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  "body": "\n## Definition\n\nLet $k \\ge 2$, and for $g = 1, \\dots, k$ let $X_{g1}, \\dots, X_{g n_g}$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution function $F_g$, the $k$ samples being independent.\nLet $N = n_1 + \\dots + n_k$.\nRank all $N$ observations together from $1$ to $N$, let $R_g$ be the sum of the ranks in sample $g$, and let $\\bar R_g = R_g / n_g$.\nThe *Kruskal–Wallis statistic* is\n$$H = \\frac{12}{N(N+1)} \\sum_{g=1}^{k} n_g \\Big( \\bar R_g - \\frac{N + 1}{2} \\Big)^2 = \\frac{12}{N(N+1)} \\sum_{g=1}^{k} \\frac{R_g^2}{n_g} - 3(N + 1).$$\nThe *Kruskal–Wallis test* of $H_0 \\colon F_1 = \\dots = F_k$ rejects $H_0$ when $H$ is large.\n\n## Remarks\n\nUnder $H_0$ the law of $H$ depends only on $n_1, \\dots, n_k$.\nAs all $n_g \\to \\infty$, $H$ [[converges in distribution|0JDAQHWA9N]] to the [[chi-squared distribution|0RFMMBHFTV]] with $k - 1$ degrees of freedom, which gives approximate critical values.\nThe number $(N+1)/2$ is the mean of the ranks $1, \\dots, N$.\n\nExample: $k = 2$, samples $(1, 2)$ and $(3, 4)$, so $N = 4$, $R_1 = 3$, $R_2 = 7$.\nThen $H = \\frac{12}{20}\\big(\\frac{9}{2} + \\frac{49}{2}\\big) - 15 = 17.4 - 15 = 2.4$.\nThe first form gives the same: $\\frac{12}{20}\\big(2(1.5 - 2.5)^2 + 2(3.5 - 2.5)^2\\big) = 0.6 \\cdot 4 = 2.4$.\n\nNon-example: repeated measurements on the same subjects under $k$ conditions are not independent samples.\nThe clause on independence fails; the Friedman test is used for such blocked data.\n",
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