What this answers

This test answers "which specific groups differ?" the same question Tukey HSD answers, but built for the situation where group variances are clearly unequal, exactly what a Welch ANOVA is designed to detect at the omnibus level first.

How it is calculated

Each pair gets its own standard error, computed from that pair's two groups' own variances divided by their own sample sizes, exactly like a Welch two-sample comparison, rather than a single pooled variance shared across every group. Each pair also gets its own Welch-Satterthwaite degrees of freedom, and the resulting q statistic is compared against the studentized range distribution at that pair-specific degrees of freedom, computed via the same exact numerical integration this site's Tukey HSD engine uses.

Worked example

For a low-variance group clustered tightly around 2 and a high-variance group spread widely from 5 to 35 but with a similar average, this test correctly uses each group's own variance rather than pooling them together, giving a result appropriate for the genuinely different spread between the two groups.

Assumption audit

Calculated from your data: each pair's own standard error, Welch-Satterthwaite degrees of freedom, mean difference, confidence interval, and adjusted p-value.

Evidence to review: whether group variances are genuinely different enough to warrant this test over Tukey HSD; if variances look similar across groups,Tukey HSD is the more standard, slightly more powerful choice.

You must verify: that a Welch ANOVA (or similar evidence of unequal variance) motivated this specific test choice, rather than picking whichever post-hoc test happens to produce a more favorable result.

What this result does not mean

A significant pairwise comparison here means that pair's difference is unlikely given each group's own variance and sample size, not that the difference is large in practical terms; check the confidence interval's width alongside the mean difference itself.

Common mistakes

Running Games-Howell as the default choice for every post-hoc comparison, rather than checking group variances first, gives up some statistical power for a robustness benefit you may not need; when variances genuinely look similar, Tukey HSD is the more standard, slightly more powerful choice. A second mistake is treating this test's per-pair degrees of freedom as a fixed value shared across every comparison, when in fact each pair gets its own Welch-Satterthwaite degrees of freedom based on that specific pair's two variances.

Limitations

This calculator computes the studentized range distribution via numerical integration, accurate to about 3 to 4 decimal places, sufficient for standard reporting but not intended to match a printed table or another software package to every decimal place.