Difference between revisions of "Median"
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+ | ''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. | ||
+ | ==Mean, Median, Mode== | ||
+ | https://youtu.be/TkZvMa30Juo?t=1504 | ||
+ | |||
+ | == 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 == | ||
+ | === 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 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>. | ||
+ | |||
+ | === 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 == | ||
+ | ===Pre-introductory=== | ||
+ | Find the median of <math>\{3, 4, 5, 15, 9\}</math>. | ||
+ | ===Introductory=== | ||
+ | [[2000 AMC 12 Problems/Problem 14]] | ||
+ | |||
+ | [[2004 AMC 12A Problems/Problem 10]] | ||
+ | |||
+ | ===Intermediate=== | ||
+ | ===Olympiad=== | ||
+ | |||
+ | {{problems}} | ||
+ | |||
+ | == See Also == | ||
+ | |||
+ | * [[Mean of a set]] | ||
+ | |||
+ | * [[Mode of a set]] |
Latest revision as of 14:30, 12 November 2023
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
[hide]Mean, Median, Mode
https://youtu.be/TkZvMa30Juo?t=1504
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
Median of a discrete distribution
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 .
Median of a continuous distribution
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
Pre-introductory
Find the median of .
Introductory
2000 AMC 12 Problems/Problem 14
2004 AMC 12A Problems/Problem 10
Intermediate
Olympiad
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