Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Measures of central tendency ▷ Median

Definition: The median of a sample or random variable is the value separating the higher half from the lower half of its values.

1) Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a random variable $X$. Then, the median of $x$ is

\[\label{eq:med-samp} \mathrm{median}(x) = \left\{ \begin{array}{cl} x_{(n+1)/2} \; , & \text{if} \; n \; \text{is odd} \\ \frac{1}{2}(x_{n/2} + x_{n/2+1}) \; , & \text{if} \; n \; \text{is even} \; , \end{array} \right.\]

i.e. the median is the “middle” number when all numbers are sorted from smallest to largest.

2) Let $X$ be a continuous random variable with cumulative distribution function $F_X(x)$. Then, the median of $X$ is

\[\label{eq:med-rvar} \mathrm{median}(X) = x, \quad \mathrm{s.t.} \quad F_X(x) = \frac{1}{2} \; ,\]

i.e. the median is the value at which the CDF is $1/2$.


Metadata: ID: D101 | shortcut: med | author: JoramSoch | date: 2020-10-15, 10:53.