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Generalized mean

From Academic Kids

A generalized mean, also known as power mean or H÷lder 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,...,an as

<math>

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

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

<math>M(s)\leq M(t)<math>

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

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

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

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) = xt. But other functions could be used, such as f(x) = ex.

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