Bonferroni Correction Calculator

Hypothesis Testing and Statistical Inference

Adjusts p-values to control the family-wise error rate when performing multiple comparisons.

Practical Examples

Explore how the Bonferroni correction works with these sample datasets.

Example 1: Drug Efficacy Study

Clinical Trial

A clinical trial testing 5 different potential side effects of a new drug. The initial alpha is 0.05.

α: 0.05, n: 5

P-Values: 0.01, 0.02, 0.03, 0.04, 0.05

Example 2: Gene Expression Analysis

Genomics Research

Analyzing 20 genes to see if their expression differs between two groups. The significance level is set to 0.01.

α: 0.01, n: 20

P-Values: 0.0004, 0.001, 0.003, 0.05, 0.1, 0.0001, 0.02, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 0.002, 0.006, 0.008, 0.011, 0.03

Example 3: Website Optimization

A/B Testing

An e-commerce site tests 3 different button colors against the original to see which improves click-through rate.

α: 0.05, n: 3

P-Values: 0.04, 0.09, 0.21

Example 4: Educational Intervention

No Significant Results

A study looks at 8 different teaching methods to see if they improve test scores. No method shows a statistically significant improvement.

α: 0.05, n: 8

P-Values: 0.12, 0.34, 0.55, 0.08, 0.23, 0.41, 0.6, 0.19

Other Titles
Understanding the Bonferroni Correction: A Comprehensive Guide
A deep dive into adjusting for multiple comparisons in statistical analysis to maintain scientific rigor and avoid false discoveries.

What is the Bonferroni Correction?

  • The Problem of Multiple Comparisons
  • Defining the Family-Wise Error Rate (FWER)
  • The Bonferroni Solution
The Bonferroni correction is a statistical method used to counteract the problem of multiple comparisons. When you conduct multiple hypothesis tests, the probability of getting a statistically significant result purely by chance (a 'false positive' or Type I error) increases. For example, if you set your significance level (alpha) to 0.05, you accept a 5% chance of a false positive for a single test. But if you run 20 independent tests, the probability of at least one false positive balloons to over 64%. This is known as the multiple comparisons problem.
Controlling the Family-Wise Error Rate (FWER)
The core idea is to control the Family-Wise Error Rate (FWER), which is the probability of making at least one Type I error in the 'family' of tests. The Bonferroni correction achieves this by using a more stringent significance level for each individual test.
The Simple Formula
The method is straightforward: you divide your initial desired alpha level by the number of tests (n) you are performing. This gives you a new, smaller 'Bonferroni corrected alpha' (α'). Each individual test must now have a p-value less than or equal to this new, corrected alpha to be considered statistically significant.

Conceptual Example

  • If α = 0.05 and you run 10 tests, your corrected alpha is 0.05 / 10 = 0.005.
  • To declare any result significant, its p-value must be less than 0.005, not the original 0.05.

Step-by-Step Guide to Using the Bonferroni Correction Calculator

  • Inputting Your Data
  • Executing the Calculation
  • Interpreting the Results
1. Enter the Initial Significance Level (α)
This is your desired alpha for the entire family of tests. It's the overall risk of a Type I error you are willing to accept. A value of 0.05 is the most common choice.
2. Specify the Number of Tests (n)
Enter the total count of all the separate hypothesis tests you have conducted.
3. Provide the P-Values
In the final field, enter the p-value obtained for each of your tests, separating each value with a comma. Ensure the number of p-values you enter matches the number of tests specified in the previous step.
4. Interpret the Output
The calculator will provide the 'Bonferroni Corrected Alpha (α')'. It will then display a table listing each of your original p-values and indicating whether it is 'Significant' or 'Not Significant' based on whether it is less than or equal to the corrected alpha. A summary count is also provided.

Example Walkthrough

  • Inputs: α = 0.05, n = 3, P-Values = 0.01, 0.02, 0.06.
  • Calculation: Corrected α' = 0.05 / 3 ≈ 0.0167.
  • Results: The p-value 0.01 is ≤ 0.0167 (Significant). The p-values 0.02 and 0.06 are > 0.0167 (Not Significant).

Real-World Applications of Bonferroni Correction

  • Medical and Pharmaceutical Research
  • Genomics and Bioinformatics
  • Marketing and A/B Testing
Genomics Research
In studies analyzing thousands of genes (e.g., microarrays), scientists test each gene for a link to a disease. Without correction, hundreds of genes could appear significant by chance alone. The Bonferroni correction helps identify the most promising candidates for further study.
Clinical Trials
When a new drug is tested, researchers often look at multiple outcomes (e.g., reduced blood pressure, lower cholesterol, fewer side effects). The Bonferroni correction ensures that a claim of the drug's effectiveness on any single outcome is statistically robust.
Neuroimaging (fMRI)
fMRI studies analyze brain activity across thousands of tiny regions called voxels. Correcting for multiple comparisons is essential to avoid spurious claims about which parts of the brain are 'lighting up' in response to a stimulus.

Application Scenario

  • A marketing team tests 10 different headlines for an ad. To confidently claim a winning headline, they must use a correction like Bonferroni to avoid acting on random variation.

Common Misconceptions and Correct Methods

  • Is Bonferroni Always the Best Choice?
  • The Assumption of Independence
  • Alternatives to Bonferroni
The 'Overly Conservative' Criticism
The primary criticism of the Bonferroni correction is that it can be overly conservative. By reducing the alpha level so much, it increases the chance of Type II errors (false negatives), where you fail to detect a real effect. This is especially true when a large number of tests are performed or when the tests are correlated (not independent).
Alternatives to Bonferroni
Because of its conservatism, other methods have been developed. The Sidak correction is slightly more powerful but requires tests to be independent. Methods that control the False Discovery Rate (FDR), such as the Benjamini-Hochberg procedure, are often preferred in exploratory fields like genomics because they are less stringent and focus on the proportion of false positives among all significant results, rather than avoiding a single false positive altogether.

When to Consider an Alternative

  • If you are running 10,000 tests on gene expression data, a Bonferroni correction might be so strict that no gene passes the significance threshold. In this case, controlling the FDR with Benjamini-Hochberg would be a more practical approach.

Mathematical Derivation and Examples

  • Boole's Inequality
  • Formal Derivation
  • Worked Numerical Example
Foundation in Boole's Inequality
The Bonferroni correction is derived from a simple probability rule known as Boole's inequality. It states that for any set of events, the probability of at least one of them occurring is no greater than the sum of their individual probabilities. P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) ≤ P(A₁) + P(A₂) + ... + P(Aₙ).
Derivation Steps
Let Eᵢ be the event of making a Type I error for test i. The FWER is the probability of at least one such error, P(∪Eᵢ). We want this FWER to be less than or equal to our chosen alpha (α). From Boole's inequality, FWER ≤ ΣP(Eᵢ). If we set the significance level for each of the n tests to be α/n, then ΣP(Eᵢ) = Σ(α/n) = n * (α/n) = α. Therefore, by setting the individual test alpha to α/n, we guarantee that the overall FWER is less than or equal to α.

Numerical Example

  • Let α = 0.05 and n = 4. The corrected significance level is α' = 0.05 / 4 = 0.0125.
  • Suppose our p-values are p₁=0.01, p₂=0.03, p₃=0.005, p₄=0.1.
  • Comparing each to α'=0.0125, we find that p₁ and p₃ are significant, while p₂ and p₄ are not.