Descriptive statistics

Outlier Calculator

Flag unusual values by three different rules — and see what removing them would actually change.

Updated August 2026Runs in your browser — nothing is uploadedVerified against R

How to read this result

Flagged is not the same as wrong. These rules find values that are far from the others. Whether that means a typo, a different population, or a genuine extreme case is a question the data cannot answer.

Chauvenet's criterion is the one that knows how big your sample is. The other three compare each point against a fixed threshold — 1.5 IQRs, 3 standard deviations — no matter how many observations you have. Chauvenet asks a different question: out of n values, how many would you expect to fall this far from the mean under a normal distribution? If the answer is less than half of one observation, the point is rejected. Because the expectation scales with n, so does the fence: at n = 10 it sits at 1.96 SD, at n = 100 at 2.81, at n = 1,000 at 3.48. A value 3 SD out is remarkable in ten observations and unremarkable in a thousand, and this is the only rule here that says so.

The four rules disagree, and the disagreement is informative. The IQR rule uses quartiles and ignores the tails. The z-score rule uses the mean and standard deviation — both of which the outlier itself inflates, which is why a single extreme value can hide from it entirely. The modified z-score uses the median and the median absolute deviation, so it does not suffer from that masking.

Small samples break the z-score rule outright. With n = 10, the largest possible |z| is about 2.85 — so a rule of "flag |z| > 3" can never flag anything, no matter how extreme the value.

Deleting an outlier needs a reason from outside the data. "It was far from the others" is not one. "The sensor was recalibrated that morning" is. If you do remove values, report that you did and show both results — which is why the comparison table above exists.

The formula

fences = Q_{1} - k IQR, Q_{3} + k IQR M_{i} = \frac{0.6745 (x_{i} - \tilde{x})}{MAD}

$k$: 1.5 for the standard box-plot rule, 3 for “extreme” outliers
$\tilde{x}$: the median
$M A D$: the median absolute deviation from the median

The 0.6745 scales the modified z-score so that it matches an ordinary z-score for normal data. The usual threshold is 3.5.

Worked example

One delivery that took three times as long

Ten delivery times in minutes: 12, 14, 11, 15, 13, 12, 14, 16, 13, 45. The 45 stands out — but does a rule agree, and what happens to the summary if it goes?

Find the quartiles.
Q1 = 12.25, Q3 = 14.75
The interquartile range.
IQR = 2.5
The fences sit 1.5 × IQR outside them.
12.25 − 3.75 = 8.5 and 14.75 + 3.75 = 18.5
Only the 45 lies outside.
1 outlier
Compare the summary with and without it.
mean 16.5 → 13.3; SD 10.12 → 1.58

outliers 1 (the 45)lower fence 8.5upper fence 18.5mean without it 13.33

Removing one value out of ten cuts the standard deviation by 84%. That is exactly why the decision needs a justification: it changes every downstream number.

Checked against Tukey's fences computed from R's quantile(type = 7).

The calculator above is loaded with these numbers by the Load the worked example button.

Assumptions, and when to use something else

The IQR rule assumes only orderable data. The z-score rule assumes roughly normal data and enough of it — below about 20 observations it is unreliable, and below 10 it can be mathematically incapable of flagging anything. The modified z-score assumes the median absolute deviation is not zero, which fails when more than half the values are identical.

None of the three tests whether a value is wrong. They test whether it is far.

You want to see the outliers in contextBox plot makerDraws the fences and marks the flagged points.
You want the fences' ingredientsFive-number summaryThe quartiles and IQR the rule is built on.
Outliers are ruining your testMann-Whitney U testRank-based tests are unaffected by how extreme an extreme value is.
You want robust summaries insteadRange and MADMedian absolute deviation and the IQR, which outliers cannot dominate.

Questions people ask

Should I remove outliers?

Only with a reason that comes from outside the data — a known measurement error, a subject who did not follow the protocol, a value that is physically impossible. Removing points because they are inconvenient changes the answer and, if unreported, misrepresents it. When in doubt, report both analyses.

Why does the z-score rule miss obvious outliers?

Because the outlier inflates the very standard deviation it is being compared against — a phenomenon called masking. With a small sample the effect is severe: at n = 10 no value can have |z| above 2.85. The modified z-score, which uses the median absolute deviation, does not have this problem.

What does 1.5 × IQR mean?

It is Tukey's convention for the box-plot whiskers, chosen so that about 0.7% of normal data falls outside — roughly 1 in 150 observations. Using 3 × IQR instead flags only “far out” values.

Can I have too many outliers?

If a rule flags 10% of your data, the rule is telling you the distribution is heavy-tailed or skewed, not that you have a hundred errors. Consider a transformation or a robust method rather than deleting a tenth of the sample.