Wilcoxon Signed-Rank
Compare paired measurements without assuming normality — W, z, p, and effect size.
W₊ = 33.5, W₋ = 2.5; the smaller is reported. 8 pairs used.
Two-tailed p = 0.0278, with a tie correction applied.
r = |z| ÷ √n, the effect size usually reported alongside a Wilcoxon test.
With fewer than about 10 pairs the normal approximation is rough — check W against an exact critical-value table instead of relying on this p-value.
The Wilcoxon signed-rank test ranks the sizes of paired differences and compares the ranks of increases against decreases. For 8 pairs with W₊ = 33.5 and W₋ = 2.5, W = 2.5 gives z = −2.20 and a two-tailed p of .028 — a significant change.
The paired test that ignores the distribution
Where the paired t-test compares the mean difference against zero and assumes those differences are normal, the Wilcoxon signed-rank test uses only their order and sign. It suits before-and-after designs with small samples, skewed differences, or ordinal outcomes — pain ratings, questionnaire scores, judged rankings.
The key idea is that it uses more information than a simple sign test. Counting how many participants improved ignores by how much; the signed-rank test ranks the absolute differences, so a large improvement contributes a bigger rank than a marginal one, while still never assuming a distribution.
Zero differences are dropped
A participant whose score did not change contributes nothing to either direction, and Wilcoxon’s original procedure removes those pairs before ranking. This reduces n, which is worth knowing: a study of 30 people where 12 showed no change is effectively an 18-pair test, and the reported n should say so.
W₊ and W₋ are the rank sums of the positive and negative differences, n the number of non-zero pairs, and t the size of each group of tied absolute differences.
Worked example: 8 pairs
Difference, rank, then sum by direction:
- 1 Take the difference for each pair. After minus before, giving differences of 2, 1, 4, 1, 4, 1, −1 and 3.
- 2 Drop any zero differences. None here, so all 8 pairs remain in the analysis.
- 3 Rank the absolute differences. Ignore the sign when ranking. The four differences of size 1 tie and share the average of ranks 1 to 4, which is 2.5.
- 4 Sum the ranks by direction. Positive differences give W₊ = 33.5; the single negative difference gives W₋ = 2.5.
- 5 Take the smaller sum. W = 2.5, well below the expected 18 for 8 pairs.
- 6 Standardise and interpret. z = (2.5 − 18) ÷ 7.04 = −2.20, two-tailed p = .028, with an effect size of r = .78.
Which paired test to use
All three answer the same design question with progressively fewer assumptions and progressively less information used.
| Test | Uses | Assumes |
|---|---|---|
| Paired t-test | Size and direction of differences | Differences roughly normal |
| Wilcoxon signed-rank | Rank order and direction | Differences symmetric about the median |
| Sign test | Direction only | Almost nothing |
Assumptions, ties, and reporting
The test is often called assumption-free, which overstates it. Interpreting the result as a shift in the median assumes the distribution of differences is roughly symmetric. Without symmetry it remains a valid test of whether positive and negative differences are balanced, but the median reading weakens.
Tied absolute differences share an average rank and reduce the variance of W, so the calculator applies the standard correction. Ties are common with coarse scales, where many participants change by exactly one point.
The normal approximation used for the p-value is dependable from about 10 pairs upwards. Below that, check W against an exact critical-value table. When reporting, give W, the number of non-zero pairs, z, the exact p, and the effect size r = |z| ÷ √n, alongside medians and interquartile ranges for each condition.