Bonferroni Correction
Adjust significance levels for multiple comparisons using Bonferroni and Sidak corrections. Control Type I error rates and identify significant p-values. Free online calculator.
About This Calculator
The Bonferroni Correction Calculator adjusts significance levels when performing multiple hypothesis tests simultaneously. When you run multiple statistical tests, the probability of obtaining at least one false positive (Type I error) increases substantially. The Bonferroni correction, named after Italian mathematician Carlo Emilio Bonferroni, is the simplest and most widely used method to control the family-wise error rate (FWER) — the probability of making one or more false discoveries among all tests.
Our calculator supports two correction methods: the classic Bonferroni correction (α/m) which divides the original significance level by the number of tests, and the Šidák correction (1 - (1-α)^(1/m)) which provides a slightly less conservative adjustment assuming independent tests. For example, with 10 tests at α = 0.05, the Bonferroni correction yields a threshold of 0.005, while Šidák gives 0.0051. You can optionally enter comma-separated p-values to instantly see which ones remain statistically significant after the correction. The interactive charts compare the original and corrected alpha levels and visualize each p-value against the adjusted threshold.
The Bonferroni correction is widely used across many fields. In India, it is commonly applied in agricultural statistics, pharmaceutical clinical trials regulated by CDSCO, and academic research across IITs and AIIMS. In the United States, the FDA requires multiple comparison adjustments in drug trials, and it is standard practice in psychology, neuroscience, and genomics research (GWAS studies). In the United Kingdom, the MHRA and academic institutions like Oxford and Cambridge apply Bonferroni corrections in clinical research, epidemiology, and social sciences. While the method is universal, practitioners should consider that Bonferroni can be conservative; for exploratory analyses with many tests, the Benjamini-Hochberg procedure may be preferred.
Frequently Asked Questions
What is the Bonferroni correction?
The Bonferroni correction is a statistical adjustment used when performing multiple hypothesis tests simultaneously. It divides the significance level (alpha) by the number of comparisons to control the family-wise error rate (FWER). For example, if you run 5 tests with alpha = 0.05, the Bonferroni-corrected threshold becomes 0.05/5 = 0.01, meaning each individual test must have a p-value below 0.01 to be considered statistically significant.
How do you calculate the Bonferroni correction?
The classic Bonferroni correction formula is corrected alpha = original alpha / number of tests. For example, with alpha = 0.05 and 10 tests: 0.05/10 = 0.005. The Sidak correction uses the formula corrected alpha = 1 - (1 - alpha)^(1/m), which is slightly less conservative. Enter your number of tests, significance level, and optional p-values into our calculator to get instant results.
What is the difference between Bonferroni and Sidak correction?
The classic Bonferroni correction (alpha/m) is simpler and more conservative, assuming independence between tests. The Sidak correction (1 - (1-alpha)^(1/m)) is slightly less conservative and provides more power while still controlling the family-wise error rate, but it assumes the tests are independent. For most practical purposes, the two methods give very similar results, with Sidak being marginally more powerful.
When should I use the Bonferroni correction?
You should use the Bonferroni correction whenever you are performing multiple hypothesis tests simultaneously, such as in ANOVA post-hoc analysis, genome-wide association studies (GWAS), A/B testing with multiple metrics, clinical trials with multiple endpoints, or any research setting where multiple comparisons increase the risk of Type I errors (false positives).
What are the limitations of the Bonferroni correction?
The Bonferroni correction is often considered too conservative, especially with many comparisons. It controls false positives at the expense of increasing false negatives (Type II errors), reducing statistical power. Alternative methods like the Holm-Bonferroni step-down procedure, Benjamini-Hochberg false discovery rate (FDR) control, or Tukey's HSD may be more appropriate depending on the research context.
Is the Bonferroni correction used in A/B testing?
Yes, the Bonferroni correction is commonly used in A/B testing when evaluating multiple metrics or running multiple variant comparisons simultaneously. For example, if testing 5 different webpage designs against 10 key performance indicators, the Bonferroni correction adjusts the significance threshold to account for the increased chance of false positives, ensuring that observed improvements are genuinely significant.
How do I interpret the corrected significance level?
The corrected significance level is the new threshold each individual p-value must beat to be considered statistically significant after adjusting for multiple comparisons. For example, if the corrected alpha is 0.01, only p-values less than 0.01 are significant. P-values above this threshold are not considered statistically significant, even if they are below the original 0.05 level. Our calculator automatically marks which p-values pass the corrected threshold.
What is the family-wise error rate (FWER) in multiple comparisons?
The family-wise error rate (FWER) is the probability of making at least one Type I error (false positive) across all hypothesis tests in a family. If you run 20 independent tests each at alpha = 0.05, the FWER is 1 - (0.95)^20 = 64%, meaning there is a 64% chance of at least one false positive. The Bonferroni correction controls the FWER by lowering the individual test threshold to alpha/m.