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Interquartile Range: Definition, Formula & Example

Instructor: Barry Rollins

Barry has taught mathematics at the college and high school level and has a master's degree in teaching secondary mathematics.

This lesson offers a definition of interquartile range as well as a description of the concept. The formula is given along with examples. Users may then test their new knowledge with a short quiz after the lesson.

Definition and Formula

Interquartile range is defined as the difference between the upper and lower quartile values in a set of data. It is commonly referred to as IQR and is used as a measure of spread and variability in a data set. This topic is often discussed in statistics with similar topics such as mean deviation and distribution.

The interquartile range formula is

IQR formula

Explanation and Examples

To calculate IQR using the formula, you simply subtract the third quartile minus the first quartile. The question then arises, what are these quartiles we are discussing?

I like to think of math in terms of money, as I rarely have any money but I would always like to have the skills to work with it just in case things change one day! Consider a data set spanning the values from one to one hundred, so that there are one hundred elements in the set: (1, 2, 3, 4, 5. . .97, 98, 99, 100)

Each penny is an element of the data set
Each penny is an element of the data set

If each of the values in the set represented pennies, I would now have all the pennies needed to count up to one dollar. To divide this set into quartiles (think of the quartiles as quarters) you need four to make one dollar. This is easiest to do to data by starting with the median. While you have likely always called it the median in math courses, it is also known as the 'second quartile' or

Q2

It is used to divide the data in half. Since our data set has an even number of values, the median could be located by averaging the two central values, in this case 50 and 51. This would give a median of 50.5. The value of the median is not as important as how it divides the data when considering IQR. We now have two data sets, 1 to 50 in the lower set and 51 to 100 in the higher set.

Once we have divided the set in half, we need to divide each of our new sets in half to find our quartiles. We can start with the lower set, 1 to 50. This is like two quarters out of our dollar. To divide this up we need to find the middle again, so we find the median of this lower set. There is an even number of elements again, so the median is in between 25 and 26, giving us 25.5 when we add those values and divide by 2. As this is the median of the lower half of the data, we call this

Q1

The same process is followed to find the upper quartile, working with the higher data set, 51 to 100. The median of this set is in between 75 and 76, so we average these two values to find 75.5. This is the third quartile, or

Q3

To finish our problem and find IQR we simply subtract the upper quartile minus the lower quartile, so we find IQR = 75.5 - 25.5 = 50.

To consider a more traditional example, look at the following data set:

(1, 3, 6, 4, 7, 5, 9, 9)

To find the interquartile range we first find the median, beginning by ordering the set:

1, 3, 4, 5, 6, 7, 9, 9

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