You are here
Homemedian of a distribution
Primary tabs
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.

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 $\alpha(\operatorname{ln}2)^{{1/\gamma}}+\mu$.
Mathematics Subject Classification
60A99 no label found6207 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections