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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution, and let $m$ be its [[median|08W2CW8PKW]].\nLet $m_0 \\in \\R$.\nThe *sign statistic* is\n$$S = \\#\\set{i : X_i \u003e m_0}.$$\nThe *sign test* of $H_0 \\colon m = m_0$ uses the fact that under $H_0$ the statistic $S$ has the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $1/2$.\nAgainst $H_1 \\colon m \u003e m_0$ it rejects $H_0$ for large $S$, against $m \u003c m_0$ for small $S$, and against $m \\ne m_0$ for $S$ far from $n/2$.\n\n## Remarks\n\nFor matched pairs $(Y_i, Z_i)$ one applies the test to the differences $X_i = Y_i - Z_i$ with $m_0 = 0$.\nContinuity of the law gives $P(X_i = m_0) = 0$; with real data, observations equal to $m_0$ are usually dropped and $n$ is reduced.\n\nExample: $n = 10$ observations, all above $m_0$, so $S = 10$.\nThe one-sided p-value is $P(S \\ge 10) = 2^{-10} \\approx 0.001$ under $H_0$, and $H_0$ is rejected at level $0.05$.\n\nNon-example: let $P(X_i \u003c m_0) = P(X_i \u003e m_0) = 1/4$ and $P(X_i = m_0) = 1/2$.\nThen $m_0$ is a median, but the law is not continuous, and $S$ has law $\\mathrm{Bin}(n, 1/4)$ instead of $\\mathrm{Bin}(n, 1/2)$.\n\nThe [[Wilcoxon signed-rank test|38SETDT2HN]] also uses the sizes of the deviations $X_i - m_0$, at the cost of assuming symmetry.\n",
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