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GeezerVison is showing mathworld.wolfram.com — Mean -- from Wolfram MathWorld

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Mean

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The name "mean" is used for several statistical quantities, e.g., harmonic mean, geometric mean, arithmetic-geometric mean, and root-mean-square. When applied to two elements a and b with 0<a<=b, these means satisfy

H(a,b)<=G(a,b)<=AGM(a,b)<=A(a,b)<=RMS(a,b).
(1)

The following table summarizes these means (again applied to two elements a and b with 0<a<=b), where K(k) is a complete elliptic integral of the first kind.

meanvalue
harmonic mean(2ab)/(a+b)
geometric meansqrt(ab)
arithmetic-geometric mean((a+b)pi)/(4K((a-b)/(a+b)))
arithmetic mean(a+b)/2
root-mean-squaresqrt((a^2+b^2)/2)

The quantity commonly referred to as "the" mean of a set of values is the arithmetic mean

x^_=1/nsum_(i=1)^nx_i,
(2)

also called the (unweighted) average. Notations for "the" mean of a set {x_i} of n values include macron notation x^_ or x^__n. The expectation value notation <x> is sometimes also used. The mean of a list of data (i.e., the sample mean) is implemented as Mean[list].

In general, a mean is a homogeneous function that has the property that a mean mu of a set of numbers x_i satisfies

min(x_1,...,x_n)<=mu<=max(x_1,...,x_n).
(3)

The term function centroid is sometimes used to refer to an analogous quantity for a function f(x) that is not necessarily a probability density function.

Central moments are moments taken about the population mean, i.e.,

mu_r=<(x-mu)^r>.
(4)

A joke told about the mean runs as follows. Two statisticians are out hunting when one of them sees a duck. The first takes aim and shoots, but the bullet goes sailing past six inches too high. The second statistician also takes aim and shoots, but this time the bullet goes sailing past six inches too low. The two statisticians then give one another high fives and exclaim, "Got him!" (This joke plays on the fact that the mean of -6 and 6 is 0, so "on average," the two shots hit the duck.)

See also

Arithmetic-Geometric Mean, Arithmetic Mean, Central Moment, Function Centroid, Geometric Mean, Harmonic Mean, Lehmer Mean, Mean Deviation, Pearson Mode Skewness, Power Mean, Reversion to the Mean, Root-Mean-Square, Sample Mean, Standard Deviation, Stolarsky Mean, Trimean, Variance Explore this topic in the MathWorld classroom

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More things to try:

  • mean value theorem
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  • geometric mean 12, 17, 31

Cite this as:

Weisstein, Eric W. "Mean." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Mean.html

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