# Median

In statistics and probability theory, median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values.

A median is only defined on one-dimensional data, and is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.

In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size); if there is such a member, there may be more than one so that the median may not uniquely identify a sample member. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid. At most, half the population have values strictly less than the median, and, at most, half have values strictly greater than the median. If each group contains less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {a, b, c} is b. If a <> b <> c as well, then only a is strictly less than the median, and only c is strictly greater than the median. Since each group is less than half (one-third, in fact), the leftover b is strictly equal to the median (a truism).

Likewise, if a < b < c < d, then the median of the list {a, b, c, d} is the mean of b and c; i.e., it is (b + c)/2.

The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

In terms of notation, some authors represent the median of a variable x either as or as There is no simple, widely accepted standard notation for the median, so the use of these or other symbols for the median needs to be explicitly defined when they are introduced.

### Other articles related to "median":

Median - History
... The idea of the median originated in Edward Wright's book on navigation (Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass ... developed a regression method based on the L1 norm and therefore implicitly on the median ... of both the sample mean and the sample median were determined by Laplace in the early 1800s ...
Wyoming County, New York - Demographics
... The median age was 37 years ... The median household income was \$39,895, and the median family income was \$45,088 ... Males had a median income of \$31,973 versus \$22,252 for females ...
Woodland Hills, Los Angeles - Demographics
... Median household income in 2000 was \$72,568 ... Median home cost in ZIP 91364 is (2007) \$944,500 and cost of living in ZIP 91364 is (2007) 76.26% higher than the U.S ... supplied these Woodland Hills neighborhood statistics population 59,661 median household income \$89,946 ...
Binomial Distribution - Mode and Median
... can be summarized as follows In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique ... If np is an integer, then the mean, median, and mode coincide and equal np ... Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉ ...
Union County, Tennessee - Demographics
... The median age was 36 years ... The median household income was \$27,335 and the median family income was \$31,843 ... Males had a median income of \$26,436 versus \$18,665 for females ...