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.