The median of a statistical distribution with distribution function 
is the value 
such 
.
For a symmetric distribution, it is therefore equal to the mean.
Given order statistics 
, 
, ..., 
, 
, the statistical median of the random sample
is defined by
 |
(1)
|
(Hogg and Craig 1995, p. 152) and commonly denoted 
or 
. The median of a list of data is implemented as Median[list].
For a normal population, the mean 
is the most efficient (in the sense that no other unbiased
statistic for estimating 
can have smaller variance) estimate
(Kenney and Keeping 1962, p. 211). The efficiency of the median, measured as
the ratio of the variance of the mean to the variance of the median, depends on the
sample size 
as
 |
(2)
|
which tends to the value 
as 
becomes large (Kenney and Keeping 1962, p. 211). Although,
the median is less efficient than the mean, it is less sensitive
to outliers than the mean
For large 
samples with population median 
,
 |  |  |
(3)
|
 |  |  |
(4)
|
The median is an L-estimate (Press et al. 1992).
An interesting empirical relationship between the mean, median, and mode which appears to hold for unimodal curves of moderate asymmetry is given by
 |
(5)
|
(Kenney and Keeping 1962, p. 53), which is the basis for the definition of the Pearson mode skewness.
