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The Wilcoxon signed ranks test
Edexcel IAL Psychology · WPS01 Social and Cognitive Psychology
The Wilcoxon signed ranks test is the inferential test in Unit 1 (2.2.12) and must be used in the cognitive practical (2.4.1). It tests whether there is a significant difference between two related sets of scores.
When to use it
Use Wilcoxon when:
- you are testing for a difference
- the design is repeated measures (or matched pairs), so scores are related
- the data are at least ordinal.
It is a non-parametric test: it works on ranks, not the raw scores.
The method
1. Find the difference for each participant (condition A - condition B).
2. Ignore any zero differences and do not count them in N.
3. Rank the differences by size, ignoring the sign: the smallest gets rank 1. Tied differences share the average of the ranks they would take.
4. Add up the ranks of the positive differences and the ranks of the negative differences.
5. T = the smaller of the two totals. N = number of differences that are not zero.
6. Compare T with the critical value for N, the significance level and one- or two-tailed. T must be equal to or less than the critical value for the result to be significant.
| N | One-tailed 0.05 (two-tailed 0.10) | One-tailed 0.025 (two-tailed 0.05) | One-tailed 0.01 (two-tailed 0.02) |
|---|---|---|---|
| 5 | 0 | - | - |
| 6 | 2 | 0 | - |
| 7 | 3 | 2 | 0 |
| 8 | 5 | 3 | 1 |
| 9 | 8 | 5 | 3 |
| 10 | 11 | 8 | 5 |
| 11 | 13 | 10 | 7 |
| 12 | 17 | 13 | 9 |
Wilcoxon signed ranks critical values as printed in Pearson WPS01 papers. The calculated T must be equal to or less than the critical value for significance.
Worked data
Words recalled in silence (A) and while repeating 'the the the' (B), 10 participants.
A: 8, 7, 9, 6, 8, 7, 9, 5, 8, 6
B: 5, 8, 6, 6, 4, 6, 5, 4, 6, 7
Differences (A - B): 3, -1, 3, 0, 4, 1, 4, 1, 2, -1. One zero is dropped, so N = 9.
Ranks of the absolute differences (ties averaged): 6.5, 2.5, 6.5, 8.5, 2.5, 8.5, 2.5, 5, 2.5.
Sum of positive ranks = 40; sum of negative ranks = 5; T = 5.
Critical values for N = 9: one-tailed p ≤ .05 is 8; two-tailed p ≤ .05 is 5. T = 5 is equal to or less than both, so the difference is significant at p ≤ .05 (and the direction, more words in silence, matches the hypothesis).
Worked example
Scores: A = [12, 10, 15, 9, 11, 14, 13, 10], B = [9, 11, 10, 9, 7, 12, 8, 6]. Calculate T and N, and decide whether the difference is significant at p ≤ .05 for a two-tailed test.
- Differences: 3, -1, 5, 0, 4, 2, 5, 4. Drop the zero: N = 7.
- Ranks of the absolute values: 3, 1, 6.5, 4.5, 2, 6.5, 4.5.
- Positive ranks total 27; negative ranks total 1; T = 1.
- Critical value (N = 7, two-tailed p ≤ .05) = 2. 1 ≤ 2, so significant.
Answer: T = 1, N = 7: significant at p ≤ .05 two-tailed; reject the null hypothesis.
Check yourself
- Participants' scores in condition A were 20, 18, 22, 17, 19, 21, 16, 18, 20, 23, 19 and in condition B 17, 18, 19, 18, 15, 20, 15, 14, 21, 18, 16. Calculate the Wilcoxon T value.
Show answer
5.0. Differences A - B: 3, 0, 3, -1, 4, 1, 1, 4, -1, 5, 3. Drop the zero (N = 10). Ranks: 6, 6, 2.5, 8.5, 2.5, 2.5, 8.5, 2.5, 10, 6. Positive total 50, negative total 5, so T = 5.
- For the same data, what is N?
Show answer
10. There is one zero difference, so N = 11 - 1 = 10.
- For those data, with N = 10, is the result significant at p ≤ .05 for a one-tailed test (critical value 11), if the results are in the predicted direction?
- Yes, because T is equal to or less than the critical value
- No, because T is greater than the critical value
- It cannot be decided without the mean
- Yes, because N is greater than 5
Show answer
Yes, because T is equal to or less than the critical value. T = 5 and the critical value is 11; T must be equal to or less than it.
- Which study design and aim suit a Wilcoxon test?
- An independent groups experiment with nominal data
- A correlation between two variables
- A case study of one patient
- A repeated measures experiment looking for a difference with ordinal data
Show answer
A repeated measures experiment looking for a difference with ordinal data. Wilcoxon is a test of difference for related scores, at least ordinal.
- What do you do with a participant whose two scores are the same?
- Give it rank 1
- Leave the zero difference out and reduce N by one
- Count it as a positive difference
- Count it as a negative difference
Show answer
Leave the zero difference out and reduce N by one. Zero differences are ignored and not counted in N.
- Two differences are both 3 and would take ranks 4 and 5. What rank does each get?
- 4
- 5
- 9
- 4.5
Show answer
4.5. Tied values share the average of the ranks: (4 + 5) / 2 = 4.5.
- Using Pearson's table, what is the critical value of T for N = 11 at p ≤ .05, two-tailed?
Show answer
10. Two-tailed 0.05 is the middle column: 10.
Sources
- Pearson Edexcel International Advanced Subsidiary/Advanced Level in Psychology specification (issue January 2026)
- Pearson International A Level Psychology component guide: WPS01
- Pearson International A Level Psychology sample assessment materials (Wilcoxon critical values table)
- Pearson WPS01/01 question paper, June 2025 (copy hosted by Physics and Maths Tutor)
- IAL Psychology revision site: Wilcoxon signed ranks
Last updated 8 November 2026. Spotted a mistake? Tell us.