Power Mean

A power mean is a mean of the form

(1)

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

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

(2)

and

			(3)
			(4)
			(5)
			(6)

	symbol	mean
		minimum
		harmonic mean
		geometric mean
		arithmetic mean
	RMS	root-mean-square
		maximum

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 red, orange, 0 black, 1 green, 2 blue, and violet.

Wolfram Web Resources

Mathematica » The #1 tool for creating Demonstrations and anything technical.	Wolfram\|Alpha » Explore anything with the first computational knowledge engine.	Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Computerbasedmath.org » Join the initiative for modernizing math education.	Online Integral Calculator » Solve integrals with Wolfram\|Alpha.	Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.	Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.	Wolfram Language » Knowledge-based programming for everyone.