Outlier Calculator

The 3σ rule has a ceiling, and small samples sit under it

The most widely taught outlier rule cannot work on the sample sizes where outliers matter most, and the reason is arithmetic rather than statistical.

In a sample of n, the largest possible z-score is (n − 1)/√n. That is a hard ceiling, and it is easy to see why: the suspect value is inside the mean and the standard deviation it is being compared against. Push it further out and it drags both after it. The z-score converges rather than diverging.

At n = 10 that ceiling is 2.846. Take nine values of 1 and one value of a million: the most extreme point scores z = 2.846, under the threshold. At n = 11 the ceiling is 3.015, so a positive becomes possible — barely, and only for a point so extreme it dominates the entire sample.

So below n = 11 the 3σ rule returns “no outliers” for every possible dataset. It is not a weak test there; it is a constant function. The tool prints the ceiling next to your n so you can see when you are underneath it.

Grubbs’ test is the rule that fixes this properly, because it compares the same statistic against a critical value that depends on n. At n = 10 the 5% critical value is about 2.29, which sits below the 2.846 ceiling — so the test can actually reject. It is the right choice for small samples, and it is what the tool falls back to.

One limitation worth carrying: Grubbs tests for exactly one outlier. Applied repeatedly it suffers from the same masking problem described below, so for multiple suspected outliers the generalised ESD procedure or a robust rule is the better route.

Two outliers can hide each other

This is the failure that makes the z-score rule dangerous rather than merely weak, because it fails silently and in the direction of missing things.

Load the tool’s default dataset. Fifteen values clustered around 11, plus 60 and 62. Obvious to any human. The 3σ rule flags nothing.

Remove 62 and the rule immediately flags 60. The second outlier was inflating the standard deviation from 12.2 to 16.5, and at that width neither point reaches three standard deviations from a mean both of them had already pulled upward. Each outlier made the other look normal. This is called masking, and it gets worse as the number of outliers grows.

The mechanism is that the mean and standard deviation are not robust. A single arbitrary value can move a mean anywhere and inflate a standard deviation without limit. Building a detector out of them means the thing being detected corrupts the detector.

The modified z-score is immune, and cheaply so. It replaces the mean with the median and the standard deviation with the median absolute deviation. Both have a breakdown point of 50% — you can replace up to half the data with arbitrary values before either budges. On the same dataset it flags both points, and it does so whether there are one, two or five of them.

Its own blind spot is worth knowing. When more than half the values are identical the MAD is zero, and dividing by it is meaningless. The tool reports that rather than substituting a small number and carrying on — a case that arises constantly with rounded or count data.

The 0.6745 constant in the modified z-score is 1/Φ⁻¹(0.75): it scales the MAD so that under normality it estimates the same quantity as a standard deviation. That is why the threshold is 3.5 rather than 3 — it was chosen to give a comparable false-positive rate, not to be stricter.

The quartile convention decides the answer

Tukey’s fences look like the objective choice, and they hide a decision that is not.

There is no single definition of a quartile. Nine numbered conventions are in common software use, plus two hinge rules from the box-plot literature — eleven in total, and they give different answers on the same data whenever n is not a multiple of four. The quartile calculator covers the differences in full.

Since the fence is Q3 + 1.5 × IQR, a different quartile is a different fence. Take 1 2 3 4 5 6 7 8 9 15. The upper fences under the eleven conventions run from 14.5 to 17.0, and exactly one of them flags the 15 — type 7, which happens to be the default in R, NumPy and Excel’s PERCENTILE. Ten conventions say the value is fine and the most widely used one says it is an outlier.

Neither is wrong. They are answering slightly different questions about where a quartile sits between order statistics. But it means “this point is an outlier by the IQR rule” is incomplete without naming the convention, and two analysts with the same data and the same method can reach opposite conclusions.

The practical habit: state which you used, and when a point sits near the fence, treat that as the finding rather than the flag. A value that one convention catches and ten do not is not a clear outlier under any of them — it is borderline, and reporting it as flagged overstates the evidence.

