Bonferroni Correction Calculator
Control the family-wise error rate (FWER) across multiple simultaneous hypothesis tests by dividing the target significance level by the number of comparisons.
Reject a single hypothesis only if its p-value falls below this threshold
- 1
FWER without correction
1 − (1 − 0.05)^5 = 0.2262Probability of at least one false positive if each test uses the uncorrected α. - 2
Bonferroni-corrected α per test
0.05 ÷ 5 = 0.010000
How does this calculator work?
When running m hypothesis tests, the Bonferroni correction sets each test's significance threshold to α/m, keeping the family-wise false-positive rate at or below α. For example, five tests at α = 0.05 require each individual p-value to be ≤ 0.01 to be declared significant. The correction is simple and conservative — more powerful alternatives exist for large test families.
Formula
How this is calculated
When you run m hypothesis tests simultaneously, each at significance level α, the probability of making at least one false rejection (a Type I error somewhere in the family) inflates well above α. Without any correction, the family-wise error rate is FWER = 1 − (1 − α)^m; for m = 20 tests at α = 0.05 that is about 64% — far from the intended 5%.
The Bonferroni correction is the simplest remedy: declare each individual test significant only if its p-value falls below α / m. This guarantees FWER ≤ α regardless of dependence structure between tests (by Boole's inequality). The exact FWER after correction is 1 − (1 − α/m)^m, which is at most α and converges to α as m grows. The corresponding two-tailed critical z-score rises with m, making the per-test threshold stricter.
The Bonferroni correction is conservative — it controls FWER strongly but at the cost of statistical power (more true effects may be missed). Alternatives such as the Holm-Bonferroni step-down procedure or the Benjamini-Hochberg false discovery rate (FDR) control are less conservative and preferred when power loss is a concern. Use Bonferroni when the number of tests is small to moderate and controlling even a single false positive is critical.
Frequently asked questions
Bonferroni uses Boole's inequality, which assumes tests are independent and that all null hypotheses are true. When tests are positively correlated — as is common in biological or social science data — the actual FWER is lower than α even without full correction, so Bonferroni over-adjusts and loses statistical power.
FWER is the probability of making one or more false rejections anywhere in the family — a strict error criterion. FDR (controlled by the Benjamini-Hochberg procedure) is the expected proportion of rejections that are false — a less strict criterion that allows more power when many true effects exist.
Yes. For two-tailed tests the corrected per-test α is α/m and the critical z = invNorm(1 − α/(2m)). For one-tailed tests use α/m directly against the one-tailed p-value. The correction formula is the same regardless of tail choice.
TG we-Calculate Editorial Team. (2026). Bonferroni Correction Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/bonferroni-correction-calculator
TG we-Calculate Editorial Team. "Bonferroni Correction Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/bonferroni-correction-calculator.
TG we-Calculate Editorial Team, "Bonferroni Correction Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/bonferroni-correction-calculator
@misc{wecalculate_bonferroni_correction_calculator, title = {Bonferroni Correction Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/bonferroni-correction-calculator}}, year = {2026}, note = {TG we-Calculate} }
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