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.
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.)
