Difference between revisions of "Median"
m (Formatting) |
(Formatting) |
||
| Line 9: | Line 9: | ||
== Median of a distribution == | == Median of a distribution == | ||
| − | === | + | === Discrete distributions === |
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>. | 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>. | ||
| − | === | + | === Continuous distributions === |
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>. | 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 == | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | === Introductory === | |
| − | + | *Find the median of <math>\{3, 4, 5, 15, 9\}</math>. | |
| − | + | *[[2000 AMC 12 Problems/Problem 14]] | |
| + | *[[2004 AMC 12A Problems/Problem 10]] | ||
| − | {{ | + | === Intermediate === |
| + | === Olympiad === | ||
| + | |||
| + | {{problem}} | ||
== Video == | == Video == | ||
| + | |||
[//youtu.be/TkZvMa30Juo Video] | [//youtu.be/TkZvMa30Juo Video] | ||
Revision as of 16:14, 19 February 2025
This article is about the median used in statistics. For other medians, check Median (disambiguation).
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.
