How to Find the Interquartile Range
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.
How to Find the Interquartile Range
The interquartile range is found by subtracting the lower quartile from the upper quartile.
Question
What is the range of the test scores?
StepbyStep:
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
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.
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.
4
Subtract the lower quartile (6) from the upper quartile (8).
8 − 6 = 2
Answer:
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.
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.
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.
A Formula to Find the Interquartile Range
The formula for finding the interquartile range is shown below:
In this formula,

Q_{3} is the upper quartile of the set.

Q_{1} is the lower quartile of the set.