Outlier Calculator
Find the values that do not belong — by the IQR fence rule, by z-score and by the modified z-score, which is the one that works when the outliers are large.
Three rules, deliberately shown together because they disagree.
How the outlier calculator works
Three rules, deliberately shown together because they disagree. The IQR rule flags anything more than 1.5 × IQR beyond a quartile. The z-score rule flags |z| > 3 — but the outlier inflates the standard deviation it is measured against, so extreme values mask themselves. The modified z-score uses the median and MAD instead, which cannot be inflated.
When the three disagree, trust the modified z-score.
Formula: fences: Q1 − 1.5 IQR, Q3 + 1.5 IQR; modified z = 0.6745 (x − median) / MAD
Worked examples
| Inputs | Outliers found (IQR rule) | Note |
|---|---|---|
| Nine ordinary values and a 102 | 1 | the IQR and modified-z rules both catch it; the plain z-score does not |
| Clean data | 0 | no outliers |
| A stricter fence | 1 | 3.0 flags only extreme outliers |
FAQFrequently asked questions
What counts as an outlier?
There is no single answer. The common rules are 1.5 × IQR beyond a quartile, |z| > 3, or a modified z-score above 3.5.
Why do the rules disagree?
Because the z-score rule uses the mean and standard deviation, which the outlier itself inflates. A single extreme value can hide behind the spread it creates — this is called masking.
Which rule should I trust?
The modified z-score, which uses the median and MAD. Neither can be inflated by the point being tested.
Should I delete outliers?
Almost never automatically. Investigate first: a data-entry error should go, a genuine extreme observation usually should not. Deleting inconvenient points is how analyses get quietly wrong.
What if my data is skewed?
Then the symmetric fences over-flag the long tail. Consider transforming the data first, or use a rule designed for skewed distributions.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.