---
tag: 5SNCPXVH1J
kind: definition
name: Quasi-arithmetic mean
msc: ["26E60", 39B22]
defines: SG5C5TVGRB
uses: [0Q1D85B8G6, 0X3WGX95FE, 011EJQWWJS, 0J2WAR83WP]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-arithmetic_mean
---

## Definition

Let $I$ be a real [[interval|0Q1D85B8G6]] and let $\varphi \colon I \to \R$ be a [[continuous|0X3WGX95FE]] and [[strictly monotone|011EJQWWJS]] function.
Let $n \ge 1$ and let $x_1, \dots, x_n \in I$.
The *quasi-arithmetic mean* of $x_1, \dots, x_n$ generated by $\varphi$ is
$$M_\varphi(x_1, \dots, x_n) = \varphi^{-1}\left( \frac{1}{n} \sum_{i=1}^{n} \varphi(x_i) \right),$$
where $\varphi^{-1}$ is the [[inverse function|0J2WAR83WP]] of $\varphi$ on the interval $\varphi(I)$.

## Remarks

The value is well defined: the average of the $\varphi(x_i)$ lies between their minimum and maximum, hence in the interval $\varphi(I)$.
Example: on $I = (0, \infty)$ with $\varphi = \ln$, $M_\varphi(x_1, \dots, x_n) = \exp\big(\frac{1}{n} \sum \ln x_i\big) = (x_1 \cdots x_n)^{1/n}$, the geometric mean.
Non-example: $\varphi(x) = x^2$ on $I = \R$ is not strictly monotone, and the formula breaks down.
For the numbers $-1$ and $1$ the value $\varphi^{-1}(1)$ is not defined, since both $-1$ and $1$ map to $1$.
Other standard cases are $\varphi(x) = x$ (arithmetic mean), $\varphi(x) = 1/x$ (harmonic mean) and $\varphi(x) = x^p$ on $(0, \infty)$ with $p \ne 0$ (power mean).
Two functions $\varphi$ and $\psi$ generate the same mean exactly when $\psi = a \varphi + b$ with real $a \ne 0$ and $b$.
The weighted version replaces $\frac{1}{n}$ by weights $w_i \ge 0$ with $\sum_i w_i = 1$.
The mean is also called the Kolmogorov mean or the generalized $f$-mean.