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 $50^{{th}}$ -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. the $(\frac{N+1}{2})$ -th data point is the median if $N$ is odd, and
  2. 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 $\alpha(\operatorname{ln}2)^{1/\gamma}+\mu$ .