bonferroni calculator

What is the Bonferroni correction?

The Bonferroni correction is a simple and widely used method for handling multiple comparisons. If you test many hypotheses at once, your chance of getting at least one false positive rises quickly. Bonferroni addresses this by making the significance threshold stricter for each individual test.

Instead of comparing each p-value to the usual 0.05, you compare it to 0.05 divided by the number of tests. This helps control the probability of making one or more Type I errors across the entire family of tests.

Why multiple testing is a problem

Suppose every null hypothesis is actually true and you run 20 independent tests at α = 0.05. Even with no real effects, you are likely to see at least one “significant” result just by chance. That inflates the family-wise error rate and can lead to overconfident conclusions.

• One test at α = 0.05: 5% false-positive risk.
• Ten tests at α = 0.05: substantially higher family-wise false-positive risk.
• Dozens or hundreds of tests: false positives become very likely without correction.

Bonferroni formulas used in this calculator

Adjusted significance threshold

α_adjusted = α / m

Where:

• α = desired family-wise significance level (for example 0.05)
• m = number of hypothesis tests

Adjusted p-value for a single test

p_adjusted = min(p × m, 1)

If p_adjusted is below your original α, the finding remains significant after correction.

How to use this Bonferroni calculator

• Enter your overall α (commonly 0.05).
• Enter the number of comparisons in your analysis.
• Optionally add one p-value or a list of p-values.
• Click Calculate to see threshold, adjusted p-values, and significance flags.

Worked example

Scenario

You run 8 planned hypothesis tests and want to keep family-wise error at 0.05.

• α = 0.05
• m = 8
• α_adjusted = 0.05 / 8 = 0.00625

Now, only p-values less than or equal to 0.00625 are significant under Bonferroni.

Interpreting adjusted p-values

If one of your raw p-values is 0.004:

• p_adjusted = 0.004 × 8 = 0.032
• 0.032 < 0.05, so it remains significant at family-wise α = 0.05

If another raw p-value is 0.012:

• p_adjusted = 0.012 × 8 = 0.096
• Not significant after correction

Strengths and limitations

Strengths

• Very easy to compute and explain.
• Strong control of family-wise error rate.
• Conservative and often preferred in confirmatory analyses.

Limitations

• Can be overly conservative, especially with many tests.
• May increase Type II error (miss true effects).
• Less powerful than stepwise methods in many settings.

Bonferroni vs. other corrections

Holm-Bonferroni

A step-down method that controls family-wise error like Bonferroni but is typically more powerful.

Benjamini-Hochberg (FDR control)

Controls false discovery rate rather than family-wise error. Often preferred in large-scale exploratory studies (for example, genomics) where discovering more true effects is important.

Common mistakes to avoid

• Using the total number of possible tests rather than the number of tests actually performed/planned.
• Mixing exploratory and confirmatory tests into one family without justification.
• Reporting only corrected significance without the raw p-values and effect sizes.
• Applying strict correction automatically when an alternative error-control strategy is more appropriate.

Quick interpretation checklist

• Define your hypothesis family before testing.
• Choose α based on study context.
• Apply α/m threshold consistently.
• Report raw p, adjusted p, confidence intervals, and effect size.
• Discuss practical significance, not just statistical significance.

Final note

This calculator gives a fast, transparent Bonferroni correction for research, A/B testing, and data analysis workflows. It is a practical first step for multiple comparison control, especially when you need a conservative and easy-to-audit method.