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, 8,
they will usually be denoted
where the subscript i in indicates simply the order in which the observations were recorded and is usually assumed not to be significant. A case when the order is significant is when the observations are part of a time series.
The order statistics would be denoted
where the subscript (i) enclosed in parentheses indicates the ith order statistic of the sample.
The first order statistic (or smallest order statistic) is always the minimum of the sample, that is,
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,
The sample range is the difference between the maximum and minimum. It is clearly a function of the order statistics:
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 m, then the sample median is and so is an order statistic. On the other hand, when n is even, n = 2m and there are two middle values, and, 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.
Read more about this topic: Order Statistic
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