Quasi-arithmetic mean

Summary

A quasi-arithmetic mean is the arithmetic mean taken after a change of scale. Given a continuous strictly monotone function $\varphi$, the mean of $x_1, \dots, x_n$ is $\varphi^{-1}\big(\frac{1}{n} \sum_i \varphi(x_i)\big)$. The arithmetic, geometric, harmonic and power means are all of this form. These means are characterized by a few natural axioms, which is the Kolmogorov–Nagumo theorem.