Median

A median is a measure of central tendency used frequently in statistics.

Median of a data set

The median of a finite set of real numbers $\{X_1, ..., X_k\}$ is defined to be $x$ such that $\sum_{i=1}^k |X_i - x| = \min_y \sum_{i=1}^k |X_i - y|$ . This turns out to be $X_{(\frac{k+1}2)}$ when $k$ is odd. When $k$ is even, all points between $X_{(\frac{k}2)}$ and $X_{(\frac{k}2 + 1)}$ are medians. If we have to specify one median we conventionally take $\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2$ . (Here $X_{(i)}, i \in \{1,...,k\}$ denotes the $k^{th}$ order statistic.) For example, the median of the set $\{2, 3, 5, 7, 11, 13, 17\}$ is 7.

Median of a distribution

Median of a discrete distribution

If $F$ is a discrete distribution, whose support is a subset of a countable set ${x_1, x_2, x_3, ...}$ , with $x_i < x_{i+1}$ for all positive integers $i$ , the median of $F$ is any point lying between $x_i$ and $x_{i+1}$ where $F(x_i)\leq\frac12$ and $F(x_{i+1})\geq\frac12$ . If $F(x_i)=\frac12$ for some $i$ , $x_i$ is defined to be the median of $F$ .

Median of a continuous distribution

If $F$ is a continuous distribution, whose support is a subset of the real numbers, the median of $F$ is defined to be the $x$ such that $F(x)=\frac12$ . Clearly, if $F$ has a density $f$ , this is equivalent to saying $\int^x_{-\infty}f = \frac12$ .

Problems

Pre-introductory

Find the median of $\{3, 4, 5, 15, 9\}$ .

Introductory

Intermediate

Olympiad

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