Difference between revisions of "Median (statistics)"
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=== Median of a discrete 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 said to lie between <math>x_i</math> and <math>x_{i+1}</math> iff <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 | + | 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 said to lie between <math>x_i</math> and <math>x_{i+1}</math> iff <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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+ | === Median of a continuous distribution === | ||
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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>. | ||
== Problems == | == Problems == |
Revision as of 06:05, 25 November 2007
A median is a measure of central tendency used frequently in statistics.
Contents
[hide]Median of a data set
The median of a finite set of real numbers is defined to be when is odd and when is even, where 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 said to lie between and iff 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
Intermediate
Olympiad
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