Difference between revisions of "Median (statistics)"

Revision as of 06:05, 25 November 2007

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 $X_{(\frac{k+1}2)}$ when $k$ is odd and $\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2$ when $k$ is even, where $X_{(i)}, i \in \{1,...,k\}$ 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 said to lie between $x_i$ and $x_{i+1}$ iff $F(x_i)\leq\frac12$ 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

This page is in need of some relevant examples or practice problems. Help us out by adding some. Thanks.

@@ Line 7: / Line 7: @@
 === Median of a discrete distribution ===
-If <math>F</math> is a discrete distribution, whose support is a subset of a countable set <math>{x_1, x_2, x_3, ...}</math>, with <math>x_i < x_{i+1}</math> for all positive integers <math>i</math>, the median of <math>F</math> is said to lie between <math>x_i</math> and <math>x_{i+1}</math> iff <math>F(x_i)\leq\frac12</math> and <math>F(x_{i+1})\geq\frac12</math>. If <math>F(x_i)=\frac12</math> for some <math>i</math>, <math>x_i</math> is defiend to be the median of <math>F</math>.
+If <math>F</math> is a [[discrete distribution]], whose [[support]] is a subset of a [[countable]] set <math>{x_1, x_2, x_3, ...}</math>, with <math>x_i < x_{i+1}</math> for all positive integers <math>i</math>, the median of <math>F</math> is said to lie between <math>x_i</math> and <math>x_{i+1}</math> iff <math>F(x_i)\leq\frac12</math> and <math>F(x_{i+1})\geq\frac12</math>. If <math>F(x_i)=\frac12</math> for some <math>i</math>, <math>x_i</math> is defined to be the median of <math>F</math>.
+=== Median of a continuous distribution ===
+If <math>F</math> is a [[continuous distribution]], whose support is a subset of the real numbers, the median of <math>F</math> is defined to be the <math>x</math> such that <math>F(x)=\frac12</math>. Clearly, if <math>F</math> has a [[density]] <math>f</math>, this is equivalent to saying <math>\int^x_{-\infty}f = \frac12</math>.
 == Problems ==

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