Wilcoxon Rank Sum Test Calculator (Mann-Whitney U)
Paste or type the raw values for two independent groups to perform a Wilcoxon rank sum test (Mann-Whitney U test). The calculator ranks the pooled data, computes the W statistic, and derives a p-value using the normal approximation — no normality assumption required.
Alternative hypothesis
Reject H₀ at α = 0.05 — the distributions differ significantly.
- 1
Total n
n₁ + n₂ = 5 + 5 = 10 - 2
Expected W under H₀ (μ_W)
n₁ × (N+1) ÷ 2 = 5 × 11 ÷ 2 = 27.5 - 3
σ_W
√(n₁ × n₂ × (N+1) ÷ 12) = √(5 × 5 × 11 ÷ 12) = 4.7871 - 4
Z statistic
(W − μ_W) ÷ σ_W = (15.5 − 27.5) ÷ 4.7871 = -2.5067Standardised rank sum — read against the standard normal to get the p-value.
How does this calculator work?
Enter two groups of numbers. The calculator pools and ranks all values, sums Group 1 ranks (W), standardises to Z = (W − n₁(N+1)/2) / √(n₁n₂(N+1)/12), and derives a p-value from the standard normal distribution. A p-value below 0.05 indicates the groups differ significantly — no normality assumption required.
Formula
How this is calculated
The Wilcoxon rank sum test (also called the Mann-Whitney U test) is the non-parametric alternative to the two-sample t-test. It tests whether the central tendency (distribution location) of two independent groups differs, without requiring the data to be normally distributed — making it appropriate for small samples, ordinal data, or data with heavy tails.
The procedure: (1) Pool all observations from both groups and sort them. (2) Assign each observation a rank from 1 to N; tied values share the average of the ranks they would occupy. (3) Compute W, the sum of ranks belonging to Group 1. Under the null hypothesis of no difference, E[W] = n₁(N+1)/2 and Var[W] = n₁n₂(N+1)/12 where N = n₁ + n₂. (4) Standardise: Z = (W − E[W]) / √Var[W], then read off the p-value from the standard normal distribution. This large-sample approximation is reliable when both groups contain at least 5 observations; for very small samples (n < 5) exact critical values from published tables are preferable.
The Mann-Whitney U statistic U = W − n₁(n₁+1)/2 counts the number of times a Group 1 observation exceeds a Group 2 observation across all pairs — an equivalent way of expressing the same result.
Frequently asked questions
Use the Wilcoxon rank sum test when your data are not normally distributed, come from a small sample where normality cannot be verified, or are measured on an ordinal scale (e.g. Likert ratings). For large samples from roughly normal populations, the t-test and Wilcoxon test give very similar results.
Yes — they are mathematically equivalent. They use different test statistics (W versus U) that are linearly related: U = W − n₁(n₁+1)/2. Both lead to the same p-value and decision.
For large enough groups (n₁, n₂ ≥ 5) the sampling distribution of W is approximately normal, so Z and the resulting p-value are reliable. For very small groups, exact p-values from the Wilcoxon distribution table are more accurate. This calculator uses the normal approximation only.
Also known as
TG we-Calculate Editorial Team. (2026). Wilcoxon Rank Sum Test Calculator (Mann-Whitney U) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/wilcoxon-rank-sum-test-calculator
TG we-Calculate Editorial Team. "Wilcoxon Rank Sum Test Calculator (Mann-Whitney U)." TG we-Calculate. 2026. https://we-calculate.com/calculator/wilcoxon-rank-sum-test-calculator.
TG we-Calculate Editorial Team, "Wilcoxon Rank Sum Test Calculator (Mann-Whitney U)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/wilcoxon-rank-sum-test-calculator
@misc{wecalculate_wilcoxon_rank_sum_test_calculator, title = {Wilcoxon Rank Sum Test Calculator (Mann-Whitney U)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/wilcoxon-rank-sum-test-calculator}}, year = {2026}, note = {TG we-Calculate} }
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