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  "body": "\n## Definition\n\nLet $X$ be a real random variable with mean $\\mu = E[X]$ and standard deviation $\\sigma \u003e 0$.\nThe standard score, or $z$-score, of a real number $x$ is\n$$z = \\frac{x - \\mu}{\\sigma}.$$\nThe standardized random variable is $Z = (X - \\mu)/\\sigma$.\n\n## Remarks\n\nThe standardized variable $Z$ has mean $0$ and variance $1$.\nIf $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$, then $Z \\sim \\mathcal{N}(0, 1)$. So $P(X \\le x) = \\Phi(z)$, where $\\Phi$ is the standard normal cumulative distribution function.\nFor a data set, one uses the sample mean and the sample standard deviation in place of $\\mu$ and $\\sigma$.\n\n## Explanations\n\n### L1 human\n\nA $z$-score says how unusual a value is, in units of the standard deviation.\nSuppose that test scores have mean $70$ and standard deviation $10$.\nThen a score of $85$ has $z = 1.5$. It is one and a half standard deviations above the mean.\n",
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