Interquartile Range: Formula and Examples

What is interquartile range? It tells you how spread out the middle of your data is. Take Q3, subtract Q1, and you’ve got it. Extreme values mostly don’t matter, which is the whole point.

Quartiles, Explained Simply

Imagine ten friends standing in a row, shortest on the left. Split the row into four equal parts. Each split point is a quartile.

Quartile	Also called	What it means
Q1	First quartile, 25th percentile	A quarter of the values sit below it
Q2	Median, 50th percentile	Half of the values sit below it
Q3	Third quartile, 75th percentile	Three quarters of the values sit below it

The IQR is the gap between Q1 and Q3. That gap holds the middle 50% of your numbers. It measures the spread of data, so a small IQR means the numbers are bunched up, and a big one means they’re scattered. Anyone reading quantitative research will see it a lot, since one wild number can’t throw it off.

Now picture a box plot. The box in the middle is the IQR. So what is interquartile range, visually? It’s just the length of that box. Add the lowest value, Q1, the median, Q3 and the highest value, and you have the five number summary.

The Interquartile Range Formula

IQR = Q3 minus Q1. That’s all. Some books also use the semi interquartile range, which is half of that.

How to Find Interquartile Range

Try these nine numbers: 2, 5, 7, 10, 12, 15, 18, 20, 25.

Check that they’re in order. They are.
Find the median. It’s 12.
Set the median aside and split the rest. The lower half is 2, 5, 7, 10. The upper half is 15, 18, 20, 25.
Q1 is the middle of the lower half, so (5 + 7) ÷ 2 = 6. Q3 is the middle of the upper half, so (18 + 20) ÷ 2 = 19.
Subtract. 19 minus 6 is 13.

Done. The middle half of this data covers 13 units.

Why Your Answer Might Not Match

Here’s the thing that trips people up. Put those same numbers in Excel and you may get 11, not 13. Both are fine. There’s more than one way to work out quartiles.

Method	Q1	Q3	IQR
Median left out of both halves (many textbooks, Excel QUARTILE.EXC)	6	19	13
Median kept in both halves (Excel QUARTILE.INC)	7	18	11

Choose one method and stick with it. If it’s homework, use whatever your teacher uses. An interquartile range calculator might pick a method for you without saying so, so check its settings.

Spotting Outliers With the IQR

An outlier is a number that sits far from the others. The 1.5 IQR rule is a quick way to catch them. Multiply the IQR by 1.5. Subtract that from Q1 to get the lower fence. Add it to Q3 to get the upper fence. Anything outside the fences gets flagged.

Statistician John Tukey made this rule popular in his 1977 book Exploratory Data Analysis. The 1.5 is just a habit people agreed on. It’s not magic.

Here’s an example with made up delivery times in minutes: 22, 25, 27, 28, 30, 31, 33, 35, 38, 90. Q1 is 27, Q3 is 35, and the IQR is 8. Times 1.5 is 12. So the lower fence is 15 and the upper fence is 47. The 90 is way over, so it gets flagged.

Measure	With the 90	Without the 90
Range	68	16
Mean	35.9	29.9
IQR	8	8

See how the range and mean swing? The IQR doesn’t budge. Still, a flagged number isn’t always wrong. Look at how you got the data first, since some data collection methods produce odd values by accident. Remove a number only when you have a real reason.

IQR, Range or Standard Deviation?

	Range	IQR	Standard deviation
What it shows	Biggest minus smallest	Spread of the middle 50%	Typical distance from the mean
Hurt by outliers?	Yes, a lot	Very little	Yes
Best for	A quick check	Skewed data or data with outliers	Evenly spread data

A handy rule: the median goes with the IQR, and the mean goes with the standard deviation. If your data forms a bell curve, the IQR is roughly 1.35 times the standard deviation. Want to compare spread between very different sets of numbers? The coefficient of variation is worth a look.

What the IQR Can’t Do

It skips the bottom 25% and the top 25% of values, so it can miss what’s happening at the edges. Two data sets can have the same IQR and look nothing alike. And with tiny samples, quartiles can be shaky.

The usual slips are mixing up the range and the IQR, forgetting to sort, and changing methods in the middle of a project.

Wrapping Up

So, what is interquartile range in one line? It’s how wide the middle half of your data is, found by subtracting Q1 from Q3. Use it for skewed data or when outliers are around, say which method you used, and think twice before deleting flagged values.

FAQs

Can the IQR be negative?

No. Q3 is never smaller than Q1, so the IQR is zero or higher.

Is the IQR affected by outliers?

Hardly at all. Outliers live in the outer quarters, and the IQR ignores those.

Does the 1.5 IQR rule always work?

No. It’s a quick check, not a final verdict. Ask why a value looks odd before you remove it.

What is interquartile range used for?

People use it to describe spread, compare groups, draw box plots and catch outliers. It works especially well with skewed data.

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