Bonferroni Correction Calculator

Bonferroni Correction

Overall Family-Wise Alpha ($\alpha$) Desired total Type I error rate across all tests (default is 0.05)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Number of Simultaneous Tests ($m$) Total number of hypothesis tests, comparisons, or regressions

Number of tests must be an integer ≥ 1.

Shortcut: Post-Hoc Pairwise Groups ($k$) Optional: Calculates $m = \frac{k(k-1)}{2}$ pairwise comparisons automatically

Multiple Testing Correction

What Is the Bonferroni Correction?

The Bonferroni Correction is a statistical procedure used to counter the **Multiple Comparisons Problem**. When running multiple statistical hypothesis tests simultaneously, the probability of obtaining at least one false positive (Type I error) inflates rapidly.

Based on Boole's inequality, the Bonferroni method strictly bounds the Family-Wise Error Rate (FWER) by dividing the overall desired significance level ($\alpha$) by the total number of tests ($m$).

When to Use It

Post-Hoc Pairwise Comparisons: Adjusting alpha when comparing multiple treatment pairs following ANOVA or Kruskal-Wallis.
Multiple Correlation Matrices: Controlling false discoveries across large feature correlation tables.
Subgroup & A/B Testing: Evaluating multiple demographic cohorts or trial endpoints without inflating Type I risk.
Conservative FWER Control: Where avoiding false positive conclusions is of paramount importance.

The Mathematical Formulas

$$\text{Bonferroni Threshold: } \alpha_{\text{Bonf}} = \frac{\alpha}{m}$$ $$\text{Adjusted P-Value: } p_{\text{adj}} = \min(1.0, \, p_{\text{raw}} \times m)$$ $$\text{Šidák Exact Correction: } \alpha_{\text{Šidák}} = 1 - (1 - \alpha)^{1/m}$$ $$\text{Uncorrected Inflation: } \text{FWER}_{\text{uncorrected}} = 1 - (1 - \alpha)^m$$

Where $m$ is the total number of hypothesis tests, $\alpha$ is the original significance threshold (commonly 0.05), and $\text{FWER}$ represents family-wise false-positive probability.

Worked Example

Scenario: A researcher runs $m = 5$ independent hypothesis tests at $\alpha = 0.05$.

Step 1: Uncorrected False Positive Risk:
$\text{FWER} = 1 - (1 - 0.05)^5 = 1 - (0.95)^5 = 1 - 0.7738 = 0.2262 \text{ (22.62\%)}$.

Step 2: Compute Adjusted Significance Cutoff:
$\alpha_{\text{Bonf}} = \frac{0.05}{5} = 0.0100 \text{ (1.00\%)}$.

Step 3: Evaluate P-Values ($p = [0.008, 0.014, 0.035, 0.048, 0.120]$):
$p_1 = 0.008 \le 0.0100 \implies \text{Statistically Significant}$
$p_2 = 0.014 > 0.0100 \implies \text{Not Significant (Adjusted } p_{\text{adj}} = 0.070)$

Conclusion: Only the first hypothesis remains significant after controlling for multiple testing inflation.

Correction Analysis Complete

Bonferroni Correction Results

Adjusted Significance Threshold (α_Bonf)

α_adj = 0.0100

Controls total family-wise false positive rate at exactly α = 0.05 across m = 5 tests.

Bonferroni α 0.0100

Šidák α (Exact) 0.0102

Number of Tests (m) 5

Original Alpha (α) 0.0500

Uncorrected FWER 22.62%

Significant Tests 1 of 5

Bonferroni Correction Calculator

Bonferroni Correction

Bonferroni Correction Results

Hypothesis Testing & Adjusted P-Values Table

Step-by-Step Mathematical Formulation

Methodological Interpretation

Alpha Threshold Decay & Type I Error Inflation Curve

Perfect for students, researchers, and professionals looking to build real statistical skills.