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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.

• Minimum (${\displaystyle M_{-\infty }}$)
• Harmonic mean (${\displaystyle M_{-1}}$) ≤
• Geometric mean (${\displaystyle M_{0}}$) ≤
• Arithmetic mean (${\displaystyle M_{1}}$) ≤
• Quadratic mean (${\displaystyle M_{2}}$) ≤
• Cubic mean (${\displaystyle M_{3}}$) ≤
• Maximim (${\displaystyle M_{+\infty }}$)

## Other means

• ${\displaystyle M_{2}\leq }$ Contraharmonic mean ${\displaystyle \leq M_{+\infty }}$ (seems to be larger than ${\displaystyle M_{3}}$ 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 ${\displaystyle \mathbb {R} }$.

• ${\displaystyle M_{-1}\leq }$ Geometric–harmonic mean ${\displaystyle \leq M_{0}\leq }$ Arithmetic–geometric mean ${\displaystyle \leq ^{?}}$ Heronian mean ${\displaystyle \leq M_{1}}$
• ${\displaystyle M_{0}\leq }$ Logarithmic mean ${\displaystyle \leq M_{1/3}}$
• ${\displaystyle M_{2/3}\leq }$ Idcentric mean ${\displaystyle \leq M_{1}}$

agm seems to be ${\displaystyle \leq M_{1/2}}$ ghm seems to be ${\displaystyle \geq M_{-1/2}}$

## Families of means

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