Interquartile Range (IQR)
Reference for IQR = Q3 - Q1 with calculation steps, box plot interpretation, and outlier detection using the 1.5 x IQR rule.
Includes worked examples.
The Formula
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
Key Notes
- Formula: IQR = Q3 − Q1: Q1 is the 25th percentile (median of the lower half) and Q3 is the 75th percentile (median of the upper half). The IQR spans the middle 50% of the data.
- Robust to outliers: Unlike the range (max − min) and standard deviation, the IQR is not affected by extreme values. It is the preferred measure of spread for skewed distributions such as income or house prices.
- Outlier detection with the 1.5×IQR rule: Any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is flagged as a potential outlier. This is the rule used to draw the whiskers on a box plot.
- Interpreting the value: A large IQR means the middle half of the data is spread out (high variability). A small IQR means the middle half is tightly clustered around the median.
- IQR vs standard deviation: Standard deviation is preferred for roughly symmetric, bell-shaped data. IQR is preferred when the distribution is skewed or contains outliers. Both together give a fuller picture.