What this calculator does
Tukey fences sit 1.5 interquartile ranges below the first quartile and above the third. For a data set with quartiles at 11.25 and 14.75, the IQR is 3.5 and the fences fall at 6 and 20, making a value of 200 unambiguously an outlier.
The 1.5 multiplier is a convention, not a derivation. Tukey chose it because it flags roughly 0.7% of values from a normal distribution, which is a useful rate: frequent enough to catch real anomalies, rare enough not to flag ordinary variation. The 3.0 multiplier marks points that are extreme even by outlier standards.
The formula
The first and third quartiles are found and their difference gives the interquartile range. The lower fence is Q1 minus 1.5 times the IQR and the upper fence is Q3 plus 1.5 times it. Values outside those bounds are listed as outliers. The far-out fences use 3 times the IQR instead, identifying points that are extreme even among the outliers.
| Term | Meaning |
|---|---|
| Tukey fence | A cutoff at 1.5 IQR beyond a quartile, the standard outlier criterion. |
| Far-out fence | The same construction at 3 IQR, marking extreme outliers. |
| IQR | The interquartile range, Q3 minus Q1, which sets the scale for the fences. |
| Outlier | A value beyond the fences. It is a flag for investigation, not a verdict that the value is wrong. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. The fences are computed from the quartiles, so extreme values do not affect where the fences fall. |
When to use it
Screening data before analysis
Identifying candidates for investigation is a standard first step, since a single data entry error can distort everything downstream.
Drawing a box plot
The whiskers of a standard box plot extend to the most extreme points inside the fences, with anything beyond drawn individually.
Setting an automated threshold
The fences give a distribution-free rule for flagging unusual values in monitoring and quality work.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Where do the fences fall for different data?
Clean data and data containing extreme values.
| Data set | Outlier fences | Outliers found | Far-out fences (×3) |
|---|---|---|---|
| 1, 2, 3, 4, 5, 6, 7, 8 | -2.500 to 11.500 | none | -7.750 to 16.750 |
| 10, 12, 14, 16, 18, 20, 22, 60 | 3.000 to 31.000 | 60 | -7.500 to 41.500 |
| 12, 15, 14, 10, 13, 200, 11, 16, 12, 9 | 6.000 to 20.000 | 200 | 0.7500 to 25.250 |
Questions
Why 1.5 times the IQR?
It is a convention Tukey chose for its practical flagging rate, not something derived from theory. For normally distributed data it flags about 0.7% of values, which is frequent enough to catch genuine anomalies without constantly flagging ordinary variation.
Should I remove outliers that are flagged?
Not automatically. A flagged value may be a data entry error, a measurement fault or a genuine extreme observation, and those call for different responses. Investigate first. Removing real extreme values because a rule flagged them will bias your results and understate true variability.
What is the difference between an outlier and a far-out point?
Distance. An outlier lies beyond 1.5 IQR from a quartile, a far-out point beyond 3 IQR. The second is sometimes called an extreme outlier and is more likely to indicate an error rather than genuine variation, though it is still a prompt to investigate rather than a verdict.
Do the fences work for skewed data?
Less well. Because they are symmetric multiples of the IQR, they flag more points on the long tail of a skewed distribution than on the short one, even when nothing is wrong. Adjusted versions exist for skewed data, and for strongly skewed data a transform first is often the better answer.
For the interquartile range on its own, see the interquartile range calculator. For half the IQR as a spread measure, see the quartile deviation calculator.