median of a distribution

Given a probability distribution (density) function $f_{X}(x)$ on $\Omega$ over a random variable $X$ , with the associated probability measure $P$ , a median $m$ of $f_{X}$ is a real number such that

1.

$P(X\leq m)\geq\frac{1}{2},$
2.

$P(X\geq m)\geq\frac{1}{2}.$

The median is also known as the $50^{\text{th}}$ -percentile or the second quartile.

Examples:

•

An example from a discrete distribution. Let $\Omega=\mathbb{R}$ . Suppose the random variable $X$ has the following distribution: $P(X=0)=0.99$ and $P(X=1000)=0.01$ . Then we can easily see the median is 0.
•

Another example from a discrete distribution. Again, let $\Omega=\mathbb{R}$ . Suppose the random variable $X$ has distribution $P(X=0)=0.5$ and $P(X=1000)=0.5$ . Then we see that the median is not unique. In fact, all real values in the interval $[0,1000]$ are medians.
•
In practice, however, the median may be calculated as follows: if there are $N$ numeric data points, then by ordering the data values (either non-decreasingly or non-increasingly),
1. (a)
  
  the $(\frac{N+1}{2})$ -th data point is the median if $N$ is odd, and
2. (b)
  
  the midpoint of the $(N-1)$ th and the $(N+1)$ th data points is the median if $N$ is even.
•

The median of a normal distribution (with mean $\mu$ and variance $\sigma^{2}$ ) is $\mu$ . In fact, for a normal distribution, mean = median = mode.
•

The median of a uniform distribution in the interval $[a,b]$ is $(a+b)/2$ .
•

The median of a Cauchy distribution with location parameter t and scale parameter s is the location parameter.
•

The median of an exponential distribution with location parameter $\mu$ and scale parameter $\beta$ is the scale parameter times the natural log of 2, $\beta\operatorname{ln}2$ .
•

The median of a Weibull distribution with shape parameter $\gamma$ , location parameter $\mu$ , and scale parameter $\alpha$ is $\alpha(\operatorname{ln}2)^{1/\gamma}+\mu$ .

Title	median of a distribution
Canonical name	MedianOfADistribution
Date of creation	2013-03-22 14:24:10
Last modified on	2013-03-22 14:24:10
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60A99
Classification	msc 62-07
Synonym	second quartile
Defines	median