Interquartile Range- Calculation and Interpretation Guide

What Is the Interquartile Range (IQR)?

The interquartile range tells you the spread of your middle 50% of data. It's the distance between the 25th percentile (Q1) and the 75th percentile (Q3).

That's it. No fancy math tricks. Just Q3 minus Q1.

Unlike range (which uses extremes), IQR ignores outliers and skewed tails. It shows you where the bulk of your data actually sits.

Why IQR Actually Matters

Standard deviation lies to you. Mean lies to you. But IQR? It shows the truth about your typical data.

Here's why: if your data has extreme values, the mean gets pulled toward them. Standard deviation inflates. You think your data is more spread out than it really is.

IQR cuts through that noise. It tells you what a "normal" observation looks like in your dataset.

How to Calculate IQR: Step by Step

Step 1: Sort Your Data

Arrange all values from smallest to largest. This is non-negotiable. Mess this up and everything else fails.

Example dataset: 3, 7, 8, 12, 15, 18, 22, 25, 30

Step 2: Find the Median (Q2)

The median splits your data in half. With 9 values, the middle value is the 5th one.

Median = 15

Step 3: Find Q1 (First Quartile)

Look at the lower half (excluding the median if odd count). Find the median of that half.

Lower half: 3, 7, 8, 12

Q1 = median of 7 and 8 = 7.5

Step 4: Find Q3 (Third Quartile)

Look at the upper half. Find the median of that half.

Upper half: 18, 22, 25, 30

Q3 = median of 22 and 25 = 23.5

Step 5: Calculate IQR

Subtract Q1 from Q3:

IQR = Q3 - Q1 = 23.5 - 7.5 = 16

Reading the Box Plot

Box plots visualize IQR automatically. Here's what you're looking at:

Left whisker: Minimum value (or 1.5 × IQR below Q1)
Box left edge: Q1 (25th percentile)
Box center line: Median (Q2)
Box right edge: Q3 (75th percentile)
Right whisker: Maximum value (or 1.5 × IQR above Q3)

Any point beyond 1.5 × IQR from the box edges? That's an outlier.

IQR vs Standard Deviation: When to Use Which

Situation	Use This
Data has outliers or skewness	IQR
Data is symmetric and clean	Standard deviation
Describing typical spread	IQR
Statistical tests requiring variance	Standard deviation
Comparing multiple datasets	Standard deviation
Reporting data with extreme values	IQR (with median)

Spotting Outliers with IQR

The outlier fence method works like this:

Lower bound: Q1 - 1.5 × IQR

Upper bound: Q3 + 1.5 × IQR

Using our example (Q1 = 7.5, Q3 = 23.5, IQR = 16):

Lower bound: 7.5 - (1.5 × 16) = 7.5 - 24 = -16.5
Upper bound: 23.5 + (1.5 × 16) = 23.5 + 24 = 47.5

Any value below -16.5 or above 47.5 is flagged as an outlier. In our dataset, nothing qualifies.

Values beyond 3 × IQR get labeled "extreme outliers" instead of just "outliers."

Common Mistakes to Avoid

Including the median in halves: When finding Q1 and Q3, exclude the median for odd-length datasets
Confusing IQR with semi-interquartile range: SIQR is just IQR ÷ 2
Using IQR for normally distributed data: You lose information about the distribution tails
Forgetting to sort first: Everything depends on ordered data

Quick Reference

Percentile	Name	What It Means
25th	Q1	25% of data falls below
50th	Median / Q2	Half of data falls below
75th	Q3	75% of data falls below
IQR	Q3 - Q1	Middle 50% spread

The Bottom Line

IQR answers one question: how spread out is the typical part of my data?

It's resistant to outliers. It's simple to calculate. And it tells you more about real-world variation than standard deviation when your data isn't clean.

Use it when you have extreme values. Use the median instead of the mean. And always, always sort your data first.