The Bonferroni correction is a method for multiple comparisons that lowers your significance threshold by dividing your alpha (usually .05) by the number of tests you run, so that running many tests does not inflate your chance of a false positive. If you have ever run six t-tests on the same dataset and worried that one "significant" result might just be luck, that fear is exactly what Bonferroni is designed to fix.
Key Takeaways
- The Bonferroni correction divides your alpha level by the number of comparisons: for 5 tests at α = .05, your new threshold is .05 / 5 = .01.
- It controls the family-wise error rate — the probability of making at least one false positive across a whole set of tests.
- Bonferroni is the most conservative common correction, meaning it strongly protects against false positives but raises the risk of missing real effects (Type II error).
- You apply it whenever you run multiple significance tests on the same data, such as post hoc comparisons after ANOVA or several correlations at once.
- In APA 7, report the adjusted alpha or the corrected p-value and state that a Bonferroni correction was applied.
Why do multiple comparisons need correcting at all?
Every significance test carries a built-in risk of a false positive. When you set α = .05, you accept a 5% chance of declaring an effect real when it isn't. Run one test, and that's fine. Run 20 independent tests, and the math changes: the probability that at least one comes up falsely significant climbs to roughly 64% (1 − 0.95²⁰). That inflated risk across a set of tests is called the family-wise error rate.
The Bonferroni correction tackles this directly by making each individual test harder to pass. Instead of judging every test against .05, you judge each one against a stricter threshold so that the overall error rate stays near 5%. Statisticians reach for it constantly in psychology, medicine, and the social sciences — anywhere researchers slice one dataset into many comparisons.
How do you calculate the Bonferroni correction?
The formula is deliberately simple. Take your original alpha and divide it by the number of comparisons (m):
Adjusted α = α / m
So if your original α = .05 and you run 4 comparisons, your new significance threshold is .05 / 4 = .0125. A test result now counts as significant only if its p value is below .0125.
There are two equivalent ways to apply it, and both are correct as long as you're consistent:
- Adjust the threshold: keep raw p values and compare each to α / m (e.g., .0125).
- Adjust the p-values: multiply each raw p value by m and compare to your original .05. (An adjusted p is capped at 1.0 — you never report a p above 1.)
Both approaches give the same yes/no decision. Software like StatRyx and SPSS typically report the adjusted p-values so you can compare everything against the familiar .05 line without doing division in your head.
When should I use a Bonferroni correction?
Use a Bonferroni correction whenever you run several significance tests on the same data and want to keep your overall false-positive rate controlled. The most common situations are:
- Post hoc comparisons after a significant ANOVA — comparing each pair of group means.
- Running multiple correlations on the same set of variables.
- Multiple t-tests comparing several outcomes between two groups.
- Testing several subgroups or several time points from one study.
You do not need it for a single planned comparison, and you should think twice about it when you have dozens of tests — because Bonferroni becomes so strict it can bury genuine effects. In those cases, less conservative alternatives (Holm, Tukey's HSD, or false discovery rate methods) are often better. If you're deciding between corrections, our guide on choosing post hoc tests after ANOVA walks through the trade-offs.
A worked example with real numbers
Suppose you ran a one-way ANOVA comparing anxiety scores across three therapy groups (CBT, mindfulness, control) in a study of 60 participants, and got a significant omnibus result, F(2, 57) = 5.84, p = .005. The ANOVA tells you the groups differ somewhere, but not which pairs differ. So you run three pairwise comparisons:
- CBT vs. mindfulness: p = .041
- CBT vs. control: p = .008
- Mindfulness vs. control: p = .190
You have m = 3 comparisons, so your adjusted threshold is .05 / 3 = .0167.
Now judge each raw p against .0167:
- CBT vs. mindfulness: p = .041 → not significant (above .0167)
- CBT vs. control: p = .008 → significant (below .0167)
- Mindfulness vs. control: p = .190 → not significant
Notice what happened: without correction, CBT vs. mindfulness (p = .041) would have looked significant. After Bonferroni, it doesn't survive. That's the correction doing its job — protecting you from over-claiming. Only the strongest difference, CBT vs. control, holds up.
How do I report a Bonferroni correction in APA 7?
State that you applied the correction, give the adjusted alpha, and report each comparison's statistics. A clean APA 7 write-up of the example above reads:
A one-way ANOVA revealed a significant effect of therapy type on anxiety, F(2, 57) = 5.84, p = .005, η² = .17. Pairwise comparisons using a Bonferroni correction (adjusted α = .0167) showed that the CBT group reported significantly lower anxiety than the control group, p = .008. No other pairwise differences were significant.
If you report adjusted p-values instead of an adjusted threshold, say so explicitly ("p values are Bonferroni-corrected") and compare them against .05. Remember APA style: italicise p, drop the leading zero, and report to two or three decimals. StatRyx generates this exact formatting automatically, so the correction, the adjusted alpha, and the APA sentence come out ready to paste into your thesis.
Bonferroni vs. other corrections: key differences
| Method | How it works | Strictness | Best for |
|---|---|---|---|
| Bonferroni | Divide α by number of tests | Most conservative | Small number of planned comparisons |
| Holm-Bonferroni | Step-down; less strict than plain Bonferroni | Moderate | Same uses, more statistical power |
| Tukey's HSD | Built for all pairwise ANOVA comparisons | Moderate | Post hoc after ANOVA |
| Šidák | Slightly less strict than Bonferroni | Moderate | Independent tests |
| False Discovery Rate (FDR) | Controls proportion of false positives | Least strict | Many tests (e.g. genomics) |
The honest trade-off: Bonferroni is the simplest and most protective, but its strictness means you sacrifice statistical power. With 10 or more comparisons, most methodologists prefer Holm or FDR to avoid missing real effects.
Is the Bonferroni correction too conservative?
It can be. Because dividing alpha by a large number produces a tiny threshold, Bonferroni raises your risk of a Type II error — failing to detect a real effect. With 20 comparisons, your threshold drops to .0025, and modest-but-genuine effects may never reach it. That's why the Holm-Bonferroni method exists: it controls the same family-wise error rate but tests your p-values in a stepwise order, gaining power without sacrificing rigor. For most researchers running 3–8 comparisons, plain Bonferroni is defensible and easy to explain to re