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# User:Charles R Greathouse IV/Averages

This is a collection of different types of averages.

## Power means

This is a list of generalized means, also known as power means or Hölder means. They include the three Pythagorean means: arithmetic, geometric, and harmonic.

## Other means

• $M_2 \le$ Contraharmonic mean $\le M_{+\infty}$ (seems to be larger than M3 for the 2-variable case)

## Two variables only

These means are usually defined for only two variables. It's unclear if they can be meaningfully extended to arbitrary finite submultisets of $\mathbb{R}$.

• $M_{-1} \le$ Geometric–harmonic mean $\le M_0 \le$ Arithmetic–geometric mean $\le^?$ Heronian mean $\le M_1$
• $M_0 \le$ Logarithmic mean $\le M_{1/3}$
• $M_{2/3} \le$ Idcentric mean $\le M_1$

agm seems to be $\le M_{1/2}$ ghm seems to be $\ge M_{-1/2}$

## Families of means

Beyond these families, most means can be extended to a weighted version.