The arithmetic mean of a set of values is the quantity commonly called "the" mean or the average. Given a set of samples 
, the arithmetic mean is
 |
(1)
|
It can be computed in the Wolfram Language using Mean[list].
The arithmetic mean is the special case 
of the power mean and is one
of the Pythagorean means.
When viewed as an estimator for the mean of the underlying distribution (known as the population mean), the arithmetic mean of a sample is called the sample mean.
For a continuous distribution function, the arithmetic mean of the population, denoted 
, 
, 
, or 
and called the population
mean of the distribution, is given by
 |
(2)
|
where 
is the expectation value. Similarly, for a discrete distribution,
 |
(3)
|
The arithmetic mean satisfies
 |
(4)
|
 |
(5)
|
and
 |
(6)
|
if 
and 
are independent statistics. The "sample
mean," which is the mean estimated from a statistical sample, is an unbiased
estimator for the population mean.
Hoehn and Niven (1985) show that
 |
(7)
|
for any constant 
.
For positive arguments, the arithmetic mean satisfies
 |
(8)
|
where 
is the geometric mean and 
is the harmonic mean (Hardy
et al. 1952, Mitrinović 1970, Beckenbach and Bellman 1983, Bullen et
al. 1988, Mitrinović et al. 1993, Alzer 1996). This can be shown
as follows. For 
,
 |
(9)
|
 |
(10)
|
 |
(11)
|
 |
(12)
|
 |
(13)
|
with equality iff 
. To show the second part of the inequality,
 |
(14)
|
 |
(15)
|
 |
(16)
|
with equality iff 
. Combining (◇) and (◇) then gives (◇).
Given 
independent random normally distributed variates 
, each with population mean 
and variance 
,
 |
(17)
|
 |  |  |
(18)
|
 |  |  |
(19)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
so the sample mean is an unbiased estimator of the population mean. However, the distribution of 
depends on the sample size. For large samples, 
is approximately normal.
For small samples, Student's t-distribution
should be used.
The variance of the sample mean is independent of the distribution, and is given by
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
 |  |  |
(26)
|
 |  |  |
(27)
|
For small samples, the sample mean is a more efficient estimator of the population mean than the statistical median, and approximately
less (Kenney and Keeping 1962, p. 211).
Here, an estimator of a parameter of a probability distribution is said to be more
efficient than another one if it has a smaller variance.
In this case, the variance of the sample mean is generally less than the variance
of the sample median. The relative efficiency of two estimators is the ratio of this
variance.
A general expression that often holds approximately is
 |
(28)
|
(Kenney and Keeping 1962).
