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  "body": "\n## Definition\n\nLet $\\hat{\\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\\theta$, with $E_\\theta[|\\hat{\\theta}|] \u003c \\infty$ for each $\\theta$. Here $E_\\theta$ is the expectation when $\\theta$ is the true parameter.\nThe bias of $\\hat{\\theta}$ at $\\theta$ is\n$$\\operatorname{Bias}_\\theta(\\hat{\\theta}) = E_\\theta[\\hat{\\theta}] - \\theta.$$\nThe estimator is unbiased if $\\operatorname{Bias}_\\theta(\\hat{\\theta}) = 0$ for each $\\theta$.\n\n## Remarks\n\nFor an i.i.d. sample with mean $\\mu$, the sample mean is an unbiased estimator of $\\mu$.\nThe sample variance with divisor $n$ has bias $-\\sigma^2/n$. With divisor $n - 1$, it is unbiased.\nAn unbiased estimator need not be good. The [[mean squared error|0V4VSCBXP3]] is variance plus squared bias, and a biased estimator can have a smaller mean squared error.\nUnbiasedness is not preserved by nonlinear maps. For example, the sample standard deviation $S_n$ is a biased estimator of $\\sigma$, even though $S_n^2$ is unbiased for $\\sigma^2$.\n\n## Explanations\n\n### L1 human\n\nAn estimator is unbiased if it is right on average, over many repetitions of the experiment.\nBias is a systematic error: a scale that always shows $1$ kg too much has a bias of $1$ kg.\n",
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