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  "body": "\n## Summary\n\nA quasi-arithmetic mean is the arithmetic mean taken after a change of scale.\nGiven a continuous strictly monotone function $\\varphi$, the mean of $x_1, \\dots, x_n$ is $\\varphi^{-1}\\big(\\frac{1}{n} \\sum_i \\varphi(x_i)\\big)$.\nThe arithmetic, geometric, harmonic and power means are all of this form.\nThese means are characterized by a few natural axioms, which is the [[Kolmogorov–Nagumo theorem|9Q5DPYQ5NP]].\n\n## Explanations\n\n### L1 human\n\nTo average numbers in a quasi-arithmetic way, first move them to a new scale, take the ordinary average there, and move the result back.\nWith the logarithm as the scale, the average of $2$ and $8$ is computed as the average of $\\ln 2$ and $\\ln 8$, which is $\\ln 4$, so the mean is $4$.\nThis is the geometric mean.\nWith the scale $\\varphi(x) = 1/x$, the same recipe gives the harmonic mean, used for averaging speeds over equal distances.\n",
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