Bonferroni Correction Calculator

The Bonferroni correction is a statistical adjustment applied when you conduct multiple hypothesis tests simultaneously. Without correction, the probability of making at least one false positive (Type I error) rises sharply with each additional test. The correction keeps the overall familywise error rate at your chosen α by dividing it by the number of comparisons performed.

The formula is simple: divide your original significance level (α) by the number of comparisons (k). For example, if α = 0.05 and you run 10 tests, the Bonferroni threshold is 0.05 / 10 = 0.005. A test result is only considered statistically significant if its p-value falls below this adjusted threshold.

For 10 simultaneous tests at α = 0.05, the Bonferroni corrected threshold is 0.05 / 10 = 0.005. This means each individual test must produce a p-value below 0.005 to be declared statistically significant, maintaining the overall familywise error rate at 5%.

Use the Bonferroni correction when you are performing multiple independent or related hypothesis tests on the same dataset and want to control the familywise error rate. It is commonly applied in clinical trials, genomics studies, ANOVA post-hoc comparisons, and any analysis where many tests run simultaneously. It is most appropriate when the number of comparisons is moderate and the tests are independent.

Both corrections address multiple comparisons, but they use different formulas. The Bonferroni correction is α / k, while the Šidák correction is 1 − (1 − α)^(1/k). The Šidák correction is slightly less conservative and is mathematically exact when tests are independent, whereas Bonferroni provides a conservative upper bound. For small numbers of comparisons, the results are nearly identical.

Yes, the Bonferroni correction is widely considered conservative, especially when tests are correlated or when the number of comparisons is very large. By requiring an extremely small p-value threshold, it increases the risk of Type II errors (false negatives), meaning real effects may go undetected. Alternatives like the Benjamini-Hochberg procedure (controlling the false discovery rate) are often preferred in large-scale studies such as genome-wide association studies.

No. Several other methods exist, including the Šidák correction, Holm-Bonferroni step-down procedure, Benjamini-Hochberg false discovery rate (FDR) correction, and Tukey's HSD for pairwise comparisons. The best method depends on whether your tests are independent, how many comparisons you make, and whether you are controlling the familywise error rate or the false discovery rate.

The Bonferroni correction is technically derived under the assumption of independent tests, but it remains a valid (conservative) correction even when tests are correlated or dependent. Because it is an inequality-based bound, it will over-correct when tests are positively correlated, making it more conservative than necessary in such cases.