Difference between revisions of "Median (statistics)"
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| − | + | ''This article is about the median used in statistics. For other medians, check [[Median (disambiguation)]].'' | |
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| + | A '''median''' is a measure of central tendency used frequently in statistics. | ||
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| + | == 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. | ||
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| + | == Median of a distribution == | ||
| + | === Discrete distributions === | ||
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| + | 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>. | ||
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| + | === Continuous distributions === | ||
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| + | 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>. | ||
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| + | == Problems == | ||
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| + | === Introductory === | ||
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| + | *Find the median of <math>\{3, 4, 5, 15, 9\}</math>. | ||
| + | *[[2000 AMC 12 Problems/Problem 14]] | ||
| + | *[[2004 AMC 12A Problems/Problem 10]] | ||
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| + | === Intermediate === | ||
| + | === Olympiad === | ||
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| + | {{problem}} | ||
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| + | == Video == | ||
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| + | [//youtu.be/TkZvMa30Juo Video] | ||
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| + | == See Also == | ||
| + | |||
| + | * [[Mean of a set]] | ||
| + | * [[Mode of a set]] | ||
| + | |||
| + | {{stub}} | ||
Revision as of 16:24, 19 February 2025
This article is about the median used in statistics. For other medians, check Median (disambiguation).
A median is a measure of central tendency used frequently in statistics.
Contents
Median of a data set
The median of a finite set of real numbers 
is defined to be 
such that 
. This turns out to be 
when 
is odd. When is even, all points between 
and 
are medians. If we have to specify one median we conventionally take 
. (Here 
denotes the 
order statistic.) For example, the median of the set 
is 7.
Median of a distribution
Discrete distributions
If 
is a discrete distribution, whose support is a subset of a countable set 
, with 
for all positive integers 
, the median of is any point lying between 
and 
where 
and 
. If 
for some , is defined to be the median of .
Continuous distributions
If is a continuous distribution, whose support is a subset of the real numbers, the median of is defined to be the such that 
. Clearly, if has a density 
, this is equivalent to saying 
.
Problems
Introductory
- Find the median of
. - 2000 AMC 12 Problems/Problem 14
- 2004 AMC 12A Problems/Problem 10
.Intermediate
Olympiad
This problem has not been edited in. Help us out by adding it.
Video
See Also
This article is a stub. Help us out by expanding it.
