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One of the numerical characteristics of probability distributions, a particular case of a [[Quantile|quantile]]. For a real-valued random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633101.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633102.png" />, a median is defined as a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633105.png" />. Every random variable has at least one median. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633107.png" /> in a closed interval, then every point of this interval is a median. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633108.png" /> is a strictly-monotone function, then the median is unique. In the symmetric case, if the median is unique, it is identical with the [[Mathematical expectation|mathematical expectation]], provided that the latter exists. The fact that a median always exists is used for centering random variables (see, for instance, [[Lévy inequality|Lévy inequality]]). In mathematical statistics, to estimate the median of a distribution in terms of independent results of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633109.png" /> one uses a so-called sample median — a median of the corresponding order statistics (cf. [[Order statistic|Order statistic]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331010.png" />, that is, of the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331012.png" /> is odd, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331014.png" /> is even.
 
One of the numerical characteristics of probability distributions, a particular case of a [[Quantile|quantile]]. For a real-valued random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633101.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633102.png" />, a median is defined as a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633105.png" />. Every random variable has at least one median. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633107.png" /> in a closed interval, then every point of this interval is a median. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633108.png" /> is a strictly-monotone function, then the median is unique. In the symmetric case, if the median is unique, it is identical with the [[Mathematical expectation|mathematical expectation]], provided that the latter exists. The fact that a median always exists is used for centering random variables (see, for instance, [[Lévy inequality|Lévy inequality]]). In mathematical statistics, to estimate the median of a distribution in terms of independent results of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m0633109.png" /> one uses a so-called sample median — a median of the corresponding order statistics (cf. [[Order statistic|Order statistic]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331010.png" />, that is, of the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331012.png" /> is odd, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063310/m06331014.png" /> is even.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>

Revision as of 16:03, 19 March 2012

2010 Mathematics Subject Classification: Primary: 60-01 Secondary: 62-01 [MSN][ZBL]

One of the numerical characteristics of probability distributions, a particular case of a quantile. For a real-valued random variable with distribution function , a median is defined as a number such that and . Every random variable has at least one median. If for all in a closed interval, then every point of this interval is a median. If is a strictly-monotone function, then the median is unique. In the symmetric case, if the median is unique, it is identical with the mathematical expectation, provided that the latter exists. The fact that a median always exists is used for centering random variables (see, for instance, Lévy inequality). In mathematical statistics, to estimate the median of a distribution in terms of independent results of observations one uses a so-called sample median — a median of the corresponding order statistics (cf. Order statistic) , that is, of the quantity if is odd, and of if is even.

References

[1] M. Loève, "Probability theory" , Springer (1977)
[2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
How to Cite This Entry:
Median (in statistics). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Median_(in_statistics)&oldid=11382
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article