Mann-Whitney U and Wilcoxon Signed-Rank Test Calculator
Mann-Whitney U test (independent two-group) and Wilcoxon signed-rank test (paired). Use these instead of the t-test when data are markedly skewed, contain extreme values, or are inherently ordinal.
How they differ from the t-test
The t-test compares means and assumes approximate normality. These tests compare ranks — values are sorted and only their position matters, not the actual magnitude. This has two consequences:
Advantage: No normality assumption. The influence of extreme values is compressed — an outlier simply ranks last rather than pulling the mean toward it.
Cost: Information about actual magnitude is discarded. When data are genuinely close to normal, rank tests are slightly less sensitive than the t-test — a somewhat larger sample is needed to detect the same difference.
Check the minimum achievable p first with small samples
This is the most important thing this page wants to flag. Rank-test p-values come from combinatorics. With very small samples, p cannot reach 0.05 even when the effect is as large as possible.
- Wilcoxon paired test: minimum two-tailed p = 2 / 2ⁿ for n pairs. At n = 5 the minimum p is 0.0625 — no effect size can make the result significant.
- Mann-Whitney: minimum two-tailed p = 2 / C(n₁+n₂, n₁) for groups of size n₁ and n₂. With three observations per group the minimum p is 0.1; with four per group it is 0.0286.
This page computes and displays the minimum achievable p for your sample size before anything else. If it already exceeds α, the experiment could not have produced a significant result regardless of the data — the correct response is to collect more observations, not to switch tests.
Exact method vs. normal approximation
With small samples and no ties, this page uses the exact method: it enumerates all possible rank assignments and directly counts the proportion more extreme than the observed statistic. With larger samples or ties, it switches to the normal approximation (with continuity correction and tie correction).
The two methods can differ substantially with small samples: for this page’s example data, the exact method gives p = 0.00216 while the normal approximation gives p = 0.00507 — a difference of more than twofold. The exact method is not strictly valid when ties are present, which is why the page switches automatically; the output states which method was used.
Effect size
Rank-biserial correlation r = 1 − 2U/(n₁n₂), ranging from −1 to 1. It quantifies how strongly values from one group tend to exceed values from the other. r = 0 means the groups are completely interleaved; |r| = 1 means they are completely separated. As with the t-test: the p-value says how unlikely the difference is by chance; the effect size says how large the difference is.
FAQ
When should I use a rank test instead of a t-test?
When data are clearly skewed, contain extreme values, or are inherently ordinal (e.g., a 1–5 rating scale). Very small samples (fewer than 5–6 per group) make it impossible to effectively verify normality, and rank tests can be a safer default — but check the minimum achievable p shown on this page first; with too few observations, the rank test cannot reach significance either.
Why can a paired test with n = 5 never be significant?
The Wilcoxon p-value comes from the combinatorics of signs: n pairs yield 2ⁿ possible sign assignments, and the most extreme case (all same sign) gives a two-tailed p = 2/2ⁿ. At n = 5 this is 2/32 = 0.0625, which already exceeds 0.05. This is not a lack of sensitivity — it is an upper bound on the available information. The solution is to collect more observations.
Why do the exact method and normal approximation differ so much?
With small samples, the rank-statistic distribution is far from normal, so the approximation can be substantially off. For this page's example data, the exact method gives p = 0.00216 and the normal approximation gives p = 0.00507 — more than twofold. The page uses the exact method automatically for small tie-free samples; the approximation is used when ties are present, because the exact method is no longer strictly valid in that case.
Does the Mann-Whitney test actually compare medians?
Strictly, no. It tests whether the probability that a randomly drawn value from one group exceeds a randomly drawn value from the other equals 0.5. This is equivalent to comparing medians only when the two distributions have the same shape and differ only in location. When the shapes differ (e.g., substantially different variances), a significant result indicates that the distributions differ, but cannot be interpreted directly as a difference in medians.
How do I interpret the rank-biserial correlation r?
r = 1 − 2U/(n₁n₂), ranging from −1 to 1. It measures how strongly values in one group tend to exceed values in the other: r = 0 means the groups are completely interleaved; |r| = 1 means every value in one group exceeds every value in the other. Read it alongside the p-value: p says how unlikely the difference is by chance; r says how large that difference is.
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