In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Given a sample of size n {\displaystyle n} , the kth order statistic is denoted x ( k ) {\displaystyle x_{(k)}} , with 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} . Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.
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
x ( 1 ) = 3 x ( 2 ) = 6 x ( 3 ) = 7 x ( 4 ) = 9 {\displaystyle {\begin{aligned}x_{(1)}&=3\\x_{(2)}&=6\\x_{(3)}&=7\\x_{(4)}&=9\end{aligned}}}
The first order statistic (or smallest order statistic) is always the minimum of the sample, that is,
X ( 1 ) = min { X 1 , … , X n } {\displaystyle X_{(1)}=\min\{\,X_{1},\ldots ,X_{n}\,\}}
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 n, the nth order statistic (or largest order statistic) is the maximum, that is,
X ( n ) = max { X 1 , … , X n } . {\displaystyle X_{(n)}=\max\{\,X_{1},\ldots ,X_{n}\,\}.}
The sample range is the difference between the maximum and minimum. It is a function of the order statistics:
R a n g e { X 1 , … , X n } = X ( n ) − X ( 1 ) . {\displaystyle {\rm {Range}}\{\,X_{1},\ldots ,X_{n}\,\}=X_{(n)}-X_{(1)}.}
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 n of observations is odd. More precisely, if n = 2m+1 for some integer m, then the sample median is X ( m + 1 ) {\displaystyle X_{(m+1)}} and so is an order statistic. On the other hand, when n is even, n = 2m and there are two middle values, X ( m ) {\displaystyle X_{(m)}} and X ( m + 1 ) {\displaystyle X_{(m+1)}} , 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.
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