Median¶

Definition¶

Let $X$ be a random variable with distribution function $F_X$ . A median of $X$ is a real number $m$ such that

\mathbb{P}(X\le m)\ge \frac{1}{2}\quad\text{and}\quad \mathbb{P}(X\ge m)\ge \frac{1}{2}.

Interpretation¶

A median is a central value that splits the distribution into two halves. Unlike the mean, it depends only on the order of the values, not on how far out the extreme ones lie.

A right-skewed density split by a vertical line at the median, about 1.68, into two shaded parts of area 1/2 each, with a dashed line at the mean 2.

The density is $f(x)=x\mathrm{e}^{-x}$ for $x\ge0$ $0$ , with CDF $F(x)=1-(1+x)\mathrm{e}^{-x}$ $x) e^{- x}$ . The solid line at $m\approx1.68$ $1.68$ , the solution of $F(m)=\tfrac{1}{2}$ $\frac{1}{2}$ , cuts the area into two parts of $\tfrac{1}{2}$ each, and the dashed line marks the mean $2$ . The long right tail pulls the mean outwards but does not move the halfway point.

Move the candidate $m$ and check both conditions. For the skewed density the median is unique; for $\operatorname{Bin}(3,\frac12)$ every $m\in[1,2]$ is a median.

The Definition uses two inequalities rather than the equation $F_X(m)=\tfrac{1}{2}$ $\frac{1}{2}$ , because that equation can have many solutions or none. For $\operatorname{Bin}(3,\tfrac{1}{2})$ the CDF equals $\tfrac{1}{2}$ on all of $[1,2)$ , and every $m\in[1,2]$ satisfies both conditions. For $\operatorname{Bern}(0.3)$ the CDF jumps from $0$ to $0.7$ at $0$ , so the equation has no solution, yet $0$ is the unique median. For a continuous, strictly increasing CDF the median is simply the quantile at level $\tfrac{1}{2}$ .

Because only the order of the values matters, medians move along with increasing transformations: if $m$ is a median of $Y$ and $g$ is strictly increasing, then $g(m)$ is a median of $g(Y)$ . This gives the median $\mathrm{e}^{\mu}$ of the lognormal distribution. For a symmetric distribution, the centre of symmetry is a median.

Key Results¶

A median need not be unique: the medians of a distribution form an interval, which contains more than one point exactly when the CDF equals $\tfrac{1}{2}$ on a whole stretch (for a fair die, every point of $[3,4]$ is a median). If the CDF jumps from below $\tfrac{1}{2}$ to above $\tfrac{1}{2}$ at a point, that point is the unique median.
For a continuous strictly increasing CDF, the median is the point where the CDF equals one half.
The median is robust to extreme values compared with the mean.

Examples¶

The median of a symmetric normal distribution is its centre.
For waiting times, the median is the time by which half of cases have occurred.
In income data, the median can better describe a typical person than the mean when there are extreme high incomes.

Prerequisites¶

Cumulative Distribution Function

Sources¶

Probability Theory for EOR · Week 6

Aliases: 0.5-quantile, second quartile, 0.5 quantile, 50th percentile Keywords: median of a random variable Symbols: element of, mu

Last updated: 22 October 2026 19:04