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  "body": "\n## Definition\n\nLet $X$ be an integrable real [[random variable|0DKRR22AEQ]] with mean $\\mu = E[X]$.\nThe variance of $X$ is\n$$\\operatorname{Var}(X) = E\\left[(X - \\mu)^2\\right] \\in [0, \\infty].$$\n\n## Remarks\n\nOther notations are $\\operatorname{V}(X)$, $\\sigma^2$ and $\\sigma_X^2$. The variance is finite if and only if $E[X^2] \u003c \\infty$.\nFor a discrete $X$ with values $x_i$, $\\operatorname{Var}(X) = \\sum_i (x_i - \\mu)^2 P(X = x_i)$. For $X$ with density $f$, $\\operatorname{Var}(X) = \\int (x - \\mu)^2 f(x) \\, dx$.\n$\\operatorname{Var}(X) = 0$ if and only if $X = \\mu$ almost surely.\nFor a fair die, $\\mu = 7/2$ and $\\operatorname{Var}(X) = 35/12$.\n\n## Explanations\n\n### L1 human\n\nThe variance tells how far, on average, the values of $X$ are from the mean. The distances are squared, so that large deviations count more and positive and negative deviations do not cancel.\nIts square root, the standard deviation, is in the same units as $X$.\n",
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