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| − | ''This article is about the median used in statistics. For other medians, check [[Median (disambiguation)]].''
| + | * [[Median (geometry)]] |
| − | | + | * [[Median (statistics)]] |
| − | A '''median''' is a measure of central tendency used frequently in statistics.
| + | {{disambig}} |
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| − | == Median of a data set ==
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| − | 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 ==
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| − | === 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>.
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| − | *[[2000 AMC 12 Problems/Problem 14]]
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| − | *[[2004 AMC 12A Problems/Problem 10]]
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| − | === Intermediate ===
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| − | === 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 ==
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| − | * [[Mean of a set]]
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| − | * [[Mode of a set]]
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| − | {{stub}} | |
