Grubbs' Test Critical Values

The |z| > 3 rule uses the same fixed cutoff no matter how many data points you have, which is too strict for large samples and too lenient for small ones. Grubbs' test uses a critical value computed specifically for your exact sample size, so it correctly flags outliers a fixed cutoff can miss — see the worked example in the linked outlier detection article, where a value with |z| = 2.65 fails the fixed cutoff but is flagged by Grubbs' test at n = 9.

Testing the single most extreme value out of n data points, rather than one pre-chosen value, means there are n opportunities for a large deviation to occur by chance. The α/(2n) correction — a Bonferroni-style adjustment built directly into the critical value — accounts for that, so the test's overall false-positive rate stays at α.

The version here tests for exactly one outlier at a time. With two or more suspected outliers, a masking effect can occur where one extreme value hides another, similar to how the standard deviation itself gets inflated — a generalized version (the Tietjen-Moore test) or an iterative approach is needed for multiple outliers.

Like the z-score rule, it assumes the data (with the suspected outlier removed) is approximately normally distributed. Applying it to strongly skewed data can flag genuine, non-erroneous extreme values as outliers.