Median

Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$. A median of $X$ is a real number $m$ with $$P(X \le m) \ge \frac{1}{2} \quad \text{and} \quad P(X \ge m) \ge \frac{1}{2}.$$

Remarks

Each real random variable has a median, and the medians form a closed bounded interval. For example, $Q(1/2)$ is a median, where $Q$ is the quantile function. The median can be different from the mean. For the exponential distribution with rate $\lambda$, the median is $\ln 2 / \lambda$ and the mean is $1/\lambda$. For a fair coin coded as $0$ and $1$, each $m \in [0, 1]$ is a median. For a sorted data set $x_1 \le \dots \le x_n$ with $n$ odd, the median is $x_{(n+1)/2}$. For $n$ even, one usually takes the mean of the two middle values.

Explanations

L1 human

The median is the middle value: half of the chance lies on each side of it. The median of the incomes in a country is often more useful than the mean. A few very large incomes do not move the median much.