Difference between revisions of "Median (statistics)"

Revision as of 06:54, 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 defiend to be the median of $F$ .

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.

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-A '''median''' is a common type of [[mean]] for a set of numbers.
+A '''median''' is a measure of central tendency used frequently in statistics.
-== Definition ==
+== Median of a data set ==
-The median of a [[finite]] [[set]] of [[real number]]s is the middle [[element]] of the set when the elements are written in order (i.e. least to greatest).  When the number of elements is even, there are two middle elements and so the [[arithmetic mean]] of the two is taken to be the median.
+The median of a [[finite]] [[set]] of [[real number]]s <math>\{X_1, ..., X_k\}</math> is defined to be <math>X_{(\frac{k+1}2)}</math> when <math>k</math> is odd and <math>\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2</math> when <math>k</math> is even, where <math>X_{(i)}, i \in \{1,...,k\}</math> denotes the <math>k^{th}</math> [[order statistic]]. For example, the median of the set <math>\{2, 3, 5, 7, 11, 13, 17\}</math> is 7.
-For example, the median of the set <math>\{2, 3, 5, 7, 11, 13, 17\}</math> is 7.  In order to find the median of the set <math>\{5, 3, 9, 7\}</math>, we would first write it in order <math>\{3, 5, 7, 9\}</math>. Then, to find the median, we take <math>\frac{5+7}{2}=6</math>.
+== Median of a distribution ==
+=== Median of a discrete distribution ===
-The median is one of several different [[mean]]s for a set of numbers.  It appears most frequently in the field of [[statistics]] and also occasionally on mathematical contests.
+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>.
 == Problems ==

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