How to Calculate Intermediary Spread (IQR)

IQR (short for “interquartile range”) is the spread between, or interquartile range, of a data set. This concept is used in statistical analysis to help draw conclusions about a set of numbers. IQR is often used for the range of variation because it eliminates most outliers in the data. Let’s learn how to determine IQR.

Table of Contents

Understanding IQR

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Know how to use IQR. Essentially, the spread represents the width or “dispersion” of the set of numbers. ^{[1] X Research Source} The interquartile range is defined as the difference between the upper quartile (25% highest) and the lower quartile (25% lowest) of the data set. ^{[2] X Research Source}

Tip: The lower quartile is usually denoted Q1, the upper quartile is Q3 – so the midpoint of the data set will be Q2 and the highest is Q4.

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Understanding quartiles. To visualize a quartile, divide the list into four equal parts. Each part will be a “quartile”. ^{[3] X Research Source} For example in data set: 1, 2, 3, 4, 5, 6, 7, 8.

1 and 2 are the first quartile – Q1
3 and 4 are the second quartile – Q2
5 and 6 are the third quartile – Q3
7 and 8 are the fourth quartile – Q4

Sort data set

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Gather your data. If you are learning about IQR to study and take a test, the test will provide a set of numbers, for example: 1, 4, 5, 7, 10. You will calculate based on these numbers. However, you may need to rearrange numbers from a table or puzzle. ^{[5] X Research Sources}

You need to make sure that each number represents one type of data: for example, the number of eggs in a particular bird’s nest or the number of parking spaces per house in a block.

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Sort the data set in ascending order. In other words, you need to arrange the numbers in order from smallest to largest. Draw conclusions from the following examples.

Even-numbered data set (A): 4 7 9 11 12 20
Odd data set (B): 5 8 10 10 15 18 23

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Divide the data into two parts. To proceed, you find the midpoint of the data – this will be one or more numbers in the middle of the series. If you have an odd number, choose the exact middle number. With an even number of data, the midpoint will be between the two central numbers.

In the even number example (set A), the midpoint between 9 and 11 looks like this: 4 7 9 | 11 12 20
In the odd number example (set B) then (10) is the midpoint. We have: 5 8 10 (10) 15 18 23

Calculate IQR

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Find the median of the top and bottom half in the data set. Median is the “midpoint” or number in the middle of a data set. ^{[6] X Research Source} In this case, you will not find the midpoint of the entire data, but only the relative medians of the upper and lower subsets. If you have an odd number of data, exclude the middle number – for example, in set B you don’t need to count 10. ^{[7] X Research Source}

In the even number example (set A):
- Median of bottom half = 7 (Q1)
- Median of upper half = 12 (Q3)
In the odd number example (set B):
- Median of bottom half = 8 (Q1)
- Median of upper half = 18 (Q3)

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Take Q3 – Q1 to find the mid spread. So you know how many numbers are between the 25th and 75th percentiles. You can use this to get an idea of how widely distributed the data is. For example, if the test has a scale of 100 and the IQR of the score is 5, you would have reason to believe that the participants have similar qualifications because the highs and lows are not too different. But if the mid-range of test scores is up to 30, you can question why some people score so high while others score so low.

In the even number example (set A): 12 – 7 = 5
In the odd number example (set B): 18 – 8 = 10

Advice

It is important that you master the knowledge, because there are many IQR calculators online, use them to check the results. ^{[8] X Research Source} Do not rely too much on calculation applications when studying! If you run into a mid-spread test, you need to know how to calculate it by hand.

Steps

Understanding IQR

Sort data set

Calculate IQR

Advice

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