StatReason

Tukey HSD Calculator

Find out which specific groups differ after a significant one-way ANOVA, comparing every pair while controlling the overall chance of a false positive across all comparisons at once, not just each comparison individually.

What this answers

This test answers "which specific pairs of groups actually differ?" after a one-way ANOVA has already told you some difference exists somewhere among three or more groups. The omnibus ANOVA test never identifies which pairs on its own; that is exactly Tukey HSD's job.

How it is calculated

Every pairwise mean difference is compared against a critical value from the studentized range distribution, computed here from the exact double-integral definition rather than looked up from a printed table, using the pooled within-group mean squared error (MSE) from all groups combined as the shared measure of noise. When groups have unequal sample sizes, this calculator applies the standard Tukey-Kramer adjustment to each pair's standard error, a documented, standard extension rather than a silent assumption of equal group sizes.

Worked example

For three groups of 1 to 3, 4 to 6, and 7 to 9, each cleanly separated from the others, every pairwise comparison is significant at the .05 level, with confidence intervals that exclude zero in every case, correctly detecting the clear separation visible in the raw numbers.

Assumption audit

Calculated from your data: the pooled MSE, the studentized range critical value for your group count and error degrees of freedom, and every pairwise difference, confidence interval, and adjusted p-value.
Evidence to review: whether group variances look reasonably similar; if one or more groups show clearly different spread, the Games-Howell Test makes fewer assumptions about equal variance and is generally the safer choice.
You must verify: that a one-way ANOVA on this same data was already significant before running pairwise comparisons; running Tukey HSD without that first gate, while not mathematically wrong, is a departure from the standard workflow this test is designed for.

What this result does not mean

A significant pairwise comparison means that specific pair's difference survives family-wise correction, not that the difference is large in practical terms; check the confidence interval's width and the raw mean difference for a sense of magnitude, since a p-value alone does not convey size.

Limitations

Tukey HSD assumes roughly equal variances across groups; when that assumption looks doubtful, prefer the Games-Howell Test instead. This calculator computes the studentized range distribution via numerical integration, accurate to about 3 to 4 decimal places, sufficient for standard reporting purposes.