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 ($M_{3}$) ≤
• Maximum ($M_{+\infty }$)

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.

• Quasi-arithmetic means (generalized ƒ-means): generalize the power means
• Lehmer means
• Stolarsky means: generalize the logarithmic mean
• Schwab-Borchardt means

Robust estimators

See also: Robust statistics, Trimmed estimator, Winsorizing

• L-estimators
  • Trimean
  • Midhinge
  • Trimmed mean
    • Interquartile mean
  • Winsorized mean
  • Modified mean
• M-estimators

Other measures of central tendency

See also: Central tendency

• Mode
• Median
• Geometric median

See also

• LogSumExp