Median |
The median is the value separating the higher half from the lower half of a data
The median is a commonly used measure of the properties of a data set in
Because of this, the median is of central importance in
The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest.
If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers
This list contains seven numbers. The median is the fourth of them, which is 6.
If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the
the median is the mean of the middle two numbers: this is , which is . (In more technical terms, this interprets the median as the fully
The formula used to find the index of the middle number of a data set of n numerically ordered numbers is . This either gives the middle number (for an odd number of values) or the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, and the median will be taken by averaging the seventh (the floor of this index) and eighth (the ceiling of this index) values. So the median can be represented by the following formula:
Type | Description | Example | Result |
---|---|---|---|
Sum of values of a data set divided by number of values: | (1 + 2 + 2 + 3 + 4 + 7 + 9) / 7 | 4 | |
Median | Middle value separating the greater and lesser halves of a data set | 1, 2, 2, 3, 4, 7, 9 | 3 |
Most frequent value in a data set | 1, 2, 2, 3, 4, 7, 9 | 2 |
One can find the median using the
There is no widely accepted standard notation for the median, but some authors represent the median of a variable x either as x͂ or as μ_{1/2}^{[1]} sometimes also M.^{[3]}^{[4]} In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced.
The median is used primarily for
The median is a popular
With an even number of observations (as shown above) no value need be exactly at the value of the median. 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
In a population, at most half have values strictly less than the median and at most half have values strictly greater than it. If each set 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, and, 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. As a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.^{[6]}
The median can be used as a measure of
A median is only defined on
The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible
When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the
For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the