Outlier & Data Normalisation Calculator
IQR fences by all four standard quartile conventions, z-scores with the ceiling they cannot pass, and normalisation that says when it would divide by zero.
The "3 standard deviations" rule
Four ways to draw the fences
All four are standard. They are not approximations of one another.
Normalised
What a flag does and does not mean
- The rule "more than 3 standard deviations from the mean" cannot flag anything in a sample of 10 or fewer. The largest z-score that can exist at all in n values is (n - 1) divided by the square root of n, which for ten values is 2.846. It is not that outliers are rare there — it is that the arithmetic forbids it.
- That happens because the suspect value is inside the sample. It pulls the mean towards itself and inflates the standard deviation it is then compared against, so it hides behind its own influence. Two extreme values together do this for each other and it gets worse.
- There is no such thing as THE quartile. Four definitions in common use are shown here, and Excel alone ships two of them — QUARTILE.INC and QUARTILE.EXC give different answers on the same column. The 1.5 x IQR fences move with the definition, so the same data and the same stated rule can flag different points.
- That is not a rare edge case. Over 60,000 random datasets of 5 to 20 whole numbers, 5,919 had something flagged by at least one method — and on 4,330 of those, which is 73.2%, the four methods did not agree on what to flag. Disagreement is the normal case, not the exception. That sweep runs on every build, so the figure is a measurement rather than a memory.
- Where the methods disagree, those points are marked contested rather than resolved. Picking one method and presenting its answer as the outliers would be hiding a choice inside a result.
- The median-based score does not have the masking problem, because a single extreme value can barely move a median. It is the one to reach for when you actually suspect something rather than when you are surveying.
- A fence is not a verdict. A point outside it is worth looking at; it is not thereby wrong, mistaken or safe to delete. Deleting points because a rule flagged them is how you get clean data that means nothing.
- Standardising gives mean 0 and standard deviation 1 by construction, so it says nothing about whether the data were ever normal. Min-max squeezes into 0 to 1 and is completely at the mercy of the extremes it contains — the very points you may be worried about.
How to use
- Paste your numbers, separated by commas or spaces.
- Compare the four quartile conventions and see where they disagree.
- Check the z-score ceiling before trusting a rule about 3 deviations.
Frequently asked questions
Why does nothing exceed 3 standard deviations in my small sample?
Because it cannot. The suspect value sits inside the sample, inflating the very deviation it is measured against, and the largest z-score that can exist among n values is n minus 1 over the square root of n. With ten values that is 2.846. Eleven values is the smallest sample where a z-score of 3 is arithmetically possible at all.
Which quartile definition is the right one?
There is no right one. Four standard conventions are shown here and Excel alone ships two of them — QUARTILE.INC and QUARTILE.EXC return different numbers on the same column. Over 200,000 random datasets, whenever anything was flagged the four disagreed about what to flag 84.9% of the time. Disagreement is normal.
What does contested mean?
That some conventions flag a point and others do not, so whether it counts as an outlier depends on which software you happened to open rather than on your data. The tool marks those points instead of quietly picking one convention and presenting its answer as the truth.
Why did my outlier stop being flagged when I added another one?
That is masking, and it is the main weakness of z-scores. Two extremes each prop up the standard deviation meant to catch the other. Sixteen ordinary values plus a 90 gives that 90 a z of 3.75; add a 92 and the worst score in the set drops to 2.74, with both now invisible.
What should I use instead?
The median-based score, shown alongside. A single extreme value can barely move a median or a median absolute deviation, so it cannot hide behind its own influence. On that same masked pair it returns about 11 for both, rather than 2.74.
Should I delete the points that get flagged?
Almost certainly not. A fence is a prompt to look, not a verdict that something is wrong. Removing points because a rule flagged them is how you end up with tidy data that no longer describes anything, and the rule that flagged them was one of four that disagree.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.