Mann–Whitney U test

Remarks

Let $R_X$ be the sum of the ranks of $X_1, \dots, X_m$ in the pooled sample of size $m + n$. Then $U = R_X - m(m+1)/2$, so the test is the same as the Wilcoxon rank-sum test. Under $H_0$, $E[U] = mn/2$ and $\operatorname{Var}(U) = mn(m + n + 1)/12$, and $U$ is asymptotically normal. The ratio $U/(mn)$ is an unbiased estimator of $P(X_1 > Y_1)$, and $U$ is a two-sample U-statistic. With ties, each tied pair counts $1/2$.

Example: $X = (3, 5)$ and $Y = (1, 4)$. The pairs with $X_i > Y_j$ are $(3, 1)$, $(5, 1)$ and $(5, 4)$, so $U = 3$. In the pooled order $1, 3, 4, 5$ the $X$ values have ranks $2$ and $4$, and $R_X - 3 = 6 - 3 = 3$ agrees.

Non-example: measurements of the same subjects before and after a treatment are not independent samples. The clause on independence fails, and the Wilcoxon signed-rank test on the differences is used instead.