The Bonferroni correction calculator above divides your family-wise alpha by the number of comparisons to give an adjusted per-test threshold, and reports how many of your p-values survive it. It also shows the family-wise error rate you were carrying before the correction, which is usually the number that persuades people the correction is necessary at all.
Arb Digital publishes this because multiple-comparison inflation is the most common way a genuinely careful analysis produces a false result. Run twenty independent tests at the five percent level with nothing real going on, and the chance of at least one significant result is about sixty-four percent. Nobody intended to cheat; the arithmetic simply was not done. The tool also runs Holm and Benjamini-Hochberg on the same p-values, because Bonferroni is rarely the best choice even when a correction is clearly needed.
What This Bonferroni Correction Calculator Does
It performs three separate jobs. First, it computes the adjusted alpha: your family-wise alpha divided by the number of comparisons. Second, it computes the family-wise error rate — the probability of at least one false positive across the whole family — both before and after correction, so the size of the problem and the size of the fix are both visible. Third, if you supply the actual p-values, it applies three procedures to them and reports which hypotheses each one rejects.
The three procedures are genuinely implemented here, not merely named. Bonferroni compares each p-value against alpha divided by the number of tests. Holm sorts the p-values and uses a threshold that relaxes as it works down the list. Benjamini-Hochberg controls a different quantity entirely — the false discovery rate — and typically rejects more hypotheses than either of the others. The per-test panel shows all three verdicts side by side for every p-value you entered.
This is a different job from our p-value calculator, which produces a single p-value from a test statistic. That page tells you what one test found; this page tells you how to read a whole set of them together. Both are needed, and the order matters: compute each p-value first, then correct the family.
How to Use It
- Set the family-wise alpha to the error rate you want across the entire set of tests. This is not a per-test figure.
- Enter the number of comparisons if you only want the adjusted threshold, or leave it and paste the p-values instead.
- Paste every p-value in the family, including the non-significant ones. The count of tests is what drives the correction, so omitting the null results understates it.
- Read the adjusted threshold in the headline and the two family-wise error rates in the grid.
- Compare the three procedures in the per-test panel before deciding which one your analysis should report.
The Formula and How It Is Calculated
The Bonferroni-adjusted threshold is simply α′ = α ÷ m, where m is the number of comparisons. Equivalently you can leave the threshold alone and multiply each p-value by m, capping the result at 1; that gives the Bonferroni-adjusted p-value, which is what most software reports. The two formulations are identical in what they reject.
The family-wise error rate before correction, assuming the tests are independent, is 1 − (1 − α)ᵐ. With alpha at 0.05 and ten tests that is 1 − 0.95¹⁰ = 0.4013, so a forty percent chance of at least one false positive somewhere in the family. Apply the correction and each test now runs at 0.005, giving 1 − 0.995¹⁰ = 0.0489 — just under the five percent you wanted. That is the correction working exactly as advertised.
Run the defaults through the per-test panel and the contrast is stark. Ten p-values, five of them below 0.05 uncorrected. Bonferroni's threshold of 0.005 keeps only one. Holm also keeps one here, because it stalls at the second-smallest p-value of 0.008 against a threshold of 0.05 ÷ 9 = 0.00556. Benjamini-Hochberg keeps three, because the third-smallest p-value of 0.012 clears its threshold of (3 ÷ 10) × 0.05 = 0.015. Same data, three defensible answers, controlling three different things. The NIST/SEMATECH e-Handbook section on multiple comparisons sets out where Bonferroni sits among the standard post-hoc procedures.
Bonferroni Is Conservative, and That Has a Cost
The Bonferroni inequality guarantees that the family-wise error rate is at most alpha, whatever the dependence structure between the tests. That guarantee is its great strength: it holds when the tests are correlated, when they are independent, and when you have no idea which. Almost no other method is that robust.
The price is statistical power. Dividing alpha by the number of tests makes each individual test much harder to pass, so real effects are missed. With fifty comparisons at a family alpha of 0.05, each test must clear 0.001, and an effect that would comfortably reach significance on its own may not survive. Worse, the penalty depends on how many other things you happened to test, which means a real finding can be erased by an unrelated hypothesis a colleague added to the same family. When the tests are positively correlated — which they usually are, because they share the same subjects and the same measurements — Bonferroni is stricter than it needs to be, and the true family-wise error rate ends up well below alpha rather than at it.
Holm and Benjamini-Hochberg, and When to Prefer Them
The Holm step-down procedure is strictly better than Bonferroni and there is rarely a reason not to use it. Sort the p-values from smallest to largest. Compare the smallest against alpha divided by m, the next against alpha divided by m − 1, the next against alpha divided by m − 2, and so on, stopping the first time a p-value fails. Everything before the stop is rejected; everything from the stop onwards is not. It controls the family-wise error rate under exactly the same conditions as Bonferroni, and it always rejects at least as many hypotheses. It is uniformly more powerful with no additional assumptions.
