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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_m$ be [[i.i.d.|0AJN7WHBJZ]] with continuous [[distribution function|0MXMS3AE13]] $F$, and let $Y_1, \\dots, Y_n$ be i.i.d. with continuous distribution function $G$, independent of the $X_i$.\nThe *Mann–Whitney statistic* is\n$$U = \\#\\set{(i, j) : 1 \\le i \\le m, \\ 1 \\le j \\le n, \\ X_i \u003e Y_j}.$$\nThe *Mann–Whitney U test* of $H_0 \\colon F = G$ rejects $H_0$ when $U$ is far from $mn/2$.\nFor a one-sided alternative it rejects only when $U$ is large, or only when $U$ is small.\nUnder $H_0$ the law of $U$ depends only on $m$ and $n$.\n\n## Remarks\n\nLet $R_X$ be the sum of the ranks of $X_1, \\dots, X_m$ in the pooled sample of size $m + n$.\nThen $U = R_X - m(m+1)/2$, so the test is the same as the Wilcoxon rank-sum test.\nUnder $H_0$, $E[U] = mn/2$ and $\\operatorname{Var}(U) = mn(m + n + 1)/12$, and $U$ is asymptotically normal.\nThe ratio $U/(mn)$ is an unbiased estimator of $P(X_1 \u003e Y_1)$, and $U$ is a two-sample [[U-statistic|NTVQB1DC56]].\nWith ties, each tied pair counts $1/2$.\n\nExample: $X = (3, 5)$ and $Y = (1, 4)$.\nThe pairs with $X_i \u003e Y_j$ are $(3, 1)$, $(5, 1)$ and $(5, 4)$, so $U = 3$.\nIn the pooled order $1, 3, 4, 5$ the $X$ values have ranks $2$ and $4$, and $R_X - 3 = 6 - 3 = 3$ agrees.\n\nNon-example: measurements of the same subjects before and after a treatment are not independent samples.\nThe clause on independence fails, and the [[Wilcoxon signed-rank test|38SETDT2HN]] on the differences is used instead.\n",
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