Outlier Calculator
Finds outliers by Tukey’s method — anything beyond 1.5 interquartile ranges from the quartiles.
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Your details
At least four values are needed to form quartiles.
Result
- Outliers found
- 87
- Fences
- 3 to 27
- Quartiles
- Q1 12 · Q3 18
- Mean with outliers
- 22.4444
- Mean without them
- 14.375
- Median (unaffected)
- 15
1 of 9 values fall outside the fences
Q1 − 1.5×IQR to Q3 + 1.5×IQR
IQR 6
Removing them moves the mean by 8.0694
The median barely moves, which is why it is preferred for skewed data
9 values, sorted…
Outlier if x < Q1 − 1.5×IQR or x > Q3 + 1.5×IQR
Q1 = 12, Q3 = 18
IQR = 18 − 12 = 6
1.5 × 6 = 9
Outliers: 87
Formula — written step by step
Nine ordinary values and one of 87…
Outlier if x < Q1 − 1.5×IQR or x > Q3 + 1.5×IQR
Q1 = 12, Q3 = 18
IQR = 18 − 12 = 6
1.5 × 6 = 9
Fences: 3 and 27
Outliers: 87
Q1 and Q3 are the lower and upper quartiles, and the interquartile range is the distance between them. Tukey chose 1.5 as a working compromise: wide enough that ordinary variation passes, narrow enough that genuine anomalies do not.
A worked example
Why the median survives what the mean does not
- Data
- 12, 15, 11, 18, 15, 20, 15, 9
- Then add
- a single value of 87
- Before — mean 14.375, median 15
- After — mean 22.44, median 15
- The mean moves by over 8
- The median does not move at all
One value shifts the mean by 56%; the median is untouched
The median has a breakdown point of 50%: half the data must be corrupted before it fails. The mean has a breakdown point of zero — a single sufficiently extreme value can drag it anywhere. This is why house prices and salaries are reported as medians.
How to use this calculator
- Paste your values, separated by commas, spaces or new lines. At least four are needed to form quartiles.
- Read the fences: anything outside them is flagged.
- Compare the mean with and without the flagged values to see how much influence they carry.
- Investigate before deleting. An outlier may be a data-entry error, or it may be the most important observation you have.
Tukey’s fences
The interquartile range is the span of the middle half of the data — from the 25th to the 75th percentile. Tukey’s rule flags anything lying more than one and a half interquartile ranges beyond either quartile as an outlier.
Its virtue is robustness. Quartiles are unaffected by how extreme the extreme values are, so the method cannot be subverted by the very points it is looking for. This is the same logic that makes a box plot readable, and the fences drawn here are the whiskers on one.
Why not just use the standard deviation
Flagging anything beyond three standard deviations is intuitive but self-defeating on small samples: an extreme value inflates the standard deviation it is being measured against, and can end up inside its own threshold.
The effect is severe with few observations. In a sample of ten, no value can exceed about 2.85 standard deviations from the mean whatever it is, so a |z| > 3 rule flags nothing at all — no matter how obviously wrong a point may be.
Finding one is not permission to remove it
An outlier is a value that does not fit the pattern of the rest. That may be a transcription error, an instrument fault, or a genuine and important observation. The statistics cannot tell you which, and deleting points because they are inconvenient is how findings become unreproducible.
The Antarctic ozone hole went unremarked for years because processing software discarded readings as implausible outliers. The honest approach is to investigate the cause, report the analysis both with and without, and say plainly what was done.
Formula & methodology
What this calculator assumes
- Uses quartiles rather than the mean and standard deviation, because an extreme value inflates the standard deviation it would be judged against and can hide inside its own threshold.
- The 1.5 multiplier is a convention chosen by Tukey, flagging roughly 0.7% of values from a normal distribution. A 3.0 multiplier marks severe outliers.
- Identifying an outlier is not grounds for deleting it. It may be a data-entry error or the most important observation in the set.
Sources
- Tukey, Exploratory Data Analysis, Addison-Wesley, 1977 — the box plot and its fences.
Ways of identifying an outlier
| Method | Rule | Best for |
|---|---|---|
| Tukey / IQR | Beyond 1.5 × IQR from a quartile | Any distribution; robust to the outliers themselves |
| Extreme IQR | Beyond 3.0 × IQR | Flagging only severe cases |
| Z-score | |z| > 3 | Large, roughly normal samples |
| Modified z | Based on the median, not the mean | Small samples where one value distorts the SD |
The z-score method has a circularity problem: the outlier inflates the standard deviation it is being judged against, so an extreme value can mask itself. The IQR method uses quartiles, which the outlier cannot move.
Frequently asked questions
Why 1.5 times the IQR?
It is a convention chosen by John Tukey as a practical balance. For normal data it flags roughly 0.7% of values, which is rare enough to be interesting without being constant noise. A 3.0 multiplier is used for severe outliers.
Should I remove outliers?
Not automatically. Investigate the cause first. Remove a value only when there is an identified reason it is invalid, and report that you did.
Why does this use quartiles rather than the mean?
Because quartiles are robust. An extreme value shifts the mean and inflates the standard deviation, so a method based on them can be defeated by the outlier it is hunting.
How many values do I need?
At least four to form quartiles, and realistically eight or more for the fences to mean much. With very few points, an unusual value may simply be ordinary variation.