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

1.
$P(X\le m)\ge \frac{1}{2},$

2.
$P(X\ge m)\ge \frac{1}{2}.$
The median is also known as the ${50}^{\text{th}}$percentile or the second quartile.
Examples:

•
An example from a discrete distribution. Let $\mathrm{\Omega}=\mathbb{R}$. 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 $\mathrm{\Omega}=\mathbb{R}$. 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 nondecreasingly or nonincreasingly),

(a)
the $(\frac{N+1}{2})$th data point is the median if $N$ is odd, and

(b)
the midpoint of the $(N1)$th and the $(N+1)$th data points is the median if $N$ is even.

(a)

•
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, $\beta \mathrm{ln}2$.

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

Canonical name  MedianOfADistribution 
Date of creation  20130322 14:24:10 
Last modified on  20130322 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 6207 
Synonym  second quartile 
Defines  median 