Bonferroni Adjustment Calculator: Formula and Examples

A Bonferroni adjustment calculator helps control the chance of making one or more false-positive findings when you test several hypotheses. Enter the number of tests and either a significance level or a p-value; the calculator returns the adjusted threshold or adjusted p-value.

Formula

The Bonferroni method divides the chosen family-wise significance level by the number of tests. To adjust a raw p-value, multiply it by the number of tests and cap the result at 1:

• Adjusted significance threshold: α_Bonferroni = α / m
• Adjusted p-value: p_adjusted = min(1, p × m)

Here, α is the family-wise significance level, a probability with no unit; m is the number of tests, a positive whole-number count; and p is the raw p-value, also unitless and between 0 and 1. The two outputs express the same decision rule: a raw p-value is significant after adjustment if p ≤ α/m, equivalently if the adjusted p-value is no greater than α.

Worked Example

Using the calculator defaults, there are m = 5 tests, α = 0.05, and a raw p-value of p = 0.012.

1. Divide the family-wise level by the number of tests: 0.05 / 5 = 0.01.
2. Multiply the raw p-value by the number of tests: 0.012 × 5 = 0.06.
3. The adjusted p-value is below 1, so the cap does not change it.

The adjusted threshold is 0.0100, and the adjusted p-value is 0.0600. Since 0.012 is greater than 0.010, this result does not meet the 0.05 family-wise significance criterion after Bonferroni adjustment.

What the Adjustment Controls

When many hypotheses are tested, the chance of at least one false positive can be higher than the significance level used for each individual test. The Bonferroni procedure sets a stricter per-test threshold to control the family-wise error rate (FWER): the probability of making one or more Type I errors across a defined group, or “family,” of tests.

For example, if five independent tests each use α = 0.05 without adjustment, the chance of at least one false positive under the global null is 1 − (1 − 0.05)⁵, about 0.226. Bonferroni instead uses 0.01 per test, which guarantees that the probability of one or more false positives across the family is no greater than 0.05. This guarantee does not require the tests to be independent; it can be conservative when tests are strongly related.

Choosing the Number of Tests

The calculation is simple, but deciding what counts as the test family takes scientific judgment. Include the hypotheses that belong to the same planned inferential question, rather than choosing a count after seeing which results are significant. A family might be several outcomes, groups, time points, or pairwise comparisons addressed in one analysis.

• Use the number of hypotheses included in the family, not the number of observations or participants.
• Count each planned comparison that produces a separate p-value.
• Decide and report the family definition and correction method transparently.
• If the analysis has distinct scientific questions, explain why they are treated as separate families rather than combining or splitting them arbitrarily.

For example, testing four outcomes at α = 0.05 gives a Bonferroni threshold of 0.0125 for each test. If the same study also makes six planned pairwise comparisons that belong to that family, the appropriate count may instead be ten. The count depends on the analysis plan and question, not simply on how many p-values happen to appear in a report.

How to Interpret the Results

The adjusted threshold is the maximum raw p-value that qualifies under the selected family-wise level. The adjusted p-value is the raw p-value expressed on a scale that can be compared with the original α. Do not compare an adjusted p-value with α/m a second time: compare it with α. Conversely, compare the raw p-value with α/m.

An adjusted p-value of 0.06 does not mean there is a 6% probability that the finding is false. A p-value is a probability, assuming the null hypothesis and the statistical model, of observing results at least as extreme as those obtained. Adjustment changes the decision procedure for multiple testing; it does not measure the probability that a hypothesis is true.

Common Problems and Limitations

• Using zero tests: m must be a positive integer. A zero or fractional count has no valid interpretation in this formula.
• Entering percentages as whole numbers: enter α = 0.05 for 5%, not 5. P-values are decimals from 0 to 1.
• Rounding too early: retain unrounded p-values and thresholds for decisions, then round for presentation. Near the cutoff, premature rounding can change the apparent result.
• Ignoring the family: an adjustment cannot compensate for a poorly defined set of tests. Explain which comparisons were included.
• Assuming adjustment repairs study design: it does not correct biased sampling, measurement problems, model misspecification, or selective reporting.

Bonferroni is easy to calculate and gives strong FWER control, but it can reduce statistical power: genuine effects may fail to pass the stricter threshold, especially with many tests or correlated outcomes. A non-significant adjusted result is not proof of no effect. Consider the effect estimate, confidence interval, study design, and the purpose of the analysis as well as the threshold.

Alternatives to Bonferroni

Other procedures may be more suitable depending on the goal. Holm’s step-down procedure controls FWER and is generally at least as powerful as the basic Bonferroni procedure. The Benjamini–Hochberg procedure instead controls the false discovery rate (FDR), the expected proportion of false discoveries among rejected hypotheses, under stated conditions. FDR control is not the same guarantee as controlling the chance of any false positive. Choose a method based on the scientific question and analysis plan; a calculator for this formula cannot determine which error criterion is appropriate.

Frequently Asked Questions

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

Do I multiply or divide the p-value?
For an adjusted p-value, multiply the raw p-value by the number of tests, capping it at 1. To get the equivalent raw-p-value cutoff, divide α by the number of tests.

Can a Bonferroni-adjusted p-value be greater than 1?
No. The adjusted p-value is capped at 1 because p-values range from 0 to 1. A raw p-value of 0.4 across five tests gives min(1, 2) = 1.

Does Bonferroni require independent tests?
No. The method controls the family-wise error rate without requiring independent tests, though it can be conservative, particularly when tests are correlated.

Is Bonferroni the same as false discovery rate correction?
No. Bonferroni controls the chance of at least one false positive in the family. FDR procedures control the expected share of false positives among the results declared significant.

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in m | Number of tests | - | 5 | 1
in a | Significance level | - | 0.05 | 0
in p | Raw p-value | - | 0.012 | 0
out adjusted_alpha | Bonferroni significance threshold | - | a / m | 4
out adjusted_p | Bonferroni-adjusted p-value | - | min(1, p * m) | 4