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  "body": "\n## Example\n\nLet $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$ with $\\sigma \u003e 0$, and let $\\Phi$ be the cumulative distribution function of $\\mathcal{N}(0, 1)$.\nSince $Z = (X - \\mu)/\\sigma \\sim \\mathcal{N}(0, 1)$, $P(|X - \\mu| \\le k\\sigma) = 2\\Phi(k) - 1$ for each $k \u003e 0$. Numerical values of $\\Phi$ give:\n$$P(|X - \\mu| \\le \\sigma) \\approx 0.6827, \\quad P(|X - \\mu| \\le 2\\sigma) \\approx 0.9545, \\quad P(|X - \\mu| \\le 3\\sigma) \\approx 0.9973.$$\nSo about $68\\%$, $95\\%$ and $99.7\\%$ of the mass of a normal distribution lie within one, two and three standard deviations of the mean.\n\n## Remarks\n\nThe rule holds for normal distributions only. For an arbitrary distribution with finite variance, Chebyshev's inequality gives only $P(|X - \\mu| \\le k\\sigma) \\ge 1 - 1/k^2$, for example $3/4$ for $k = 2$.\nThe interval $\\mu \\pm 1.96\\sigma$ contains $95\\%$ of the mass. It is the base of the usual $95\\%$ confidence intervals.\n\n## Explanations\n\n### L1 human\n\nFor data that follow a bell curve, almost all values are within three standard deviations of the mean.\nIf adult heights have mean $170$ cm and standard deviation $10$ cm, then about $95\\%$ of the heights are between $150$ cm and $190$ cm.\n",
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