What Is 1.5 IQR Rule?
The 1.5 IQR rule is a quick test for outliers. Work out the interquartile range, multiply it by 1.5, and subtract that from Q1 and add it to Q3. Any value outside those two fences is flagged as an outlier.
Paste your values and the calculator sorts them, finds Q1 and Q3 as the medians of the lower and upper halves, and applies the rule. You can switch the multiplier to 3 to find only extreme outliers.
A flagged value deserves a look, not automatic deletion. It may be an error, or it may be a real and important observation.
How to Use the 1.5 IQR Rule Calculator
- Paste your values.
- Pick the multiplier, 1.5 for outliers or 3 for extreme outliers.
- Read the fences and the list of values outside them.
1.5 IQR Rule Formula
IQR
Fences
1.5 IQR Rule Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| Q1 | First quartile, the median of the lower half | |
| Q3 | Third quartile, the median of the upper half | |
| IQR | Interquartile range | Q3 − Q1 |
| Fences | Q1 − 1.5 IQR and Q3 + 1.5 IQR | Values outside are outliers |
1.5 IQR Rule Worked Example
With the example values loaded in the calculator above, the steps are:
- Sorted data: 4, 7, 9, 10, 11, 12, 13, 15, 40.
- Q1 = 8, Q3 = 14 (medians of the lower and upper halves). IQR = 14 − 8 = 6.
- Lower fence = 8 − 1.5 × 6 = -1. Upper fence = 14 + 1.5 × 6 = 23.
- A value is an outlier if it is below -1 or above 23.
Result: IQR (Q3 − Q1) = 6; Lower fence (Q1 − 1.5 × IQR) = -1; Upper fence (Q3 + 1.5 × IQR) = 23; Outliers = 40.
Worked example
Take 4, 7, 9, 10, 11, 12, 13, 15, 40. The lower half is 4, 7, 9, 10 with median 8, and the upper half is 12, 13, 15, 40 with median 14, so Q1 = 8 and Q3 = 14.
IQR = 14 − 8 = 6. The lower fence is 8 − 9 = −1 and the upper fence is 14 + 9 = 23. The value 40 is above 23, so it is an outlier.
Why 1.5
The factor 1.5 is a convention. It comes from the box plot whiskers and works well as a screening rule. Use 3 to pick out only extreme outliers.
Where 1.5 IQR Rule Is Used
Screening data
Flag unusual values before an analysis.
Box plots
Find where the whiskers end and which points are drawn as outliers.
Statistics coursework
Check outlier answers with the quartile and fence steps shown.
Common 1.5 IQR Rule Mistakes to Avoid
- Look at the box plot as well as the numbers.
- Check flagged values for typing errors before anything else.
1.5 IQR Rule Terms and Definitions
- IQR
- The interquartile range, the spread of the middle half of the data.
- Fence
- A cut-off beyond which values are flagged as outliers.
1.5 IQR Rule FAQs
What is the 1.5 IQR rule?
Values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are called outliers.
Why 1.5?
It is a widely used convention, not a law. It flags unusual points without catching too many ordinary ones.
Should I remove outliers?
Not automatically. Check whether they are errors or real, and report what you did.
What does multiplying by 3 do?
It widens the fences so only extreme outliers are flagged.