Generalized mean

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In mathematics, a generalized mean, also known as power mean or Hölder mean (named after Otto Hölder), is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

Contents

Definition

If p is a non-zero real number, we can define the generalized mean with exponent p (or power mean with exponent p) of the positive real numbers $x_1,\dots,x_n$ as:

While for p equal to 0 we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero):

Furthermore, for a sequence of positive weights wi we can define weighted power means as follows:

For the sake of simplicity, we might assume that the weights are normalized so that they sum up to 1 (which can be easily done by dividing each weight by their sum), thus allowing some terms in the above formulae to be omitted:

The unweighted means can be easily produced by assuming that all weights equal 1/n. For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes):

Properties

• Like most means, the generalized mean is a homogeneous function of its arguments $x_1,\dots,x_n$. That is, if b is a positive real number, then the generalized mean with exponent p of the numbers $b\cdot x_1,\dots, b\cdot x_n$ is equal to b times the generalized mean of the numbers $x_1,\dots, x_n$.
• Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.