Does a large C mean that the group's mean is wrong?

No. C uses within-group variances and does not test mean accuracy.

Can C be used with unequal group sizes?

The classical calibration assumes equal sizes; use a specifically justified extension or another suitable method.

Is a flagged variance proof of bad data?

No. It indicates inconsistency with the specified model and calls for investigation of data, conditions and assumptions.

AI Glossary / Statistics and probability / Intermediate

Cochran's C test

Definition

Cochran's C test is an upper-tail variance-outlier test that examines whether the largest of several sample variances is unusually large relative to their total. In its classical form, it assumes independent, normally distributed observations and equal sample sizes across groups. It is especially associated with comparisons of laboratory repeatability. The test concerns variability within groups, not differences between their average results.

What it means

A group can be inconsistent even when its mean is close to the others. Cochran's statistic compares that group's spread with the combined spread estimates. A large value suggests that one variance dominates, under the equal-population-variance null model. It is not a general measure of data quality and does not establish why a group varies more. In an AI-assisted measurement workflow, a flagged group could reflect inconsistent acquisition, changing inputs, genuine heterogeneity or a processing issue that needs investigation.

How it works

Calculate the sample variance within each of k groups using the same convention. Form C as the largest variance divided by the sum of all k variances. Compare C with a suitable upper-tail reference value based on the number of groups, the common within-group degrees of freedom and the chosen significance level. The fact that the largest variance is selected from several groups is part of the test, so an ordinary two-group variance cutoff is not interchangeable. Inspect the underlying observations and assumptions before interpreting the result. If every variance is zero, the ratio is undefined rather than evidence from a successful test.

Three practical examples

Laboratory comparison: investigate a participant laboratory whose repeated measurements are much more variable than those of peers.
Measurement development: compare repeatability across controlled instrument setups with equal numbers of independent replicate observations.
AI workflow review: examine unusually variable repeated outputs under carefully standardized conditions, provided the normality and independence assumptions are credible.

Think of it this way

Think of comparing the consistency of several archers. Their average arrow positions might differ, but this question asks whether one archer's arrows are unusually scattered relative to everyone else's. The comparison does not identify the cause of the scatter or tell you whose average aim is closest to the target.

Worked example

Consider three illustrative groups with three observations each. Group A has 9, 10 and 11, giving sample variance 1. Group B has 19, 20 and 21, also giving sample variance 1. Group C has 30 minus the square root of 8, 30, and 30 plus the square root of 8, giving sample variance 8. The sample means are 10, 20 and 30, but those means do not enter C. The statistic is 8/(1 + 1 + 8) = 0.8. A significance decision requires the reference distribution for three groups and two within-group degrees of freedom; the number 0.8 alone is not a universal rejection rule. Multiplying every observation by 10 multiplies each variance by 100 and leaves C unchanged at 0.8.

Illustrative case study

Fictional case: Pine Laboratory compares repeated readings from several stations processing standardized samples. One station has visibly larger scatter. The analyst checks replication counts and measurement conditions before applying a variance-outlier procedure. The result prompts inspection of calibration records and sample handling; it does not automatically justify deleting the station's data. A documented investigation finds that the station's operating conditions differed from the intended comparison. The team reports that limitation and repeats the controlled comparison. This invented case illustrates responsible follow-up, not a claim that the test diagnoses equipment faults.

Common misconceptions and mistakes

Do not confuse Cochran's C with Cochran's Q for related binary responses or a meta-analysis heterogeneity statistic. The classical C procedure examines an unusually large variance, not an unusually small one. Equal sample sizes matter because variance estimates with different degrees of freedom have different sampling behavior. Extensions for unequal sizes are not obtained merely by using the same cutoff. Non-normality and dependence can invalidate the calibration. Repeatedly testing and removing groups can change overall error rates and should not become an undocumented way of obtaining a desired dataset.

Business use

Businesses running measurement programs can use a variance-outlier check to prioritize investigation of inconsistent repeatability. Specify comparable test conditions, replication and the follow-up process before inspecting the results. Retain the original measurements and document any justified exclusions. Compare this narrow question with broader variance-homogeneity procedures when the concern is several differing groups rather than a single dominant variance. For AI-generated outputs, first ask whether numerical variance and a normal sampling model actually represent the business concern; categorical quality judgments may require a different analysis.

Check your understanding

Why do the very different group means in the worked example not affect C?

Answer: Each sample variance is calculated around its own group mean. Translating an entire group changes its mean but leaves its variance unchanged. C therefore measures relative within-group spread, not agreement between group centers.

Related terms

Learn next

distinguish precision from accuracy and learn how sample variance depends on replication. Compare variance-outlier tests with broader tests of variance homogeneity. Examine residual plots and measurement protocols before applying a normal-theory reference distribution, and study how multiple investigations affect the interpretation of repeated significance tests.

Sources

NIST Dataplot, Cochran Variance Outlier Test: https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/cochvari.htm Royal Society of Chemistry Analytical Methods Committee, Using the Grubbs and Cochran tests to identify outliers: https://www.rsc.org/images/TB%2069_tcm18-247571.pdf NIST/SEMATECH, Detection of Outliers: https://itl.nist.gov/div898/handbook/eda/section3/eda35h.htm

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