Mean

In statistics, mean has two related meanings:

Generalized mean

The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the Arithmetic, Geometric and Harmonic Means.

Related Topics:
Generalized mean - Power mean - Hölder mean

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: ar{x}(m) = sqrt{rac{1}{n}sum_{i=1}^n{x_i^m}}

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By choosing the appropriate value for the parameter m we can get the arithmetic mean (m = 1), the geometric mean (m -> 0) or the harmonic mean (m = -1)

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This could be generalised further as

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: ar{x} = f^{-1}left({rac{1}{n}sum_{i=1}^n{f(x_i)}} ight)

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and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x)=x, the geometric mean with f(x)=log(x), and the harmonic mean with f(x)=1/x.

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