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  "body": "\n## Definition\n\nLet $(x_1, y_1), \\dots, (x_n, y_n)$ be $n \\ge 2$ pairs of real numbers, with $x_1, \\dots, x_n$ distinct and $y_1, \\dots, y_n$ distinct.\nLet $r_i$ be the rank of $x_i$ among $x_1, \\dots, x_n$ and $s_i$ the rank of $y_i$ among $y_1, \\dots, y_n$, with rank $1$ for the smallest.\n*Spearman's rank correlation coefficient* is the sample [[correlation coefficient|04CTY6XYWN]] of the pairs $(r_i, s_i)$.\nWith $d_i = r_i - s_i$ it equals\n$$\\rho_s = 1 - \\frac{6 \\sum_{i=1}^{n} d_i^2}{n(n^2 - 1)}.$$\n\n## Remarks\n\nThe closed formula holds because the ranks of each variable are the numbers $1, \\dots, n$, whose mean and variance are known.\nFor random variables $X$ and $Y$ with continuous [[distribution functions|0MXMS3AE13]] $F$ and $G$, the population version is the correlation of $F(X)$ and $G(Y)$, which equals $12 \\, E[F(X) G(Y)] - 3$.\n\nExample: $x = (1, 2, 3)$ and $y = (1, 3, 2)$.\nThe ranks are $r = (1, 2, 3)$ and $s = (1, 3, 2)$, so $d = (0, -1, 1)$ and $\\rho_s = 1 - 6 \\cdot 2 / (3 \\cdot 8) = 1/2$.\nFor $y = (1, 4, 9)$ instead, $d = 0$ and $\\rho_s = 1$.\n\nNon-example: for data with ties, such as $y = (1, 1, 2)$, the ranks are not defined by the clause above.\nThe usual fix gives tied values the average of their ranks and takes the Pearson correlation; the formula with $\\sum d_i^2$ is then no longer exact.\n\nThe [[Kendall rank correlation coefficient|DYFY16SFHZ]] is another rank correlation, based on pairs of observations.\n",
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