Definition:Median (Statistics)

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This page is about median in the context of statistics. For other uses, see median.

Definition

Let $S$ be a set of ordinal data.

Let $S$ be arranged in order of size.

The median is the element of $S$ that is in the middle of that ordered set.


Suppose there are an odd number of elements of $S$ such that $S$ has cardinality $2 n - 1$.

The median of $S$ in that case is the $n$th element of $S$.


Suppose there are an even number of elements of $S$ such that $S$ has cardinality $2 n$.

Then the middle of $S$ is not well-defined, and so the median of $S$ in that case is the arithmetic mean of the $n$th and $n + 1$th elements of $S$.


Continuous Random Variable

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have probability density function $f_X$.

A median of $X$ is defined as a real number $m_X$ such that:

$\ds \map \Pr {X < m_X} = \int_{-\infty}^{m_X} \map {f_X} x \rd x = \frac 1 2$

That is, $m_X$ is the first $2$-quantile of $X$.

Hence it is also the $50$th percentile of $X$.


Examples

Arbitrary Example $1$

Let $S = \set {1, 7, 31}$ be a set of raw data.

The median of $S$ is $7$.


Arbitrary Example $2$

Let $S = \set {2, 5, 9, 16}$ be a set of raw data.

The median of $S$ is $\dfrac {5 + 9} 2 = 7$.


Also see

  • Results about medians can be found here.


Sources

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