WilcoxonRangsummenTest

The Wilcoxon rank-sum test, also known as the Mann–Whitney U test, is a nonparametric statistical test used to assess whether two independent samples originate from the same distribution. It is commonly applied to determine if one sample tends to have larger values than the other. The null hypothesis states that the distributions are identical in location, while the alternative asserts a stochastic difference in one direction (or two-sided, depending on the specification).

The test requires independent samples and at least ordinal data. It does not assume normality of the underlying distributions, making it robust to outliers and non-normal shapes. It is often interpreted as a test for differences in central tendency when the distribution shapes are similar, but strictly it tests for stochastic difference between the two populations.

Calculation proceeds by ranking all observations from both samples together, with tied values receiving average ranks. Let W denote the sum of ranks for one sample (or equivalently U, the Mann–Whitney statistic). The two standard expressions are related by U1 = n1 n2 + n1(n1+1)/2 − W1. For small samples, exact p-values are available. For large samples, the distribution of U (or W) is approximated by a normal distribution with mean n1 n2 / 2 and variance n1 n2 (n1 + n2 + 1) / 12, with optional tie corrections and a continuity correction.

Interpretation centers on whether the observed test statistic is unusually large or small under H0; a small p-value leads to rejecting the null in favor of a difference in distributions, with the direction inferred from which sample tends to have larger values. The test is distinct from the Wilcoxon signed-rank test (paired data) and from the parametric t-test. It is widely implemented in statistical software under names such as wilcox.test or Mann–Whitney U test.