Media generalizada

La media generalizada es una abstracción de los diversos tipos de media (geométrica, aritmética, armónica, etc). Se define como:^[1]​

${\displaystyle M_{m}(x_{1},\dots ,x_{n})={\begin{cases}\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{m}\right)^{\frac {1}{m}}&{\mbox{si}}\,m\not =0\\\left(\prod _{i=1}^{n}{x_{i}}\right)^{\frac {1}{n}}&{\mbox{si}}\,m=0\end{cases}}}$

En donde ciertos valores del parámetro m se corresponden con otro tipo de medias:

${\displaystyle M_{\infty }(x_{1},\dots ,x_{n})=\max(x_{1},\dots ,x_{n})}$

${\displaystyle M_{2}(x_{1},\dots ,x_{n})=}$ media cuadrática

${\displaystyle M_{1}(x_{1},\dots ,x_{n})=}$ media aritmética

${\displaystyle M_{0}(x_{1},\dots ,x_{n})=}$ media geométrica

${\displaystyle M_{-1}(x_{1},\dots ,x_{n})=}$ media armónica

${\displaystyle M_{-\infty }(x_{1},\dots ,x_{n})=\min(x_{1},\dots ,x_{n})}$

1. ↑ Cf. "Media generalizada". Merigó. The Generalized Hybrid Averaging Operator and its Application in Decision Making