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Chauvenet's criterion
Definition
Chauvenet's criterion is a traditional rule for flagging an unusually distant observation under an assumed normal distribution. It compares the two-sided normal-tail probability associated with the observation's standardized distance from the mean with 1/(2n), where n is the sample size. Falling below that threshold suggests a point deserves investigation; it does not prove the observation is erroneous.
What it means
The rule asks whether fewer than half an observation would be expected beyond that distance in a sample of the given size under the normal model. Larger datasets naturally offer more opportunities for extreme values, so the cutoff changes with sample size. This is a model-based screening convention, not a universal definition of bad data or a standard fixed-significance hypothesis test. Its usefulness depends on the normality assumption and on how the center and spread are estimated.
How it works
Calculate the sample mean and a documented standard-deviation estimate. For a candidate value x, form z = absolute value of x minus the mean, divided by the standard deviation. Obtain p = 2[1 - Phi(z)], the combined upper and lower normal-tail probability. The criterion flags the observation when n × p is less than 0.5, equivalently when p is below 1/(2n). Software may differ in standard-deviation conventions or iterative removal procedures. Record those choices, inspect the measurement context and assess sensitivity rather than repeatedly deleting points until a preferred result appears.
Three practical examples
- Laboratory data review: flag an extreme repeated measurement for checks of transcription, instrument operation and experimental conditions.
- Sensor-quality assessment: use a normal-model screen as one diagnostic for a stable measurement channel, while retaining the original records.
- Statistical education: compare a sample-size-dependent outlier rule with a fixed three-standard-deviation rule and discuss their assumptions.
Think of it this way
Think of judging an unusually tall wave against a model of ordinary waves and the number observed. A rare wave is less surprising in a very long record. But if the model ignores storms, a rule based on ordinary conditions may flag genuine storm behavior rather than a measurement mistake.
Worked example
Take ten illustrative measurements: 9.8, 9.9, 10.0, 10.1, 10.2, 9.8, 10.0, 10.1, 9.9 and 15.0. Using the sample standard deviation with denominator n - 1, their mean is 10.48 and standard deviation is approximately 1.59360. For 15.0, z = (15 - 10.48)/1.59360, approximately 2.83635. The two-sided standard-normal tail probability is approximately 0.004563. Multiplying by ten gives 0.04563, below 0.5, so the criterion flags 15.0. It does not identify why that value differs. To see the role of sample size, consider a separate standardized distance of exactly three. Its two-sided normal-tail probability is about 0.002700. With n = 100, n × p is about 0.270 and the rule flags it; with n = 1,000, the product is about 2.700 and it does not. These are screening calculations using normal-tail reference values, not posterior probabilities that the measurements are wrong.
Illustrative case study
Fictional case: Birch Calibration flags an unusually high instrument reading using Chauvenet's criterion. Staff check the original record and find that the value was measured immediately after an undocumented change in test conditions. They preserve the reading, clarify whether it belongs to the intended analysis population and report results under justified inclusion rules. They do not delete it simply because the rule flagged it. This fictional case illustrates investigation and documentation without asserting that the statistical flag established a fault.
Common misconceptions and mistakes
Extreme values may be valid observations from a heavy-tailed distribution, a mixture or a changing process. The sample mean and standard deviation are themselves sensitive to unusual observations, so several outliers can mask one another. A flagged observation is not automatically a data-entry error, and an unflagged observation is not guaranteed correct. The normal-tail calculation using estimated parameters is not an exact calibrated test for every finite sample. Repeated application changes the procedure and can remove legitimate observations. A fixed three-sigma rule and Chauvenet's criterion are not equivalent across sample sizes.
Business use
Businesses should use the criterion only within a documented data-quality workflow that separates detection from disposition. Establish whether the process is stable and whether normal-tail reasoning fits the measurement. Check records, units, timing and subgroup membership before excluding a point. In AI pipelines, indiscriminate outlier removal can erase rare failures, important minority patterns or precisely the anomalies a model needs to learn. Compare conclusions with robust alternatives or sensitivity analyses, and retain both raw data and reasons for any changes.
People also ask
Does the rule require a normal distribution?
Its usual tail-probability calculation assumes one; another distribution needs a different justification.
Is the cutoff always three standard deviations?
No. It depends on sample size.
Should a flagged value be deleted automatically?
No. Investigate its validity and relevance, and use a prespecified, defensible treatment of unusual observations.
Check your understanding
Q: Why does a three-standard-deviation observation fail the criterion in the larger hypothetical sample?
Answer: A: The expected number of such extreme observations increases with sample size. A value that is rare in 100 observations is less surprising when there are 1,000 opportunities.
Related terms
- Outlier
- Normal distribution
- Masking
- Robust statistics
- standardized residual
- tail probability
- Grubbs' test and sensitivity analysis
Learn next
reproduce the example using both common standard-deviation denominators and record the difference. Then add a second extreme value and inspect how it changes the mean, spread and flags. Compare with a robust summary while keeping the original measurements intact.
More in Statistics and probability
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- CERNO test - The CERNO test, short for Coincident Extreme Ranks in Numerical Observations, is a rank-based enrichment test.
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- Chi-square test - A chi-square test is a statistical hypothesis test that compares a calculated statistic with a chi-squared reference distribution.
- Chi-squared distribution - The chi-squared distribution is a family of continuous probability distributions used to describe certain sums of squared standardized quantities.
Sources
Sources: University of Virginia Research Computing, Chauvenet criterion calculation: https://learning.rc.virginia.edu/courses/cpp-introduction/project4/ ; NIST, “Detection of outliers in measurements,” assumptions and context: https://www.nist.gov/publications/detection-outliers-measurements ; NIST Engineering Statistics Handbook, outlier detection and interpretation: https://www.itl.nist.gov/div898/handbook/eda/section3/eda35h.htm .
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