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Interquartile Range (Mathematics Glossary)

What Is the Interquartile Range?

The interquartile range is the range of the middle half of a set of data.

The interquartile range is the difference between the upper quartile and the lower quartile.

Dictionary Definition

The Merriam-Webster dictionary defines the interquartile range as "the range of values of the variable in a statistical distribution that lies between the upper and lower quartiles."

Understanding the Interquartile Range

We can divide a set of numbers, that are in numerical order, into four quartiles:

  • The middle quartile (or median), Q2, is the middle number in the set. It divides the set in two halves.

  • The lower quartile, Q1, is the middle number of the bottom half. It divides the bottom half in two.

  • The upper quartile, Q3, is the middle number of the top half. It divides the top half in two.
The interquartile range is the difference between the upper quartile and the lower quartile. It is the range of the middle half of the data:

Finding the Interquartile Range

Imagine we wanted to find the interquartile range of the numbers given below:



To find the interquartile range, find the lower quartile (3) and the upper quartile (9).
1 2 3 4 5 6 7 8 9 10 11
Subtract the lower quartile from the upper quartile:
Interquartile Range = 9 - 3 = 6
Read more about how to find the interquartile range

A Formula to Find the Interquartile Range

The formula for finding the interquartile range is shown below:



In this formula,
  • IQR is the interquartile range.

  • Q3 is the upper quartile.

  • Q1 is the lower quartile.

Why Is the Interquartile Range Useful?

There are a lot of differences in the things we choose to measure. People have many different heights or ages or incomes or test scores.

The interquartile range is useful because they can summarize how spread out the data is.

Rather than recording the height of every adult in a group, it is useful to know how spread out the heights are. The interquartile range just looks at the middle half of heights (the half of people whose heights are nearer the middle height). It looks how spread out this middle half is:

The Interquartile Range as a Measure of Spread

The interquartile range is a measure of spread. It is one way of measuring how spread out numbers in a set are.

Note: The other useful summary is a 'measure of location' (also called a 'measure of central tendency'): where most of the numbers are bunched around. An average is a measure of location.
Interactive Test
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Here's a second test on the interquartile range.
Here's a third test on the interquartile range.
Note

What Is a Quartile?

A quartile is one of the three numbers that divide the numbers in a set into four equal groups.
  • The lower quartile is the middle number between the smallest number and the median.

  • The median is the middle number of the whole set of numbers.

  • The upper quartile is the middle number between the median and the highest number.

The Interquartile Range and Outliers

Sometimes in a group of numbers, a few of them are much larger...

3, 5, 2, 7, 150

...or much smaller than the rest:

1, 502, 847, 564, 980

These relatively large (150) or small (1) numbers are called outliers.

The interquartile range is not that affected by outliers.

Consider the numbers below. The lower quartile and the upper quartile are shown in bold:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
The interquartile range is 6 (= 9 - 3).

If instead the numbers have a large outlier:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100
The interquartile range is still 6. The interquartile range is robust to outliers.

This is unlike the range, which is very much affected by outliers. In the first set of number the range increases from 10 (= 11 - 1) to 99 (= 100 - 1).