wilcoxon signed ranks
- No test is possible
- Spearman's rank
Show answer
Spearman's rank. Relationship plus ordinal data: Spearman's.
- Using p ≤ .10 instead of p ≤ .05 makes which error more likely?
- Type II error
- Type I error
- Observer bias
- Sampling bias
Show answer
Type I error. A lenient level makes false positives more likely.
- Which hypothesis needs a one-tailed test?
- There will be a relationship between sleep and aggression
- Sleep and aggression may be related in some way
- There is no relationship between sleep and aggression
- Students who sleep less will report more aggressive tendencies
Show answer
Students who sleep less will report more aggressive tendencies. A directional prediction uses a one-tailed test.
- Look up the critical value for N = 15, two-tailed, p ≤ .05 in the table. Enter it.
Show answer
0.521. 0.521.
Sources
- Pearson Edexcel International Advanced Subsidiary/Advanced Level in Psychology specification, Issue 5 (February 2021)
- Pearson IAL Psychology component guide: WPS02 (Issue 2, November 2024)
- Pearson WPS03/01 question paper, January 2022 (formulae and critical values tables)
Last updated 8 November 2026. Spotted a mistake? Tell us.