How to Find the IQR

Reviewed by David Jia

Last Updated: May 16, 2024 Fact Checked

The IQR is the "interquartile range" of a data set. It is used in statistical analysis to help draw conclusions about a set of numbers. The IQR is often preferred over the range because it excludes most outliers. Read on to learn how to find the IQR!

Method 1

Method 1 of 3:

Organizing the Data Set

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If you're learning this for a class and taking a test, you might be provided with a ready-made set of numbers, e.g. 1, 4, 5, 7, 10. This is your data set – the numbers that you will be working with. You may, however, need to arrange the numbers yourself from some sort of table or word problem. ^{[1]
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Make sure that each number refers to the same sort of thing: for instance, the number of eggs in each nest of a given bird population, or the number of parking spots attached to each house on a given block.
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2
Organize your data set in ascending order. In other words: arrange the numbers from lowest to highest. Take your cue from the following examples. ^{[2]
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- Even number of data example (Set A): 4 7 9 11 12 20
- Odd number of data example (Set B): 5 8 10 10 15 18 23
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3
Divide the data in half. To do this, find the midpoint of your data: the number or numbers in the very center of the set. If you have an odd amount of numbers, choose the exact middle number. If you have an even amount of the numbers, the midpoint will rest between the two middlemost numbers. ^{[3]
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- Even example (Set A), in which the midpoint lies between 9 and 11: 4 7 9 | 11 12 20
- Odd example (Set B), in which (10) is the midpoint: 5 8 10 (10) 15 18 23

Method 2

Method 2 of 3:

Calculating the IQR

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1
Find the median of the lower and upper half of your data. The median is the "midpoint," or the number that is halfway into a set. ^{[4]
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Research source} In this case, you aren't looking for the midpoint of the entire set, but rather the relative midpoints of the upper and lower subsets. If you have an odd number of data, do not include the middle number – in Set B, for instance, you would not figure in one of the 10s. ^{[5]
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- Even example (Set A):
  - Median of lower half = 7 (Q1)
  - Median of upper half = 12 (Q3)
- Odd example (Set B):
  - Median of lower half = 8 (Q1)
  - Median of upper half = 18 (Q3)
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2
Subtract Q3 - Q1 to determine the IQR. ^{[6]
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Research source} Now you know how many numbers lie between the 25th percentile and the 75th percentile. You can use this to understand how widely-spread the data is. For instance, if a test is scored out of 100, and the IQR of the scores is 5, you can assume that most of the people taking it had a similar grasp of the material because the high-low range is not very large. If the IQR of the test scores is 30, however, you might start to wonder why some people scored so high and others scored so low.
- Even example (Set A): 12 - 7 = 5
- Odd example (Set B): 18 - 8 = 10

Method 3

Method 3 of 3:

Understanding the IQR

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Essentially, it is a way of understanding the spread or "dispersion" of a set of numbers. The interquartile range is defined as the difference between the upper quartile (the highest 25%) and the lower quartile (the lowest 25%) of a data set. ^{[7]
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Tip: The lower quartile is usually written as Q1, and the upper quartile is Q3 – which would technically make the halfway point of the data set Q2, and the highest point Q4.
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2
Understand quartiles. To visualize a quartile, chop a list of numbers into four equal parts. Each of these parts is a "quartile." ^{[8]
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Research source} Consider the set: 1, 2, 3, 4, 5, 6, 7, 8.
- 1 and 2 are the first quartile, or Q1
- 3 and 4 are the second quartile, or Q2
- 5 and 6 are the third quartile, or Q3
- 7 and 8 are the fourth quartile, or Q4
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In order to find the difference between the upper and lower quartile, you'll need to subtract the 25th percentile from the 75th percentile. ^{[9]
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The formula is written as: Q3 – Q1 = IQR.

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Question

How do I find the IQR when the upper and lower quartile are the same number?

Community Answer

Find the IQR by subtracting Q1 from Q3. If they are the same number, the IQR equals zero.

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Can the IQR be negative?

Community Answer

No, if it is done correctly, all numbers should be in ascending order, therefore the first quartile would be smaller or the same as the third quartile.

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How do I find IQR of 1,3,4,5,8,9,10,11,12,14,16?

Community Answer

In 1,3,4,5,8,9,10,11,12,14,16 the median for the lower quartile is 4 and the median for the upper quartile is 12. Subtract Q3 by Q1 and that would make it 12-4 and 12-4= 8.

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Tips

It is important to learn to do this on your own, but there are a number of online IQR calculators that you can use to check your work. ^{[10]
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Research source} Don't get too reliant on a calculator app if you are learning this for a class! If you are asked about IQR on the test, you'll need to know how to find it by hand.

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How to Find the IQR

Steps

Organizing the Data Set

Calculating the IQR

Understanding the IQR

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