Mann-Whitney U test
Also known as: Mann-Whitney-Wilcoxon test, Wilcoxon rank-sum test, Mann-Whitney U Testi
The Mann-Whitney U test is the nonparametric alternative to the independent samples t-test, comparing two independent groups by ranking all observations together rather than relying on their means. It was introduced by H. B. Mann and D. R. Whitney in 1947 and does not require the data to be normally distributed.
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When to use it
Use it to compare exactly two independent groups when the outcome is at least ordinal and normality cannot be assumed — for instance with skewed continuous data, Likert-type scales, or small samples. Observations must be independent, and at least ten cases overall are recommended; below that the normal approximation to U becomes unreliable and a permutation test is preferable. If you wish to interpret the result specifically as a difference in medians, the two groups should have similar distribution shapes.
Strengths & limitations
- Distribution-free: no normality assumption, so it is robust to skew and outliers.
- Works with ordinal data and ranked continuous outcomes where the t-test is inappropriate.
- Simple, widely accepted, and available in every analysis package.
- Interpreting the result as a median comparison requires the two groups to have similar distribution shapes.
- The normal approximation to U is unreliable for very small samples (fewer than about ten observations).
- Limited to comparing exactly two independent groups.
Frequently asked
Does the test compare medians or distributions?
Strictly it tests whether one group is stochastically larger than the other. It can be read as a median comparison only when the two groups share a similar distribution shape; otherwise it reflects a broader difference between the distributions.
How does it relate to the Wilcoxon rank-sum test?
They are the same procedure. Mann and Whitney (1947) formulated the U statistic, while Wilcoxon (1945) introduced the equivalent rank-sum statistic, so the two names refer to one test.
What is the minimum sample size?
At least ten observations overall is recommended. With fewer cases the normal approximation to the U distribution breaks down and the p-value becomes unreliable; an exact permutation test is the safer choice.
How do I report an effect size?
Convert the test statistic to a standardised Z and report r = |Z| / √N, where N is the total sample size. Reporting a 95% confidence interval alongside it is recommended.
Sources
- Mann, H. B. & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18(1), 50–60. DOI: 10.1214/aoms/1177730491 ↗
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE. ISBN: 978-1446249185
How to cite this page
ScholarGate. (2026, June 1). Mann-Whitney U test. ScholarGate. https://scholargate.app/en/statistics/mann-whitney-u
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Independent t-testStatistics↔ compare
- Kruskal-Wallis testStatistics↔ compare
- Permutation TestStatistics↔ compare
- Wilcoxon signed-rank testStatistics↔ compare