Statistical Median

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.

REFERENCES:

Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, 1995.

Huang, J. S. "Third-Order Expansion of Mean Squared Error of Medians." Stat. Prob. Let. 42, 185-192, 1999.

Kenney, J. F. and Keeping, E. S. "The Median," "Relation Between Mean, Median, and Mode," "Relative Merits of Mean, Median, and Mode," and "The Median." §3.2, 4.8-4.9, and 13.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 32-35, 52-54, 211-212, 1962.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 694, 1992.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995.