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

## Other means

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

• $M_{-1}\leq$ Geometric–harmonic mean $\leq M_{0}\leq$ Arithmetic–geometric mean $\leq ^{?}$ Heronian mean $\leq M_{1}$ • $M_{0}\leq$ Logarithmic mean $\leq M_{1/3}$ • $M_{2/3}\leq$ Idcentric mean $\leq M_{1}$ agm seems to be $\leq M_{1/2}$ ghm seems to be $\geq M_{-1/2}$ ## Families of means

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