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 $\mathrm{\Omega }=ℝ$. 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 $\mathrm{\Omega }=ℝ$. 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, $\beta \ln 2$.

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