How to Find IQR
Uncover the mystery of the middle with interquartile range.
Published:
Jan 2025
Key takeaways
- • A not-so-scary acronym – IQR stands for interquartile range.
- • Find the middle of the road – IQR is the middle 50 percent of any data set
- • Median is your friend – Finding the median is a vital skill for discovering IQR.
Mysterious acronyms can be a little spooky, but IQR is nothing to fear. IQR stands for interquartile range, the numbers between the first and fourth quarters of a group of numbers.
Read on to learn how to find IQP and the many applications of this handy statistical skill.
Definition of IQR
Interquartile Range (IQR) is the range of numbers in the middle half of a data set.
Imagine the numbers in a data set lined up in ascending order, lowest to highest. You could divide that line into fourths or quarters, quarter one, quarter two, and so on. The interquartile range consists of the numbers in quarters two and three, the middle 50 percent of the data set.
Think of quarters one and four as the bread and the IQR, quarters two and three, as the meat and cheese right in the middle.
Importance of IQR in Statistics
A lot of statistical analysis is about understanding what is “typical.” Usually the middle of a data set gathered by surveys or observations represents what is “typical.”
The IQR can help you see how spread out your data set’s middle is and allow you to focus on the data you need.
Steps to Calculate the Interquartile Range
Fortunately if you have a good grip on finding the median, calculating IQR should be a breeze.
1. Arrange the Data Set in Ascending Order
When you get data from a survey, a math problem, or wherever, the numbers will likely be in some random order.
The first step is to clean that up by ordering the numbers from lowest to highest. This is a vital first step since everything depends on the numbers being in proper order. Go ahead and count the number of values (numbers) as you do this. You’ll need that, too.
Let’s apply this to a real-world example. Imagine that the following numbers represent the number of M&Ms each kid at a summer camp got in their trail mix.
We need to first put those numbers in order.
6, 7, 7, 9, 10, 11, 11, 14, 15
2. Find the Median
After you have your numbers in ascending order, you need to find the median. This is the middle value of an odd set of numbers or the average of the middle two values if you have an even set of numbers.
You can easily find the median by dividing the total number of values by two and then counting up to that number from the start of your data set. This will split the data set into two.
With our M&Ms from earlier, the median will be the fifth number. We know this because we have ten values total, and to find the median, we divide that total by two. That gets us five.
Now we just count five numbers from the left to discover the median.
6, 7, 7, 9, 10, 11, 11, 14, 15
10 is the median for the entire data set.
3. Find the Median of Both Halves
Repeat steps one and two to find the median for both halves of your data. After you find the median for the first half, that number is your first quartile or Q1. When you repeat the process for the second half, you’ll get the Q3. You need both for the last step.
Do not include the median of the entire set in either half.
Back to the trail mix data. Our first half is the set of numbers to the left of the median.
6, 7, 7, 9, 10, 11, 11, 14, 15
The median of that half would be the average of the two middle values since we are working with an even number. Both values are seven, so the median is seven.
Now let’s repeat this process with the second half.
6, 7, 7, 9, 10, 11, 11, 14, 15
11 + 14 = 25, and 25 divided by 2 is 12.5.
That makes 12.5 our median for the second half
4. Subtract Q1 from Q3 to find the IQR
The final step is nice and straightforward. Simply subtract the Q1 from the Q3. The value you end up with is your interquartile range.
Let’s finally find the IQR of our trail mix M&Ms.
Our Q3 is 12.5 and our Q1 is 7. We need to subtract the Q1 from the Q3.
12.5 – 7 = 5.5
Therefore, our IQR is 5.5!
Using IQR to Identify Outliers
One of the best uses of IQR is to help identify and work around outliers in data sets.
Understanding Outliers
Outliers are numbers in a data set that are outside the expected range of numbers. They are either very high or very low compared to most of the data.
Calculating the Upper and Lower Bounds
Determine where outliers are likely to exist in a data set by using the medians of your Q1 and Q3 and IQR. The median of the first quartile minus 1.5 times the IQR will determine the lower bound of the data set.
On the other hand, the median of the third quartile plus 1.5 times the IQR will make the upper bound. Numbers below and above these bounds can be considered outliers.
Examples of Identifying Outliers with IQR
Outliers tend to appear in larger data sets, so let’s add some more campers to our M&M data set from earlier.
1, 4, 5, 6, 6, 7, 7, 9, 10, 11, 11, 14, 15, 15, 19
If we go through the process of finding the IQR like before, we discover that our median is 9 now, Q1 is 6, Q3 is 14, and the IQR is 8. Now let’s find the lower and upper bounds.
The lower bound is the Q1 minus 1.5 times the IQR.
Our Q1 is 6, and 1.5 times 8 is 12. So the lower bound is 6 – 12.
That gets us a negative number, and since you cannot have negative M&Ms, our data set does not have a lower bound since the numbers we’re working with are so small.
The upper bound is Q3 plus 1.5 times the IQR.
If we do the math, we find that our upper bound is 26. If any camper had more than 26 M&Ms, they would be considered an outlier.
Applications of IQR in Real-World Scenarios
IQR is more than just a math problem to practice. It has some impressive value in the world of research and statistics.
IQR in Business Analytics
IQR is great for helping businesses get a sense of how big their average customer base is or understand other numerical truths about the “typical” nature of production and the like.
For example, finding the IQR of how many iPhones a factory produces in a day can help manufacturers set timelines for producing the next model of phone. This information can also reveal when a factory is underproducing or exceeding expectations.
IQR in Scientific Research
When researchers crunch the numbers they gather through experiments or surveys, they typically want to learn what is normal or to be expected from what they are studying.
Outliers can skew the numbers, which may affect how researchers make sense of the data. IQR helps researchers define outliers, so they can best analyze and find results from their data.
Conclusion
IQR is an important part of making sense of numbers. With a good grasp of median and a little arithmetic, you can find and use IQR like a pro.
FAQs about IQRs
The correct formula for IQR is Q3 – Q1, where Q3 is the median value of the upper half of a number set and the Q1 is the median of the lower set.
To find the Q1 and Q3, first find the median of the entire data set. Do this by counting the number of values and then dividing that by two. If you have an odd number of values, the number right in the middle of the data set is your median. If you have an even number, the average of the two middle numbers in the median. To find the Q1 and Q3, find the median of the lower half and the upper half of the data sets, excluding the median of the entire set from both halves.
On a TI-83 calculator, you can quickly find the IQR. Start by entering the data set. Press the STAT button and then select EDIT. You will be given a table with columns labeled L1, L2, etc. Enter your data set into L1. After that, press STAT and scroll over to CALC. Select it and then select 1-Var Stats. Press enter, and if you scroll down on the next screen, you’ll see the Q1 and Q3 waiting for you. After that, simply subtract the Q1 from the Q3 to find your IQR.
