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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $X_1, \\dots, X_n$ be real random variables with [[sample mean|0QNRBHWV35]] $\\bar{X}_n$.\nThe sample variance is the [[statistic|036756NFD6]]\n$$S_n^2 = \\frac{1}{n - 1} \\sum_{i=1}^{n} \\left(X_i - \\bar{X}_n\\right)^2.$$\nThe sample standard deviation is $S_n = \\sqrt{S_n^2}$.\n\n## Remarks\n\nIf the $X_i$ are i.i.d. with [[variance|007QED3E0Y]] $\\sigma^2 \u003c \\infty$, then $E[S_n^2] = \\sigma^2$. So $S_n^2$ is an unbiased estimator of $\\sigma^2$.\nThe divisor $n - 1$ instead of $n$ is Bessel's correction. The version with divisor $n$ has expectation $\\frac{n-1}{n} \\sigma^2$, so it has the [[bias|01BD2TFFP4]] $-\\sigma^2/n$.\nThe reason is that $\\sum_i (X_i - \\bar{X}_n)^2 \\le \\sum_i (X_i - \\mu)^2$: the deviations from $\\bar{X}_n$ are on average smaller than those from the true mean $\\mu$.\nFor i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ variables, $(n - 1) S_n^2 / \\sigma^2$ has the chi-squared distribution with $n - 1$ degrees of freedom, and it is independent of $\\bar{X}_n$.\n\n\n\n## Explanations\n\n### L1 human\n\nThe sample variance is the average squared distance of the data from their mean, but with the divisor $n - 1$.\nFor the data $4$, $7$, $10$, the mean is $7$, the squared deviations add up to $18$, and the sample variance is $18/2 = 9$.\n",
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