Significance letters (compact letter display), explained

The small letters above bars in papers — a, ab, b — are the compact letter display (CLD): a summary of every pairwise comparison in one glyph per group — it reproduces the significance pattern exactly, though not the p-values or effect sizes behind it (reviewers sometimes want those too, which the results table provides). The rule is simple: groups sharing a letter are not significantly different; groups sharing none are.

How the letters are made

First, a multiple-comparison procedure tests every pair of groups while controlling the overall error rate — classically Tukey’s HSD, with the Tukey–Kramer extension handling unequal group sizes. The resulting yes/no matrix of differences is then compressed into the fewest letters that reproduce it exactly. Done by hand this is genuinely fiddly — four groups already have six pairs — which is why letters are so often copied wrongly between drafts; MinutePlot computes and places them on the figure automatically.

Reading edge cases

“ab” means the group is statistically indistinguishable from both the a groups and the b groups — typically a middle group the experiment lacked power to separate. Letters can also be non-transitive: A may differ from C while B differs from neither; the display “a, ab, b” states exactly that, and it is a fact about evidence, not an error.

What to write in the caption

“Different letters indicate significant differences between groups (Tukey HSD, p < 0.05); groups sharing a letter do not differ significantly.”

MinutePlot generates this caption sentence with every lettered figure, matched to the test actually used.

When not to use Tukey

Tukey’s HSD controls the family-wise error rate — right for confirmatory all-pairs comparisons. In screening settings with very many comparisons, some fields prefer to control the false discovery rate instead (Benjamini–Hochberg), which preserves power; MinutePlot’s post-hoc options are Tukey HSD, Games–Howell and Dunn, so this is background rather than a setting.