Sample median

The sample median , also known as the ( random ) empirical median , is a special function in mathematical statistics , a branch of mathematics . It is an estimator and is used in estimation theory to estimate the median of an unknown, underlying probability distribution .

The sample median should not be confused with the median of descriptive statistics or the median in terms of probability theory . These two are indicators of a sample or a distribution, whereas the sample median is a function . See the Delimitation section for details .

definition

Let random variables and the order statistics of the random variables be given. Then is called ${\ displaystyle n}$ ${\ displaystyle X_ {1}, X_ {2}, \ dots, X_ {n}}$ ${\ displaystyle X _ {(1)}, X _ {(2)}, \ dots, X _ {(n)}}$

{\ displaystyle \ mu _ {1/2}: = {\ begin {cases} X _ {({\ tfrac {n + 1} {2}})} & {\ text {if}} n {\ text {odd }} \\ {\ tfrac {1} {2}} (X _ {({\ tfrac {n} {2}})} + X _ {({\ tfrac {n} {2}} + 1)}) & {\ text {falls}} n {\ text {even}} \ end {cases}}}

the sample median.

Demarcation

The differences between the sample median and the median of a sample or a probability distribution result from the context in which they are used.

The median in the sense of probability theory is an indicator of a probability distribution and thus an indicator of a (quantity) function. In mathematical statistics, an unknown but clearly determined probability distribution is assumed. An estimate should be made on the basis of samples that were generated according to this probability distribution. This is done with the help of suitable estimators. If one now wants to estimate the median of the probability distribution, one chooses the sample median as the estimation function. The median of a sample can then be understood as the realization of the sample median (as a random variable). Thus, the median of a sample (estimated value ) is related to the sample median (estimating function ) like a function value is related to a function. In particular, the sample median can also be examined for the classic quality criteria of the estimation function such as faithfulness to expectations . This is not meaningfully possible for the median of a sample.

Web links

AV Prokhorov: Median (in statistics) . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
MS Nikulin: Order statistic . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Eric W. Weisstein : Statistical Median . In: MathWorld (English).

Individual evidence

^ ^A ^b Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 245 , doi : 10.1515 / 9783110215274 .
↑ Norbert Henze: Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 330 , doi : 10.1007 / 978-3-658-03077-3 .

[Georgii245-1] A ^b Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 245 , doi : 10.1515 / 9783110215274 .

[2] Norbert Henze: Stochastics for beginners . An introduction to the fascinating world of chance. 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 330 , doi : 10.1007 / 978-3-658-03077-3 .