One-way ANOVA with post-hoc letters: a worked example

One-way ANOVA compares the means of three or more independent groups in a single test. The F-test tells you whether any groups differ; the post-hoc comparisons and their compact letters tell you which — which is what your figure ultimately needs to say.

The data

Grain yield (g per pot) under four fertiliser regimes, six pots each:

r = replicate: r1 is replicate 1, r2 replicate 2, and so on.

This is MinutePlot’s one-way layout: one row per group, replicates across the columns — paste it in exactly like this.

The test

ANOVA partitions the total variation into between-group and within-group parts; a large ratio of the two (the F statistic) means the group means differ more than chance allows. Here: F(3, 20) = 86.9, p < 0.001 — fertiliser affects yield. Assumptions (normal residuals by Shapiro–Wilk, similar group variances) hold for these data; MinutePlot reports the checks alongside every analysis.

Which groups differ? Tukey’s HSD and the letters

An overall p-value never answers the real question. Tukey’s honestly significant difference test compares every pair while keeping the overall false-positive rate at 5%, and the results compress into compact letter display: groups sharing a letter are not significantly different.

Read it directly: both high-nitrogen regimes beat low N, which beats the control; adding phosphorus on top of high N gave no further detectable gain (the two share the letter a).

The finished figure: means, error bars, and the letters placed automatically — this is the panel a journal expects.

Styling and exporting this figure: see Figure styling and Exporting & downloading.

Report it like this

“Fertiliser regime affected yield (one-way ANOVA, F(3, 20) = 86.9, p < 0.001). Yields under High N and N + P exceeded Low N, which exceeded the control (Tukey HSD, p < 0.05); different letters in Fig. 1 indicate significant differences.”