Bonferroni Correction Calculator
The formula
What the family-wise error control tells you
You ran many tests on one dataset, so the chance of at least one false positive inflates; Bonferroni divides the significance budget by the number of tests.
Calculation: Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1). Enter your inputs and every substitution is shown below the result.
Reading the result
Ten tests at α=.05 leave each test only .005 to play with — a raw p of .006 just misses, which is the point of the arithmetic.
Rules of thumb: Report the correction method and the full family size alongside every adjusted p
Limitations and best practice
Counting only the tests you report and not the ones you abandoned is the common cheat; the family is everything you looked at, not what survived.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Is Bonferroni too conservative?
Does it fix p-hacking?
Source & methodology
Bonferroni (1936); Dunn (1961) simultaneous comparisons.
Bonferroni Correction explained: formula, worked example and honest limits
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. There are plenty of pages about Bonferroni Correction; almost none of them show the working. This one does: the formula, every input, a fully computed example, and a sensitivity analysis calculated live from this engine’s own defaults.
Most confusion about Bonferroni Correction comes from mixing up the input you control with the input you merely observe. The formula treats them identically, but your decisions should not: one deserves measurement, the other deserves a defensible estimate.
Research arithmetic demands traceability, and Bonferroni Correction delivers it: every substitution of Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1) is printed, so 0.072 can be reproduced, cited and defended — the three things a bare calculator output cannot promise.
What makes a calculation page trustworthy is that its copy is derived from its own arithmetic: here adjusted p (Bonferroni) = 0.072, recomputed every time a field changes, and the sensitivity section ranks the inputs that move the result — Smallest raw p-value first, at ≈10.0 % of the headline for a ten-percent error either way in that input — which is the thread the rest of this guide pulls on.
How Bonferroni Correction is computed, line by line
One formula, published in full: Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1). Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Experiment-wise α (example value 0) is a plain measured quantity — the unit shown in the label is the unit the formula expects. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Number of comparisons (example value 12) is a plain measured quantity — the unit shown in the label is the unit the formula expects. If the true figure falls between counts, run both and read the spread as your uncertainty.
Smallest raw p-value (example value 0) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
The default Bonferroni Correction calculation, written out in full
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Experiment-wise α, Number of comparisons, Smallest raw p-value. Run to the end, the adjusted p (Bonferroni) resolves to 0.072 — Fails after correction: with 12 looks, this is exactly the false positive the method exists to catch on the bands this tool uses.
- α per test = 0.05 ÷ 12 = 0.0042
- Adjusted p = 0.006 × 12 = 0.072
Per-test alpha allowed — 0.004. This is the number experienced users quote when the headline alone would mislead.
Raw p × m — 0.072. Read next to the headline, this is the figure that shows the shape of the trade-off.
Effective confidence needed — 99.58 %. Compare it against the main result of 0.072 before drawing conclusions.
Chance of ≥1 false positive uncorrected — 46.0 %. This is the number experienced users quote when the headline alone would mislead.
Overwrite any input above and every line of this example updates with it — the substitution is live, not a screenshot of somebody else’s numbers.
The Bonferroni Correction corridor: your result across a realistic band
Single-point answers invite false confidence, so the same engine was run across a band around the input that matters most here. Each row is a full recalculation of Bonferroni Correction, not an interpolation.
| Smallest raw p-value setting | Adjusted p (Bonferroni) |
|---|---|
| 80 % of the example value | 0.058 |
| 90 % of the example value | 0.065 |
| example inputs | 0.072 |
| 110 % of the example value | 0.079 |
| 120 % of the example value | 0.086 |
That spread — 0.058 at the low end, 0.086 at the high — is what a careful read of Bonferroni Correction looks like: a band you can state out loud, with the reason for each edge. Reproduce it in the form above by editing Smallest raw p-value alone; the substitution steps will follow your figures exactly.
