Interquartile Range (IQR)
Calculate the spread of the middle 50% of data.
A robust measure of variability that ignores outliers.
The Formula
IQR = Q3 - Q1
The interquartile range measures the spread of the middle half of your data. Unlike range or standard deviation, IQR is not affected by extreme outliers.
Variables
| Symbol | Meaning |
|---|---|
| IQR | Interquartile range |
| Q1 | First quartile — 25th percentile (median of the lower half) |
| Q3 | Third quartile — 75th percentile (median of the upper half) |
Outlier detection: Any value below Q1 - 1.5×IQR or above Q3 + 1.5×IQR is considered an outlier.
Example 1
Data: 2, 4, 5, 7, 8, 10, 12, 15, 18
Q1 = median of {2, 4, 5, 7} = (4+5)/2 = 4.5
Q3 = median of {10, 12, 15, 18} = (12+15)/2 = 13.5
IQR = 13.5 - 4.5 = 9
Example 2
Test scores: 55, 60, 65, 70, 72, 75, 80, 85, 90, 95, 100
Q1 = 65, Q3 = 90
IQR = 90 - 65 = 25
Outlier boundaries: 65 - 37.5 = 27.5 and 90 + 37.5 = 127.5
No outliers in this data set (all values between 27.5 and 127.5)
When to Use It
Use the interquartile range when:
- Measuring data spread without being influenced by outliers
- Creating box plots (box-and-whisker diagrams)
- Identifying outliers using the 1.5×IQR rule
- Comparing variability between data sets with different distributions