# What Is The Median

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"What Is The Median"

```					                            What Is The Median
What Is The 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), and, 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.
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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.

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 location parameter.

When the median is used as a location parameter in descriptive statistics, there are several
choices for a measure of variability: the range, the interquartile range, the mean absolute
deviation, and the median absolute deviation. Since the median is the same as the second
quartile, its calculation is illustrated in the article on quartiles.

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.

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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 efficiency of candidate estimators shows that the sample
mean is more statistically efficient than the sample median when data are uncontaminated by
data from heavy-tailed distributions or from mixtures of distributions, but less efficient
otherwise, and that the efficiency of the sample median is higher than that for a wide range of
distributions.

Math.Tutorvista.com                                                   Page No. :- 4/4
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