MannWhitneyUTest

The MannWhitneyUTest, commonly called the Mann-Whitney U test or the Wilcoxon rank-sum test, is a nonparametric method for assessing whether there is a difference between two independent groups on a single ordinal or continuous outcome. It does not assume normality of the data and is appropriate when the sample distributions are not normally distributed or when sample sizes are small.

The test operates by ranking all observations from both samples combined, handling ties by assigning average ranks. Let the two samples have sizes n1 and n2, and let R1 be the sum of ranks for group 1. The U statistics are U1 = n1*n2 + n1(n1+1)/2 − R1 and U2 = n1*n2 − U1. The test statistic is U = min(U1, U2). For small samples, exact critical values of U determine significance. For larger samples, a normal approximation is used with mean μ_U = n1*n2/2 and variance σ_U^2 = n1*n2(n1+n2+1)/12, often with a continuity correction applied. The resulting p-value indicates whether the two distributions differ.

Assumptions of the test include two independent samples, data measured at least on an ordinal scale, and random sampling within each group. While it is robust to non-normal data and unequal variances, the test assumes similar-shaped distributions if one aims to interpret a difference in central tendency as a shift in location.

Interpretation centers on whether the observed value is unlikely under the null hypothesis that the two populations are stochastically identical. A small p-value suggests a difference in distributions between the groups.