Bonferroni Calculator: Adjusting for Multiple Tests

A Bonferroni calculation tool adjusts a significance level or a set of p-values to account for multiple hypothesis tests. Enter the number of tests and either the original significance level or the observed p-value; the calculation gives the adjusted cutoff or corrected p-value.

Formula

The Bonferroni method divides the chosen family-wise significance level by the number of tests to get a per-test cutoff. To adjust an observed p-value instead, multiply it by the number of tests, limiting the result to 1:

• Per-test cutoff: α_Bonferroni = α / m
• Adjusted p-value: p_adjusted = min(1, p × m)

Here, α is the original significance level (a probability, such as 0.05), m is the number of tests (a positive whole-number count), and p is the observed p-value (between 0 and 1). The outputs are probabilities and have no physical units. The input defaults are α = 0.05, m = 5, and p = 0.01.

Worked Example

Using exactly the default inputs:

1. Divide the original significance level by the number of tests: 0.05 / 5 = 0.01.
2. Multiply the observed p-value by the number of tests: 0.01 × 5 = 0.05. This is already below 1, so the cap does not change it.

The calculator returns a Bonferroni significance cutoff of 0.0100 and an adjusted p-value of 0.0500. For this example, the observed p-value equals the adjusted cutoff; because the decision convention is usually p ≤ cutoff, it meets that threshold, subject to the study’s prespecified analysis plan.

What the Bonferroni Method Does

When researchers run several tests, the chance of getting at least one false positive across the group can exceed the significance level used for each individual test. The Bonferroni correction is a simple way to control the family-wise error rate: the probability of making one or more false rejections within a defined family of tests.

The method is based on the union bound: the probability of at least one false positive is no greater than the sum of the individual false-positive probabilities. Setting each test’s cutoff to α / m therefore keeps the family-wise error rate at or below α, regardless of how the tests are related. The method does not require tests to be independent, though that conservative guarantee can make it harder to detect real effects.

When to Use the Calculator

Use this calculation when a study’s analysis plan treats several hypothesis tests as one family and specifies Bonferroni adjustment. Examples include comparing several treatment groups with a common control, testing multiple outcomes, or examining a set of planned contrasts. The important question is not simply how many tests appear in a report: define the relevant family of tests based on the research question and analysis plan.

• Choose α to match the planned family-wise error rate, often 0.05 in examples, but not automatically in every field or study.
• Count the tests included in that family. Use a positive whole number, and do not count the same test twice.
• Enter a p-value from an individual test if you want its Bonferroni-adjusted p-value.
• Interpret the adjusted result alongside the study design, effect estimate, uncertainty interval, and other relevant evidence.

How to Read the Results

The cutoff and adjusted p-value are two equivalent ways to make the same Bonferroni decision. Compare the raw p-value with α / m, or compare the adjusted p-value with α. For instance, with α = 0.05 and m = 5, the per-test cutoff is 0.01. A raw p-value of 0.008 passes that cutoff; its adjusted value is 0.008 × 5 = 0.04, which is at or below 0.05. A raw p-value of 0.02 becomes 0.10 and does not pass.

An adjusted p-value is not the probability that the null hypothesis is true, nor is it the probability that a particular result is a false positive. It is a transformed quantity used for a multiple-testing decision under the chosen correction. Values above 1 are conventionally reported as 1, which is why the calculator caps the adjusted value.

Common Mistakes and Limitations

• Using the wrong test count: The correction depends directly on m. Include the tests belonging to the prespecified family, not an arbitrary subset selected after seeing the results.
• Applying the correction twice: Either compare raw p-values with α / m or compare Bonferroni-adjusted p-values with α. Do not divide the cutoff and also multiply an already adjusted p-value.
• Rounding too early: Keep full precision during calculations and round only for display. A displayed value near the cutoff may need checking at greater precision.
• Assuming a nonsignificant result proves no effect: A stricter cutoff can reduce statistical power. Failure to pass it is not proof that the null hypothesis is true.
• Defining the family after looking at results: The interpretation of error control depends on which tests count as one family. Decide this in advance where possible and report the approach clearly.

Bonferroni is easy to calculate and makes relatively few assumptions about dependence among tests, but it can be conservative, especially when many tests are performed or outcomes are correlated. It also does not correct poor study design, biased measurements, or selective reporting.

Frequently Asked Questions

How do I calculate the Bonferroni correction?
Divide the chosen significance level α by the number of tests m. For α = 0.05 and 10 tests, the per-test cutoff is 0.005.

How do I calculate a Bonferroni-adjusted p-value?
Multiply the raw p-value by the number of tests and cap the result at 1: min(1, p × m). For p = 0.008 across 5 tests, the adjusted p-value is 0.04.

Does Bonferroni require independent tests?
No. Its family-wise error guarantee follows from the union bound and applies whether or not the tests are independent, although the correction may be conservative.

What should count as the number of tests?
Count the comparisons that belong to the analysis family being controlled. That definition depends on the research question and analysis plan; it is not always every test in an entire paper.

Is Bonferroni the same as Benjamini–Hochberg?
No. Bonferroni controls the family-wise error rate, limiting the chance of one or more false positives in the family. Benjamini–Hochberg is commonly used to control the false discovery rate, a different error measure.

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in alpha | Original significance level | - | 0.05 | 0
in m | Number of tests | - | 5 | 1
in p | Observed p-value | - | 0.01 | 0
out adjusted_alpha | Bonferroni significance cutoff | - | alpha / m | 4
out adjusted_p | Bonferroni-adjusted p-value | - | min(1, p * m) | 4