Sensitivity: which inputs change Bonferroni Correction the most
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the adjusted p (Bonferroni). It is the fastest honest answer to “what matters most in Bonferroni Correction?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Smallest raw p-value +10 % → Bonferroni Correction = 0.079 · −10 % → 0.065 (the adjusted p (Bonferroni) shifts ≈10.0 %)
- Nudge Number of comparisons by ±10 % and Bonferroni Correction lands between 0.065 and 0.079 (sensitivity ≈10.0 %)
Smallest raw p-value is the dominant lever here: ten percent either way moves the result by about 10.0 %, from 0.065 to 0.079. Spend your verification effort there first, and Number of comparisons is effectively tied with it (≈10.0 % for the same nudge), so verify the pair together. The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
What a Bonferroni Correction of 0.072 means in practice
The value above only becomes useful the moment you know what it is being measured against. The default case lands in the “Fails after correction: with 12 looks, this is exactly the false positive the method exists to catch” band, and bands are deliberately wide because real situations overlap. The sensitivity list shows how far a modest input error moves you along that ladder: for Smallest raw p-value, ±10 % covers 0.065 to 0.079. The practical reading rules: ask what would have to change to move you one band. If the answer is “a small, plausible input difference”, the band itself is not the finding — the uncertainty is.
Related measures answer related questions, and the distinctions matter: Bonferroni Correction is the quantity, Adjusted p (Bonferroni) is this tool’s reading of it, and the decision that follows belongs to neither of them separately.
How people get Bonferroni Correction wrong
Counting only the tests you report and not the ones you abandoned is the common cheat; the family is everything you looked at, not what survived. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Comparing a Bonferroni Correction computed here with a figure from another source that uses different conventions. Same name, different denominator — the disagreement is usually in the definitions, not the data.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Smallest raw p-value first, everything else second.
A final trap is presentation: reporting Bonferroni Correction to more decimal places than the inputs support. The tool prints what the arithmetic gives; you should quote only what your least precise field justifies.
Before you trust your Bonferroni Correction: a short checklist
- Confirm the example values are plausible for your own situation before replacing them.
- Note the date and the inputs beside the number so the figure can be audited later.
- Check Smallest raw p-value before anything else: it is the measured dominant input, ≈10.0 % of Bonferroni Correction per ten percent.
- Recompute once with the dominant input deliberately set low, and see whether the decision changes.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
A routine for decision-grade Bonferroni Correction
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — start from evidence. The inputs behind Bonferroni Correction are experiment-wise α, number of comparisons, smallest raw p-value. Write them down with their source next to each one; if you cannot name where a figure came from, that is the input most likely to be wrong. The inputs are the part of the work that nobody else can do for you.
Step 2 — enter the conservative case first. Resist the flattering scenario. Type the inputs you would defend in front of a sceptical reader, read the Adjusted p (Bonferroni) (0.072 for the example values), and note the verdict band. Half of a good calculation is knowing which input you would hate to be wrong about.
Step 3 — measure what deserves measuring. The ranked levers above tell you where a careful figure pays off. Everything else can stay approximate, which is a licence to stop hunting for data you do not need. Precision is a promise you make about your data; make sure your inputs can keep it.
Step 4 — ask what would have to be true. Take the result you want and work backwards through the inputs until you find the assumption that produces it; then decide honestly whether that assumption is defensible. A result worth acting on survives having its two shakiest inputs halved.
Step 5 — schedule the next run. In Science & Research the inputs drift quietly. Put a reminder a quarter out, re-enter the same fields, and read the change rather than the level. Numbers are cheap to produce and expensive to retract; the extra minute is the cheapest insurance available.
From calculation to decision: where Bonferroni Correction earns its keep
Most Statistics & Data Science disagreements are arithmetic disagreements in disguise. When Adjusted p (Bonferroni) of 0.072 sits on the table with its working shown, the conversation stops being about who sounds more confident and starts being about which input is wrong. A figure nobody can reproduce is a figure nobody will act on twice.
The audience for this page is people who need one trustworthy number before their next decision. In professional Statistics & Data Science practice the pattern is always the same — screen with Bonferroni Correction, then invest slow analysis only in whatever survives the screen. Estimates improve most when the worst case is written down in ink.
Treat Bonferroni Correction like a smoke alarm rather than a painting — its value is in being tested periodically, not in being admired once. The inputs are the part of the work that nobody else can do for you.
