Difference between revisions of "Median (statistics)"
(→Median of a distribution) |
(→Median of a data set) |
||
Line 2: | Line 2: | ||
== 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_{(\frac{k+1}2)}</math> when <math>k</math> is odd and <math>\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2</math> when <math>k</math> is even, where <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 and <math>\frac{X_{(\frac{k}2)} + X_{(\frac{k}2 + 1)}}2</math> when <math>k</math> is even, where <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 distribution == |
Revision as of 06:12, 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
such that
. This turns out 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
This page is in need of some relevant examples or practice problems. Help us out by adding some. Thanks.