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  "body": "\n## Definition\n\nLet $X$ and $Y$ be real random variables on one probability space with finite and positive variances.\nThe correlation coefficient of $X$ and $Y$ is\n$$\\rho_{X, Y} = \\operatorname{Corr}(X, Y) = \\frac{\\operatorname{Cov}(X, Y)}{\\sigma_X \\, \\sigma_Y},$$\nwhere $\\operatorname{Cov}$ is the [[covariance|0DJ8W3HP2D]] and $\\sigma_X$, $\\sigma_Y$ are the [[standard deviations|0VTN5RKNC5]].\n\n## Remarks\n\nBy the Cauchy–Schwarz inequality, $-1 \\le \\rho_{X, Y} \\le 1$.\n$|\\rho_{X, Y}| = 1$ if and only if $Y = aX + b$ almost surely for some reals $a \\ne 0$ and $b$. The sign of $\\rho$ is the sign of $a$.\nThe coefficient does not depend on units: $\\rho_{aX + b, \\, cY + d} = \\rho_{X, Y}$ if $a c \u003e 0$.\nFor a sample of pairs $(x_i, y_i)$, the sample correlation coefficient $r$ uses sample means and sample sums of squares instead of expectations.\n\n## Explanations\n\n### L1 human\n\nThe correlation coefficient tells how close the points $(X, Y)$ are to a straight line.\nA value near $1$ means a strong increasing linear relation, and a value near $-1$ means a strong decreasing one.\nA value near $0$ means that there is no linear relation.\n",
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