StatReason

Grubbs Test

Formally test whether the single most extreme value in your data is a statistical outlier under an approximate-normality assumption, rather than eyeballing a fence rule.

What this answers

This test answers "is the single most extreme value in my dataset an outlier, formally, under a stated significance level, assuming the rest of the data is approximately normal?" Unlike a generic fence rule, this gives an explicit critical value and a clear reject-or-not decision for exactly one candidate point at a time.

How it is calculated

The G statistic is the absolute distance of the most extreme value from the sample mean, divided by the sample standard deviation. This is compared against a critical value derived from the t distribution with n minus 2 degrees of freedom, using a tail probability that depends on your chosen alpha, your sample size, and whether you are testing for an extreme value on either side (two-sided) or specifically in one direction you named in advance (one-sided). If G exceeds the critical value, the most extreme point is flagged as a statistically significant outlier at that alpha level.

Worked example

For the values 10, 11, 9, 10, 11, and 50, the value 50 is by far the most extreme, pulling the mean upward and inflating the standard deviation, but still standing out enough to produce a large G statistic relative to the two-sided critical value for n equal to 6 at alpha .05 (a published value of about 1.887). Because the computed G exceeds that critical value here, the calculator concludes 50 is a statistically significant outlier at the .05 level.

Assumption audit

Calculated from your data: the mean, standard deviation, the identified most extreme value, the resulting G statistic, and the matching critical value for your chosen alpha and sidedness.
Evidence to review: whether the remaining data, after setting aside the candidate outlier, looks approximately normal using the Box Plot Maker or a Q-Q-style check; Grubbs' test assumes approximate normality and is unreliable when that assumption clearly fails.
You must verify: that you are testing a single candidate point identified in advance, not repeatedly reapplying this test to hunt for a second or third outlier after removing the first, which invalidates the stated significance level through masking effects.

What this result does not mean

A flagged outlier under Grubbs' test is a statistically extreme point given a normal-model assumption, not automatic proof of a data error; a genuinely rare but real event can also produce a significant G statistic. A result that is not significant does not mean the point is definitely legitimate, only that the evidence against it, under this specific model, did not clear your chosen threshold.

Limitations

This test only evaluates one extreme value at a time and should not be applied repeatedly to the same dataset to strip out multiple points; doing so invalidates the stated alpha level through the same masking problem that affects every single-outlier test. With small samples, the critical value is wide and only very extreme points will be flagged at all.