Home Statistical Dictionary Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

A nonparametric test for whether two independent samples come from the same distribution, based on ranking the combined data rather than assuming Normality; the most widely used alternative to the two-sample t-test.

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In Plain English

The Mann-Whitney U test (also called the Wilcoxon rank-sum test) compares two independent groups without assuming their data follows a Normal distribution. Instead of working with the actual values, it pools both groups together, ranks every observation, and then checks whether one group's ranks tend to be systematically higher or lower than the other's. It's the go-to alternative to the two-sample t-test whenever the data is skewed, has outliers, or is measured on an ordinal scale.

Definition

The Mann-Whitney U test (Mann & Whitney, 1947), equivalent to the Wilcoxon rank-sum test (Wilcoxon, 1945) developed independently and shown to be mathematically identical, tests $H_{0}$ : two independent samples are drawn from the same distribution (or, under an added assumption of identically shaped distributions, that they share the same median), by pooling both samples, ranking the combined data, summing the ranks for one group, and comparing this rank sum to its expected value under the null. Because it operates entirely on ranks, the test is fully nonparametric and does not require Normality, making it the standard alternative to the two-sample t-test whenever that test's assumptions are doubtful, though a Mann-Whitney result technically reflects a difference in overall distribution (stochastic dominance) and only reduces cleanly to a difference in medians when the two groups' distributions have the same shape and spread.

Formula

U_{1} = R_{1} - \frac{n _{1} ( n _{1} + 1 )}{2}

Notation

R_{1}

—Sum of ranks assigned to group 1 within the combined, pooled sample

n_{1}

—Sample size of group 1

Properties

Wilcoxon and Mann/Whitney developed mathematically equivalent tests independently within a few years of each other, which is why the literature and software both use either name interchangeably, R's wilcox.test() and Python's scipy.stats.mannwhitneyu() compute the same underlying test.
A statistically significant Mann-Whitney result formally establishes stochastic dominance, that one group's values tend to be larger than the other's, rather than a difference specifically in medians, the common interpretation of 'different medians' is only strictly valid when the two groups' distributions have the same shape and spread, differing only in location.
The Mann-Whitney U test's relative efficiency compared to the t-test is famously high, even when the data genuinely is Normally distributed (where the t-test is optimal), the Mann-Whitney test retains about 95.5% of the t-test's power, and it can be far more powerful than the t-test when the data is heavy-tailed or contains outliers.

At a Glance

Introduced byHenry Mann and Donald Whitney, 1947; equivalent to Wilcoxon's 1945 rank-sum test

Testswhether two independent samples come from the same distribution

Nonparametric alternative tothe two-sample (independent) t-test

Common name confusionMann-Whitney U and Wilcoxon rank-sum are the same test under two names

Used in These Tests

Mann-Whitney U Test

Last updated August 8, 2026← Back to the Dictionary