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  "body": "\n## Definition\n\nLet $n \\ge 2$ and $k \\ge 1$, and let $X_1, \\dots, X_n$ be random vectors in $\\R^k$, written as columns.\nThe *sample mean* is the [[statistic|036756NFD6]]\n$$\\bar{X} = \\frac{1}{n} \\sum_{i=1}^{n} X_i \\in \\R^k.$$\nThe *sample covariance matrix* is the $k \\times k$ matrix\n$$Q = \\frac{1}{n - 1} \\sum_{i=1}^{n} \\left(X_i - \\bar{X}\\right)\\left(X_i - \\bar{X}\\right)^T.$$\nIts entry $Q_{jl}$ is the sample covariance of the $j$-th and the $l$-th coordinates.\n\n## Remarks\n\nExample: for the observations $(0, 0)^T$ and $(2, 2)^T$, the mean is $(1, 1)^T$ and the deviations are $(-1, -1)^T$ and $(1, 1)^T$.\nEach deviation gives the matrix with all entries $1$, so $Q$ has all four entries equal to $2$.\nFor $k = 1$, $\\bar{X}$ is the [[sample mean|0QNRBHWV35]] and $Q$ is the [[sample variance|0JN9P4A944]].\nIf the $X_i$ are i.i.d. with mean vector $\\mu$ and finite [[covariance matrix|02CBHV4ZZQ]] $\\Sigma$, then $E[\\bar{X}] = \\mu$ and $E[Q] = \\Sigma$.\nNon-example: the matrix with divisor $n$ in place of $n - 1$ is also used, but it is not unbiased; its expectation is $\\frac{n-1}{n}\\Sigma$.\nThe matrix $Q$ is symmetric and positive semidefinite, and it is singular when $n \\le k$.\n",
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