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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be real random variables, for example an [[i.i.d.|0AJN7WHBJZ]] sample.\nThe sample mean is the [[statistic|036756NFD6]]\n$$\\bar{X}_n = \\frac{1}{n} \\sum_{i=1}^{n} X_i.$$\nFor observed values $x_1, \\dots, x_n$, the sample mean is the number $\\bar{x} = (x_1 + \\dots + x_n)/n$.\n\n## Remarks\n\nIf the $X_i$ are i.i.d. with mean $\\mu$, then $E[\\bar{X}_n] = \\mu$ by [[linearity of expectation|0P7PTC8DDE]]. So $\\bar{X}_n$ is an unbiased estimator of $\\mu$.\nIf also $\\operatorname{Var}(X_i) = \\sigma^2 \u003c \\infty$, then $\\operatorname{Var}(\\bar{X}_n) = \\sigma^2 / n$.\nFor i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ variables, $\\bar{X}_n \\sim \\mathcal{N}(\\mu, \\sigma^2/n)$.\nBy the law of large numbers, $\\bar{X}_n \\to \\mu$ almost surely.\n\n## Explanations\n\n### L1 human\n\nThe sample mean is the ordinary average of the data.\nFor the measurements $4$, $7$ and $10$, the sample mean is $21/3 = 7$.\n",
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