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  "body": "\n## Summary\n\nPropagation of uncertainty computes how the uncertainties of input quantities carry over to a quantity computed from them.\nInputs are modelled as random variables with known means, [[variances|0477R9PYAE]] and [[covariances|0KBFAWYRTE]].\nFor a smooth function one usually linearizes it at the means.\nThe output covariance is then $J \\Sigma J^{T}$, where $J$ is the [[Jacobian matrix|047NDTEPCV]] and $\\Sigma$ the input covariance matrix.\nIt is the standard tool for error bars in experimental science; for guaranteed bounds one uses [[interval arithmetic|QPKKTE1ZY0]] instead.\n\n## Explanations\n\n### L1 human\n\nEvery measurement comes with an uncertainty, such as $2.00 \\pm 0.01$ m.\nIf one computes an area or a speed from measured values, the result is uncertain too, and one wants to know by how much.\nFor a sum of two independent measurements, the standard deviations combine like the sides of a right triangle: $\\sigma^2 = \\sigma_1^2 + \\sigma_2^2$.\nFor a product, the relative uncertainties combine in the same way.\nFor example, a rectangle measured as $2.00 \\pm 0.02$ by $3.00 \\pm 0.03$ has area $6.00$ with relative uncertainty about $\\sqrt{0.01^2 + 0.01^2} \\approx 1.4\\%$, so about $\\pm 0.08$.\n",
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