Difference between revisions of "Median"
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A '''median''' is a measure of central tendency used frequently in statistics. | A '''median''' is a measure of central tendency used frequently in statistics. | ||
| + | ==Mean, Median, Mode== | ||
| + | https://youtu.be/TkZvMa30Juo?t=1504 | ||
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== Median of a data set == | == 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. | 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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{{problems}} | {{problems}} | ||
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| + | == See Also == | ||
| + | |||
| + | * [[Mean of a set]] | ||
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| + | * [[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
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
This page is in need of some relevant examples or practice problems. Help us out by adding some. Thanks.