The 1.5 multiplier is itself a convention. Tukey chose it because on normal data it flags about 0.7% of points — frequent enough to be useful, rare enough not to be noise. The 3.0 multiplier marks “far out” values at about 1 in 500,000. Neither number is derived from anything; they are calibrated judgements that have held up.

What to do with one once you have found it

Detection is the easy half. The decision that follows is where the analysis is actually won or lost, and the default should not be deletion.

First, check whether it is an error. A weight of 700 kg, a date in 1900, a decimal point in the wrong place, a missing value coded as −999. These are the only outliers with an uncontroversial fix, and they are more common than genuine extremes. Look at the raw record, not the statistic.

If it is real, it is data. Deleting a real observation because it is inconvenient is fabrication, however routine it has become. The extreme values in a dataset are frequently the most informative ones — the fraud, the failure, the breakthrough patient, the outage.

The honest move is to report both analyses. Run it with and without, and say what changed. If the conclusion holds either way, the outlier does not matter and you have shown it. If it does not hold, that is the single most important sentence in the write-up — your result depends on one observation, and readers need to know.

Prefer methods that do not need the decision. The median instead of the mean, a Spearman correlation instead of Pearson, a rank-based test instead of a t-test. Robust methods let outliers stay in the dataset without letting them drive the answer, which sidesteps the judgement call entirely.

Never delete on the strength of a rule alone. Every threshold on this page flags some fraction of perfectly ordinary data by construction — the 1.5 × IQR rule flags about 0.7% of any normal sample. In a dataset of 10,000 that is 70 points that are outliers by definition and unremarkable in fact.

A point can be ordinary on every variable and extreme jointly

Every rule on this page examines one variable at a time. That misses a whole class of outlier, and it is the class most likely to matter.

Take height and weight. A person 150 cm tall is unremarkable. A person weighing 110 kg is unremarkable. Someone who is both is extremely unusual — and no single-variable rule can see it, because neither number is out of range on its own.

The failure gets worse as variables multiply. With twenty variables there are 190 pairs and a vast interior in which a point can sit far from the data cloud while landing mid-range on every axis. Screening column by column will pass it every time.

Mahalanobis distance is the standard answer. It measures distance from the centre in units that account for how the variables covary, so it stretches the notion of “far” along the directions the data actually varies in. Under multivariate normality its square follows a chi-square distribution with degrees of freedom equal to the number of variables, which gives a threshold rather than a judgement.

It inherits the masking problem, and worse. Mahalanobis distance uses the mean vector and the covariance matrix, both of which outliers corrupt — and the corruption compounds across dimensions. Robust versions replacing them with a minimum covariance determinant estimate exist for exactly that reason, and they are the multivariate analogue of preferring the median to the mean.

The cheap first step is a scatter plot. For two or three variables, looking at the data finds joint outliers immediately and needs no threshold at all — the same argument the correlation page makes about Anscombe’s quartet.

Sources and methodology

References for the four rules and the ceiling.

• NIST/SEMATECH e-Handbook — Grubbs’ test and its critical values
• Iglewicz & Hoaglin on the modified z-score — the source of the 0.6745 scaling and the 3.5 threshold
• Shiffler, “Maximum Z Scores and Outliers”, The American Statistician 42(1) — the (n − 1)/√n ceiling
• Quartile Calculator — the eleven conventions, in full
• Standard Deviation Calculator — the statistic an outlier corrupts
• How we calculate — methodology

Method. All four rules run on every dataset rather than one being chosen for you, because their disagreement carries information a single verdict discards. Grubbs’ critical value is derived from the t distribution at n − 2 degrees of freedom rather than read from a table, so it is available at any n and any significance level. Masking is detected by re-running the z rule with the single most extreme point removed and reporting when the count goes up. The suite asserts the (n − 1)/√n ceiling against a brute-force search over extreme datasets, that the default dataset flags one point with one outlier and zero with two while the modified z-score flags both, and that on [1…9, 15] exactly one of the eleven quartile conventions flags the 15. That engine is verified on every change against 50 assertions. The count and the per-case breakdown are published on the formula verification page.

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