Statistics Definitions > Order Statistics

## Order Statistics Definition

Order statistics are ordered values from a sample. They include:

- The
**minimum**of a set of numbers: the first number when the items are placed in order. - The
**first quartile**(sometimes called the lower quartile), Q1: the value at the 25th percentile. - The
**median**: the middle number. - The
**third quartile**(sometimes called the upper quartile), Q3: the value at the 75th percentile. - The
**maximum**: the last number. - The
**Interquartile Range**: the difference between the first and third quartiles (Q3 – Q1).

**Any item in a set can be given an order**. The notation is X_{i} where *i* is the order place, when all numbers are placed in ascending order, from smallest to largest. For example:

- X
_{2}is the second smallest item. - X
_{99}is the 99th item in order.

The range of a data set is a function of the order. This means that the range (maximum – minimum) is dependent on the ordered numbers in the set, and that it will differ for every set. The median and the midrange are also functions of the order and so will differ depending on what numbers are in the set.

**Example**: What are the sample minimum and maximum for the following list? 13, 17, 39, 21, 24, 45, 47, 31.

**Answer**: Placing the numbers in order, you get 13, 17, 21, 24, 31, 39, 45, 47. The minimum is 13 and the maximum is 47.

## A More Technical Definition

Mathematically, order statistics satisfy the equation:

X

_{(1)}≤ X_{(2)}≤ …X_{(n)}.

This makes sense if you read it out aloud: x_{2}(the second value) is greater than x_{1}(the first value), and so on.

This can also be written as:

X_{(1:n)} ≤ X_{(2:n)} ≤ · · · ≤ X_{(n:n)}.

**The two equations are equivalent.** Note that these equations are** different from a list of values**, like:

x_{1}, x_{2},…x_{n}.

This list of value has no order (in other words, the greater than or less than symbols are missing) and is an example of **unordered statistics.**