Bonferroni Correction — frequently asked questions
What affects Bonferroni Correction the most?
Measured on this engine, Smallest raw p-value is the dominant lever: moving it ±10 % shifts the adjusted p (Bonferroni) across 0.065 to 0.079 — about 10.0 % of the headline, with Number of comparisons next at ≈10.0 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Do number of comparisons and smallest raw p-value matter equally?
Effectively. Measured on this engine they land at ≈10.0 % and ≈10.0 % for the same ten-percent nudge, so the honest advice is to get both right; chasing precision in only one of them buys almost nothing.
What is a typical Bonferroni Correction?
There is no honest universal average for Bonferroni Correction, but there is a reference point: the defaults on this page produce 0.072 (Fails after correction: with 12 looks, this is exactly the false positive the method exists to catch). Replace the inputs with your own and compare — that comparison is the useful meaning of “typical”.
How do I calculate Bonferroni Correction by hand?
Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1) is the entire method. Work the inputs in that order — α per test = 0.05 ÷ 12 = 0.0042 — and you will reach 0.072 on the default figures, the same number this tool prints.
What happens to Bonferroni Correction if smallest raw p-value changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the adjusted p (Bonferroni) goes across 0.065 to 0.079, a swing of roughly 10.0 % around the 0.072 baseline. That ratio is the practical definition of how much this input matters.
Is the Bonferroni Correction result of 0.072 good or bad?
Good and bad are comparisons, so the honest test is three questions: where does your own Bonferroni Correction sit against the 0.072 reference above, how far would a plausible input error move it (≈10.0 % per ten percent on Smallest raw p-value), and what does the alternative option compute to?
Can Bonferroni Correction be negative or empty?
Yes, and that is informative rather than broken: a negative Bonferroni Correction usually means the inputs encode an outflow or a shortfall, and the verdict band above labels exactly that situation instead of hiding it.
Is Bonferroni Correction accurate?
The arithmetic is exact: Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1) is evaluated with full floating-point precision and every substitution is printed in the worked example. What limits accuracy is input quality and the model’s assumptions, so treat the 0.072 result as the correct answer to the question you typed — and check the assumptions section before acting on it.
Is Bonferroni too conservative?
For truly independent tests, yes, slightly. Holm and Benjamini–Hochberg give the same family-wise or false-discovery control with strictly more power.
Does it fix p-hacking?
No — it prices the multiplicity you declare. Undeclared tests stay uncorrected, which is why pre-registration matters more than any correction.
Do I need an account to use this Bonferroni Correction calculator?
No, and no data leaves the tab. The engine runs in the page itself, which is why the worked example, the bands and the sensitivity ranking all respond instantly to your keystrokes.
The vocabulary of Bonferroni Correction
- Per-test alpha allowed
- 0.004 in the default Bonferroni Correction calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Raw p × m
- 0.072 on the example inputs — a companion figure printed beside the headline of Bonferroni Correction so that no result is read in isolation.
- Effective confidence needed
- 99.58 % on the example inputs — a companion figure printed beside the headline of Bonferroni Correction so that no result is read in isolation.
- Adjusted p (Bonferroni)
- The primary output of this tool: 0.072 for the default inputs, computed exactly from Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1).
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Fails after correction: with 12 looks, this is exactly the false positive the method exists to catch”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Bonferroni Correction moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Methodology, sources and limits
There is no black box on this page — the engine evaluates Per-test α = α ∕ m; Bonferroni-adjusted p = min(p × m, 1). Source of record: Bonferroni (1936); Dunn (1961) simultaneous comparisons. Any other implementation of this model should return the same figure for the same inputs. If one does not, the difference is in its assumptions — which this page prints and that one probably does not.
Nothing here replaces a qualified professional, and nothing here is approximate either — the arithmetic is exact, and the honesty is in the inputs you supply. What that buys you is reproducibility — two people, two browsers, the same arithmetic, and a printed trail from inputs to result that either of them can audit.
Next steps after the calculation
Save the number together with its inputs and today’s date; a Bonferroni Correction figure without context is unusable the moment an input changes.
This page was written for people who need one trustworthy number before their next decision, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.