Mean

There are several statistical quantities called means, e.g., harmonic mean, geometric mean, arithmetic-geometric mean, and root-mean-square. When applied to two elements  and  with , these means satisfy

(1)

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

mean	value
harmonic mean
geometric mean
arithmetic-geometric mean
arithmetic mean
root-mean-square

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

(2)

also called the (unweighted) average. Notations for "the" mean of a set  of  values include macron notation  or . The expectation value notation  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  of a set of numbers  satisfies

(3)

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

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

(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  and 6 is 0, so "on average," the two shots hit the duck.)