Quick answer
The 1.5×IQR rule flags values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR. With Q1 = 3 and Q3 = 8 (IQR 5) the fences are −4.5 and 15.5, so a value of 100 is an outlier.
Enter at least four values. The calculator returns the outliers, the quartiles, the IQR and the fences.
Related tools: Standard Deviation Calculator: Sample or population standard deviation with variance and mean. Z-Score Calculator: Standardize a value using a mean and standard deviation. Mean Median Mode Range Calculator: Mean, median, mode and range of a data set.
How to use the Outlier Calculator
- Type data set, for example 4, 8, 15, 16, 23, 42.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Data set | Text, such as 4, 8, 15, 16, 23, 42 | — |
Formula and method
Formula
lower fence = Q1 − 1.5·IQR; upper fence = Q3 + 1.5·IQR
IEEE-754 double precision; shown to at most 6 decimal places.
Quartiles are the medians of the lower and upper halves (the median is excluded from both halves when n is odd). Values outside the fences are outliers.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Data set = 1, 2, 3, 4, 5, 6, 7, 8, 9, 100.
| Result | Value |
|---|---|
| Outliers | 100 |
| Q1 | 3 |
| Q3 | 8 |
| IQR | 5 |
| Lower fence | -4.5 |
| Upper fence | 15.5 |
Example 2
Inputs: Data set = 10, 12, 12, 13, 12, 11, 14, 13.
| Result | Value |
|---|---|
| Outliers | None |
| Q1 | 11.5 |
| Q3 | 13 |
| IQR | 1.5 |
| Lower fence | 9.25 |
| Upper fence | 15.25 |
The first line is the headline result (outliers). Below it the outlier Calculator lists q1, q3, iQR, lower fence and upper fence, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
The fences
1.5 × 5 = 7.5, so the lower fence is 3 − 7.5 = −4.5 and the upper fence 8 + 7.5 = 15.5. Anything outside is flagged. This is Tukey's rule, used for box plots.
What to do
An outlier is not automatically an error. Check whether it is a data-entry mistake, a rare but genuine value or a sign of a different population. Removing outliers changes the mean and standard deviation, so document the choice.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Data set: entering "1, 2, 3" shows "Enter at least 4 numbers." beside that field and no result.
Choosing between the Outlier Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Standard Deviation Calculator](/statistics/standard-deviation-calculator/) | data set and data type | Population standard deviation (σ) |
| [Z-Score Calculator](/statistics/z-score-calculator/) | value, mean and standard deviation | Z-score |
| [Mean Median Mode Range Calculator](/statistics/mean-median-mode-range-calculator/) | data set | Mean |
| [Median Calculator](/statistics/median-calculator/) | data set | Median |
| [Relative Standard Deviation Calculator](/statistics/relative-standard-deviation-calculator/) | data set and data type | Relative standard deviation |
Background: descriptive statistics
Population statistics divide by n and sample statistics divide by n − 1 (Bessel's correction), because a sample tends to understate variability. Use the population version when your data is every value in the group, and the sample version when it is a subset.
Defaults differ between tools, and so do quartile conventions, which can give slightly different values for small data sets; this page states the method it uses.
Assumptions and limitations
- Needs at least four values
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for data set to confirm this.
Frequently asked questions
What is the IQR?
The IQR is Q3 − Q1, the spread of the middle half of the data; for Q1 = 3 and Q3 = 8 it is 5.
Is 1.5 a rule of nature?
No, 1.5 is Tukey's convention that works well for roughly bell-shaped data; other multipliers such as 3 are used for far outliers.
What are extreme outliers?
Extreme outliers lie more than 3 × IQR beyond the quartiles, compared with 1.5 × IQR for ordinary outliers.
What does the Outlier Calculator calculate?
Find outliers with Tukey's 1.5 × IQR fences. It returns Outliers, Q1, Q3, IQR and Lower fence.
What formula does the Outlier Calculator use?
lower fence = Q1 − 1.5·IQR; upper fence = Q3 + 1.5·IQR. Quartiles are the medians of the lower and upper halves (the median is excluded from both halves when n is odd).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Needs at least four values.
What is an example of a outlier result?
With data set 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, the outliers is 100.
Related calculators
- Standard Deviation Calculator — Sample or population standard deviation with variance and mean.
- Z-Score Calculator — Standardize a value using a mean and standard deviation.
- Mean Median Mode Range Calculator — Mean, median, mode and range of a data set.
- Median Calculator — Median of a data set.
- Relative Standard Deviation Calculator — Relative standard deviation (RSD) as a percentage.
- Variance Calculator — Sample or population variance.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.