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Generalized mean

A generalized mean or power mean is an abstraction of the arithmetic, geometric and harmonic means.

If t is a non-zero real number, we can define the generalized mean with exponent t of the positive real numbers a₁,...,a_n as

$M(t) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^t \right) ^ {\frac{1}{t}}$

The case t = 1 yields the arithmetic mean and the case t = −1 yields the harmonic mean. As t approaches 0, the limit of M(t) is the geometric mean of the given numbers, and so it makes sense to define M(0) to be the geometric mean. Furthermore, as t approaches ∞, M(t) approaches the maximum of the given numbers, and as t approaches −∞, M(t) approaches the minimum of the given numbers.

In general, if −∞ ≤ s < t ≤ ∞, then

$M(s)\leq M(t)$

and the two means are equal if and only if a₁ = a₂ = ... = a_n. Furthermore, if a is a positive real number, then the generalized mean with exponent t of the numbers aa₁,..., aa_n is equal to a times the generalized mean of the numbers a₁,..., a_n.

This could be generalized further to the generalized f-mean:

$M = f^{-1}\left({\frac{1}{n}\sum_{i=1}^n{f(x_i)}}\right)$

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), the harmonic mean with f(x) = 1/x, and the generalized mean with exponent t with f(x) = x^t. But other functions could be used, such as f(x) = e^x.

Generalized mean

See also