This Bonferroni correction calculator adjusts your significance level when you run multiple tests — the family-wise error guardrail — and shows the slightly less conservative Šidák correction beside it.
| Uncorrected family error (1 test false-positive risk ×m) | – |
| Bonferroni per-test α (α/m) | – |
| Šidák per-test α (1−(1−α)^(1/m)) | – |
| Adjusted p-value rule | – |
Equivalent framing: multiply each raw p-value by m (capping at 1.0) and compare against the original α — same test, two bookkeeping styles. Bonferroni assumes nothing about correlations between tests, which is its strength (always valid) and its cost (conservative when tests overlap).
The Problem: 20 Tests at α = 0.05 Expect One False “Discovery”
Multiple comparisons are the tax on exploration — run enough Two-Proportion Z-Tests across variants and noise wins a trophy. Part of our Statistics Calculators collection.
Worked Example (Verified)
- m = 20 tests, α = 0.05 (defaults): uncorrected family-wise error = 1 − 0.95²⁰ = 64.2% — more likely than not to crown a false positive; Bonferroni per-test α = 0.0025; Šidák = 0.00256, a whisper more powerful under independence
- Genomics scale: 1,000,000 SNP tests → Bonferroni α = 5×10⁻⁸, which is literally the genome-wide significance threshold the field standardized on
- A single pre-registered primary endpoint needs NO correction — corrections tax families of tests, not lone hypotheses
Bonferroni and Its Rivals
- Holm's step-down: same family-wise guarantee, strictly more powerful — order p-values, test smallest against α/m, next against α/(m−1)... the upgrade most statisticians recommend by default.
- Benjamini–Hochberg (FDR): controls the false discovery rate instead — allows some false positives among many real ones; the omics-scale workhorse.
- Šidák: exact under independence, a hair less strict (0.00256 vs 0.0025 at m = 20).
- The deepest fix is design: pre-specify few comparisons, spend α where the question lives.
Related Tools
- Two-Proportion Z-Test Calculator — the tests being corrected
- P-Value Calculator — from test statistics
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate the Bonferroni correction?
Divide the family-wise α by the number of tests: α_per-test = α/m. For 20 tests at α = 0.05: 0.0025. Equivalently, multiply each p-value by 20 (capping at 1) and compare to 0.05. This calculator also shows the Šidák value and the uncorrected family error (64.2% here).
Why is correction necessary?
Each test at α = 0.05 carries a 5% false-positive risk; 20 independent null tests compound to a 64% chance of at least one false alarm. Bonferroni caps the family-wise risk at 5% by making each test stricter — the price of looking everywhere is proving more anywhere.
Is Bonferroni too conservative?
It can be, especially with many correlated tests (the correction pretends worst-case independence). Holm's step-down is uniformly more powerful with the same guarantee; FDR methods trade the guarantee for discovery power at scale. Bonferroni remains the simplest defensible choice for small families of confirmatory tests.
Does every p-value in a paper need correction?
No — correction applies to the family of tests whose joint error you care about. One pre-specified primary hypothesis stands alone; exploratory fishing across 40 outcomes doesn't. The honest move is declaring the family (and the method) in advance.
What is the genome-wide significance threshold?
5×10⁻⁸ — literally Bonferroni for ~1 million independent common-variant tests at α = 0.05. It's the most famous applied Bonferroni number in science, and this calculator reproduces it: enter m = 1,000,000.
Is this Bonferroni correction calculator free?
Yes — free, instant, no sign-up.