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How to Find the Interquartile Range (Mathematics Lesson)

Finding the Interquartile Range

We can find the interquartile range of a set of numbers.

Imagine a teacher had set their mathematics class a test. They are concerned that some students are doing much better than others.

They wanted to find the interquartile range of test scores in the class, to see if some students are doing much better than others.

finding the interquartile range of the test scores of students

How to Find the Interquartile Range

The interquartile range is found by subtracting the lower quartile from the upper quartile.

An Example Question

What is the range of the test scores?

the test scores of the students
Step 1

List the numbers in numerical order (going from the smallest to the largest number).

7 10 8 6 4 → 4 6 7 8 10
Step 2

Find the lower quartile of the set of numbers.

4 6 7 8 10

Don't forget: The lower quartile is the middle number of the lower half of the set of numbers.

The lower quartile of the test scores is 6.

Step 3

Find the upper quartile of the set of numbers.

4 6 7 8 10

Don't forget: The upper quartile is the middle number of the upper half of the set of numbers.

The upper quartile of the test scores is 8.

Step 4

Subtract the lower quartile (6) from the upper quartile (8).

8 - 6 = 2

The interquartile range of the test scores is 2.

the interquartile range of the test scores is 2

Finding the Upper and Lower Quartiles

There are different methods for finding the quartiles. They give different values for the lower quartile and the upper quartile, and so different answers for the interquartile range.

  • In the example above, the Tukey method has been used.

    In the Tukey method, the middle number of the whole set (also called the median) is included in both the lower half and the upper half of the set.

    The lower quartile is the middle of the lower half and the upper quartiles is the middle of the upper half.

    finding the interquartile range using the Tukey method
    Interquartile range = 8 - 6 = 2
  • Another method is the Moore and McCabe method. The median is excluded from both the lower half and the upper half.

    finding the interquartile range using the Moore and McCabe method

    Because each half has an even number of numbers, the middle of each half is in between two numbers.

    The lower quartile is halfway between 4 and 6: it is 5. The upper quartile is halfway between 8 and 10: it is 9.

    Interquartile range = 9 - 5 = 4

The interquartile range will be different depending on which method is used to find the lower quartile and the upper quartile.

Read more about the methods for finding the quartiles

A Real Example of How to Find the Interquartile Range

The slider below gives another example of finding the interquartile range.

A Formula to Find the Interquartile Range

The formula for finding the interquartile range is shown below:

the formula for the interquartile range is given. I Q R equals Q 3 minus Q 1

In this formula,

  • Q3 is the upper quartile of the set.

  • Q1 is the lower quartile of the set.
Interactive Test
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Here's a second test on finding the interquartile range.
Here's a third test on finding the interquartile range.
Note

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.

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.