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Power Mean

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A power mean is a mean of the form

M_p(a_1,a_2,...,a_n)=(1/nsum_(k=1)^na_k^p)^(1/p),
(1)

where the parameter p is an affinely extended real number and all a_k>=0. A power mean is also known as a generalized mean, Hölder mean, mean of degree (or order or power) p, or power mean.

The following table summarizes some common named means that are special cases of the generalized mean, where

M_0(a_1,a_2,...,a_n)=lim_(p->0)M_p(a_1,a_2,...,a_n)
(2)

and

M_(-infty)(a_1,a_2,...,a_n)=lim_(p->-infty)M_p(a_1,a_2,...,a_n)
(3)
=min(a_1,a_2,...,a_n)
(4)
M_infty(a_1,a_2,...,a_n)=lim_(p->infty)M_p(a_1,a_2,...,a_n)
(5)
=max(a_1,a_2,...,a_n).
(6)
M_psymbolmean
M_(-infty)minminimum
M_(-1)Hharmonic mean
M_0Ggeometric mean
M_1Aarithmetic mean
M_2RMSroot-mean-square
M_inftymaxmaximum
GeneralizedMeans

The plots above visualize the generalized mean by plotting the special values

M_p(x,1)={x for p=-infty; (2x)/(1+x) for p=-1; sqrt(x) for p=0; (1+x)/2 for p=1; sqrt((1+x^2)/2) for p=2; 1 for p=infty
(7)

with p=-infty red, -1 orange, 0 black, 1 green, 2 blue, and infty violet.

See also

Arithmetic Mean, Geometric Mean, Harmonic Mean, Mean, Pythagorean Means, Root-Mean-Square

Portions of this entry contributed by David W. Cantrell

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References

Borwein, J. M. and Borwein, P. B. "General Means and Iterations." Ch. 8 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Bullen, P. S. "The Power Means." Ch. 3 in Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 121, 2003.

Referenced on Wolfram|Alpha

Power Mean

Cite this as:

Cantrell, David W. and Weisstein, Eric W. "Power Mean." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PowerMean.html

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