Median (statistics): Difference between revisions

Latest revision as of 16:25, 19 February 2025

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

Discrete distributions

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 $F$ .

Continuous distributions

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 $\int^x_{-\infty}f = \frac12$ .

@@ Line 1: / Line 1: @@
-== Definition ==
+A '''median''' is a measure of central tendency used frequently in statistics.
-The '''median''' of a set of numbers is the middle element in a 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 average of the two is taken to be the median. These show up frequently on contest problems, and also in [[statistics]]. For example, to find the median of the set {5, 3, 9, 7}, we would first write it in order {3, 5, 7, 9}. Then, to find the median, we take <math>\displaystyle\frac{5+7}{2}=6</math>
+== Median of a data set ==
+The median of a [[finite]] [[set]] of [[real number]]s <math>\{X_1, ..., X_k\}</math> is defined to be <math>x</math> such that <math>\sum_{i=1}^k |X_i - x| = \min_y \sum_{i=1}^k |X_i - y|</math>. This turns out to be <math>X_{(\frac{k+1}2)}</math> when <math>k</math> is odd. When <math>k</math> is even, all points between <math>X_{(\frac{k}2)}</math> and <math>X_{(\frac{k}2 + 1)}</math> are medians. If we have to specify one median we conventionally take <math>\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2</math>. (Here <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.
+== Median of a distribution ==
+=== Discrete distributions ===
+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 any point lying between <math>x_i</math> and <math>x_{i+1}</math> where <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>.
+=== Continuous distributions ===
+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 ==
-Find the median of {3, 4, 5, 15, 9}.
-(add various problems)
+=== Introductory ===
+*Find the median of <math>\{3, 4, 5, 15, 9\}</math>.
+*[[2000 AMC 12 Problems/Problem 14]]
+*[[2004 AMC 12A Problems/Problem 10]]
+=== Intermediate ===
+=== Olympiad ===
+{{problem}}
+== Video ==
+[//youtu.be/TkZvMa30Juo Video]
+== See Also ==
+* [[Mean of a set]]
+* [[Mode of a set]]
+{{stub}}

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Median (statistics): Difference between revisions

Latest revision as of 16:25, 19 February 2025

Contents

Median of a data set

Median of a distribution

Discrete distributions

Continuous distributions

Problems

Introductory

Intermediate

Olympiad

Video

See Also