What this calculator does
Paste in a list of numbers and find out which ones are unusually far from the rest. Those values are called outliers. The most widely used rule for spotting them comes from the statistician John Tukey. It builds two fences from the quartiles, the points that split your sorted data into four equal parts. Anything below the lower fence or above the upper fence is flagged. Because the fences come from quartiles rather than the mean (the average), the outliers themselves can't drag the fences outward.
This calculator finds the quartiles and the interquartile range (IQR), which is the spread of the middle half of your data. It sets inner fences at 1.5 IQRs and outer fences at 3 IQRs, then lists every value outside them. You can change both multipliers and choose how the quartiles are calculated.
Where it's used
Why this math matters outside the classroom:
- Data entry errors. An analyst spots a recorded age of 250 far above the upper fence and checks the record for a typo.
- Lab timings. A student whose pendulum timings are 2.1, 2.2, 2.2, 2.3, 2.3, 2.4 and 4.0 s finds 4.0 beyond the upper fence and redoes that trial.
- Housing reports. A housing report sets aside a $12 million mansion far beyond the upper fence before describing typical prices.
- Sensor monitoring. Engineers treat temperature readings below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR as possible sensor faults.
- Card charges. A bank flags a $9,000 charge on a card whose usual purchases sit far below the upper fence of the customer's spending.
Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSS.ID.A.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Enter or paste at least four numbers, separated by commas, spaces or new lines.
- Choose the quartile method. Median of halves is the common classroom method. Linear interpolation matches the QUARTILE.INC function in spreadsheets.
- Keep the fence multiplier at 1.5 and the extreme multiplier at 3 for the standard rule, or change them if your field uses different values.
- Read the outliers, the extreme outliers and the four fences, then check the working for the quartiles and fence arithmetic.
What the inputs mean
- Data set
- Separate with commas, spaces, or new lines. Don't use thousands separators.
- Quartile method
- Quartiles can be defined more than one way, and the fences move with them. Pick the method your course or software uses.
- Fence multiplier (k)
- The inner fences sit k IQRs beyond Q1 and Q3. Smaller k flags more values; larger k flags fewer.
- Extreme multiplier
- The outer fences use this larger multiplier. Values beyond them are reported as extreme outliers. It must be at least as large as k.
Formula and method
IQR = Q3 − Q1 lower fence = Q1 − k × IQR, upper fence = Q3 + k × IQR (k = 1.5) outer fences = Q1 − 3 × IQR and Q3 + 3 × IQR outlier: x < lower fence or x > upper fence
Q1 is the first quartile, the value a quarter of the way up your sorted data. Q3 is the third quartile, three quarters of the way up. The IQR is the gap between them, the width of the middle half of the data. k is the fence multiplier, and x is any single value in your list. Stepping 1.5 IQRs out from each quartile gives a range that comfortably contains typical values. For normally distributed data (the familiar bell curve), only about 0.7% of values fall outside these fences. So a value beyond them is genuinely unusual.
The outer fences at 3 IQRs mark values that are far beyond what the bulk of the data suggests. A value exactly on a fence is not counted as an outlier.
Worked example
Eleven readings: 12, 15, 14, 18, 16, 13, 17, 15, 44, 16 and 1, with median-of-halves quartiles.
- Sort: 1, 12, 13, 14, 15, 15, 16, 16, 17, 18, 44. The median is the 6th value, 15.
- Lower half 1, 12, 13, 14, 15 gives Q1 = 13. Upper half 16, 16, 17, 18, 44 gives Q3 = 17.
- IQR = 17 − 13 = 4.
- Lower fence = 13 − 1.5 × 4 = 7. Upper fence = 17 + 1.5 × 4 = 23.
- Outer fences: 13 − 3 × 4 = 1 and 17 + 3 × 4 = 29.
- 1 is below 7 and 44 is above 23, so both are outliers. 44 is also beyond 29, so it is extreme; 1 sits exactly on the lower outer fence, so it is not.
Result: Outliers: 1 and 44, with fences at 7 and 23. Only 44 is an extreme outlier.
Understanding the result
An outlier is a prompt to look closer, not an automatic reason to delete a value. It may be a typing or measurement error, or it may be a real and important observation.
If you remove outliers, report that you did and why. Results such as the mean and standard deviation can change a lot when an extreme value is dropped.
With the interpolation method in the example, Q1 = 13.5 and Q3 = 16.5, so the fences tighten to 9 and 21 and both 1 and 44 become extreme outliers. Small data sets are sensitive to the method.
Assumptions and limitations
- The rule makes no assumption about the shape of the distribution, though the 1.5 multiplier was chosen with roughly symmetric data in mind.
- Values exactly equal to a fence are treated as inside it.
- For strongly skewed data, many legitimate values in the long tail can be flagged. Consider transforming the data or using a skew-adjusted method.
- With very small data sets the quartiles are unstable, so fences can shift a lot if one value changes.
- This finds single-variable outliers only; it can't detect unusual combinations of several variables.
Frequently asked questions
Why 1.5 times the IQR?
It is a convention proposed by Tukey that balances catching real anomalies against flagging ordinary values. For normally distributed data it marks well under 1% of values. Some analysts use a larger multiplier, such as 2.2, to flag fewer values.
What is the difference between an outlier and an extreme outlier?
An outlier lies beyond the inner fences, 1.5 IQRs from the quartiles. An extreme outlier lies beyond the outer fences, 3 IQRs from the quartiles. Every extreme outlier is also an outlier.
Can I use z-scores to find outliers instead?
You can flag values with a z-score beyond ±3. A z-score counts how many standard deviations a value sits from the mean. But the mean and standard deviation are themselves pulled by the outliers, which can hide them. The fence method uses quartiles, which are more resistant.
Why does my textbook give different fences?
Most likely it calculates quartiles a different way. Try switching the quartile method; the fences depend directly on Q1 and Q3.
How this calculator is checked
Method: Tukey's fences (Q1 − k·IQR, Q3 + k·IQR, with k = 1.5 inner and 3 outer) from exploratory data analysis; quartiles by median of halves or QUARTILE.INC-equivalent interpolation. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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