# The Median (PDF)

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```					                                  The Median
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.

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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.

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The sample median

Even though sorting n items requires O(n log n) operations, selection algorithms can
compute the kth-smallest of n items (e.g., the median) with only O(n) operations.[8]

In individual series (if number of observation is very low) first one must arrange all the
observations in ascending order. Then count(n) total number of observation in given
data.

If n is odd then Median (M) = value of ((n + 1)/2)th item term.
If n is even then Median (M) = value of [((n)/2)th item term + ((n)/2 + 1)th item term ]/2

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