Benjamini-Hochberg answers a different question. Instead of controlling the probability of any false positive, it controls the expected proportion of false positives among the hypotheses you reject — the false discovery rate. Sort ascending again, find the largest position i where the p-value is at most (i ÷ m) × alpha, and reject everything up to that position. This is the right tool for exploratory and screening work, where you are generating candidates for follow-up and can tolerate some proportion of them being wrong. It is the wrong tool when a single false positive is itself the failure, which is the situation Bonferroni and Holm are built for. All three are computed in the panel above, and our p-value calculator produces the inputs they need.
What Counts as a Family, and Why It Is the Hard Part
The number you divide by is not an objective fact about the data. It is a decision about which tests belong together, and it is where the real judgement in multiple-comparison correction lives. Ten outcome measures in one trial are clearly a family. Are the same ten measures across three subgroups thirty comparisons, or three families of ten? Is a secondary analysis run six months later part of the original family or a new one?
There is no formula for this, but there is a discipline: decide the family before you see the results and write it down. Choosing the family afterwards, once you know which tests were significant, converts the correction into a formality that guarantees whatever answer you wanted. The most common form of this is quietly excluding the tests that found nothing, which shrinks m and loosens the threshold on exactly the tests you were hoping to keep. That is why the field above asks for every p-value in the family rather than the interesting ones. Pre-specifying the comparisons also lets you size the study for them, which our sample size calculator and effect size calculator handle.
Where Corrections Are Not the Right Answer
Three situations call for something other than a correction. The first is a single pre-specified primary outcome with the rest declared exploratory in advance: there is one confirmatory test, the family has one member, and no correction is needed. Declaring that structure up front is far more powerful than correcting after the fact.
The second is a genuinely structured comparison, such as several groups against one control or a set of ordered doses, where purpose-built procedures exploit the structure and beat a blanket division. The ANOVA calculator covers the omnibus test that usually precedes those, and our t-test calculator and chi-square calculator produce the individual comparisons. The third is estimation rather than testing: reporting effect sizes with confidence intervals sidesteps much of the problem, because a set of intervals communicates uncertainty without forcing a binary verdict on each one. Penn State's STAT 500 applied statistics course covers the hypothesis-testing framework these adjustments sit inside.
Arb Digital designs marketing experiments with the comparisons declared up front, so the results survive scrutiny rather than dissolving under it.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Counting only the significant tests — the correction divides by every test in the family, and dropping the null results inflates your findings.
- Choosing the family after seeing the results — the family must be defined before the analysis, or the correction guarantees the answer you wanted.
- Using Bonferroni when Holm is available — Holm controls the same error rate under the same assumptions and always rejects at least as much.
- Treating Benjamini-Hochberg as a milder Bonferroni — it controls the false discovery rate, a different quantity, and is not a substitute when any false positive is unacceptable.
- Applying the correction to the test statistic — adjust the threshold or the p-values, never the underlying statistic.
Related Free Tools From Arb Digital
Produce the p-values with the p-value calculator, compare two groups with the t-test calculator, run the omnibus test with the ANOVA calculator, test frequencies with the chi-square calculator, or report the range with the confidence interval calculator. The full free online tools hub lists every statistics tool we publish.
Frequently Asked Questions
It divides the family-wise significance level by the number of comparisons, so each individual test uses a stricter threshold. This keeps the chance of any false positive across the whole family at or below the level you chose.
Divide your family-wise alpha by the number of tests. With an alpha of 0.05 and ten comparisons, each test must reach 0.005 to count as significant.
It is the probability of making at least one false positive across an entire set of tests. For independent tests it is one minus the probability that every test avoids a false positive.
Often, yes. It holds under any dependence structure, which makes it safe but strict, and it loses real effects when the tests are correlated or numerous. Holm gives the same guarantee with more power.
Holm sorts the p-values and relaxes the threshold as it works down the list, stopping at the first failure. It controls the same error rate under the same assumptions and always rejects at least as many hypotheses.
When you are screening many candidates and can tolerate a known proportion of false positives among your discoveries. It controls the false discovery rate rather than the chance of any false positive.
Define the family before you look at the results and include every test in it, significant or not. There is no formula for this decision, which is exactly why it has to be recorded in advance.
This page explains a statistical adjustment for educational purposes. It is not a substitute for a pre-registered analysis plan, and the choice of family is a judgement the calculator cannot make for you.
