Editing Order statistic (section)

== Notation and examples ==

For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are

:6, 9, 3, 7,

the order statistics would be denoted

:<math>x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=7,\ \ x_{(4)}=9,\,</math>

where the subscript {{math|({{italics correction|''i''}})}} enclosed in parentheses indicates the {{math|{{italics correction|''i''}}}}th order statistic of the sample.

The '''first order statistic''' (or '''smallest order statistic''') is always the [[minimum]] of the sample, that is,

:<math>X_{(1)}=\min\{\,X_1,\ldots,X_n\,\}</math>

where, following a common convention, we use upper-case letters to refer to random variables, and lower-case letters (as above) to refer to their actual observed values.

Similarly, for a sample of size {{math|''n''}}, the {{math|{{italics correction|''n''}}}}th order statistic (or '''largest order statistic''') is the [[maximum]], that is,

:<math>X_{(n)}=\max\{\,X_1,\ldots,X_n\,\}.</math>

The [[sample range]] is the difference between the maximum and minimum.  It is a function of the order statistics:

:<math>{\rm Range}\{\,X_1,\ldots,X_n\,\} = X_{(n)}-X_{(1)}.</math>

A similar important statistic in [[exploratory data analysis]] that is simply related to the order statistics is the sample [[interquartile range]].

The sample median may or may not be an order statistic, since there is a single middle value only when the number {{math|''n''}} of observations is [[Even and odd numbers|odd]].  More precisely, if {{math|1=''n'' = 2''m''+1}} for some integer {{math|''m''}}, then the sample median is <math>X_{(m+1)}</math> and so is an order statistic.  On the other hand, when {{math|''n''}} is [[even and odd numbers|even]], {{math|1=''n'' = 2''m''}} and there are two middle values, <math>X_{(m)}</math> and <math>X_{(m+1)}</math>, and the sample median is some function of the two (usually the average) and hence not an order statistic.  Similar remarks apply to all sample quantiles.