False Discovery Rate Calculator

Run twenty hypothesis tests at the usual 5% significance level and you should expect one false positive by chance alone. Run a thousand and you should expect fifty. The false discovery rate (FDR) is the tool statisticians use to keep that flood of chance findings under control: instead of guarding against any single error, it controls the expected share of false positives among the findings you actually report. This page gives you two tools in one calculator: paste your p-values to get the full Benjamini-Hochberg decision table, or enter study-level counts to estimate FDR under four different correction methods.

Quick Answer

How to use the false discovery rate calculator

Use the first tab when you have the p-values themselves, and the second tab when you only have study-level counts. Both result panels stay hidden until you click Calculate on that tab.

1. P-value BH table: paste every p-value from the family of tests, separated by commas, spaces, semicolons or line breaks, and set the FDR level q (0.05 is standard). The tool sorts the values, compares each against its critical value (rank ÷ tests) × q, and marks the tests significant up to the largest rank that passes. The p-values come from your own tests, whether that is a t-test, an ANOVA read through an F statistic calculator, or a genomics pipeline.
2. Study-level FDR: enter the total tests and the number you called significant, pick alpha (or a custom alpha), a correction method, an assumed pi0 and a study type. You get the estimated FDR, expected false and true discoveries, the adjusted p-value threshold, the uncorrected family-wise error rate, a power estimate based on the assumed share of real effects, a quality band and one sentence of guidance matched to the study type.
3. Read the two tabs differently: the BH table is a decision about your data. The study estimate is an expectation built on assumptions, useful for planning and for spotting a discovery list that is probably mostly noise.

False discovery rate vs false positive rate vs family-wise error rate

Three error rates get confused because all three count false positives. They divide that count by three different things, and they answer three different questions.

The false positive rate belongs to a single test or classifier and equals 1 minus specificity: it never looks at how many discoveries you made. If that is the number you need, the false positive rate calculator computes it from false positives and true negatives. FDR belongs to a list of rejected hypotheses, which is why the two can disagree sharply: screen 10,000 truly negative cases and 10 real effects with a test whose FPR is 5%, and the significant pile holds roughly 500 false alarms against 10 real findings, an FDR near 98% from a test that looks fine on FPR alone. FWER is the strictest of the three: at alpha 0.05 across 50 tests it reaches 92.31%, which is why procedures that control it, like Bonferroni, discard so many real findings in large families.

The Benjamini-Hochberg procedure, step by step

Benjamini-Hochberg controls the expected FDR at the level q you choose. It is a step-up procedure: one pass down the sorted list decides the whole family.

Step 3 is the part people get wrong by hand: the test at rank k does not need every rank below it to pass individually. Because the largest passing rank sets the cut, a p-value can be carried over the line by a larger one behind it. The worked example below is the pre-filled data in the first tab, and the table is exactly what the tool prints (critical values in the exponential format the tool uses, and p-values as typed, so 0.010 shows as 0.01):

Rank 4 passes exactly at its critical value (0.02 against 2.000e-2), rank 5 fails, and no later rank passes, so k = 4 and the tool reports 4 of 10. Raise q to 0.10 and the same list returns 8 of 10, which is the honest price of tolerating twice the false discovery share. The procedure comes from Benjamini and Hochberg’s 1995 paper in the Journal of the Royal Statistical Society, and it assumes independent tests or the positive-dependence case the authors and later work describe. For arbitrary dependence, that is what the Benjamini-Yekutieli option on the study tab and in the method table below is for.

The study-level estimate: one study, four methods

The second tab starts from a base estimate, FDR = (tests × alpha) ÷ significant results, and each method adjusts it differently. The pre-filled study (50 tests, 20 significant, alpha 0.05, assumed pi0 0.7) run under all four methods gives:

Every method also reports the same two context figures for this study: a family-wise error rate of 92.31% with no correction, and a power estimate of 35.8%. The spread in the table is the lesson. Benjamini-Yekutieli multiplies the base estimate by the harmonic sum of 50 (about 4.499), because it refuses to assume independence, and the product is capped at 100% like every estimate on this tab. The Bonferroni row in this estimator multiplies the base figure by the number of tests and then caps at 100%, which is why it saturates here: read it as “at least this strict”, not as a measured 100%. The Storey-style row scales the base by pi0, the share of true nulls, which you enter directly as an assumption (0.7 in the pre-filled study), so assuming fewer true nulls lowers the estimate. It is a Storey-style estimate only: Storey’s method estimates pi0 from the p-value distribution itself, and a count-only tab has no p-values to estimate it from, so treat this row as what the FDR would be if pi0 really were 0.7, not as a Storey q-value. Expected false and true counts are the method’s FDR times the 20 discoveries and what remains of them, rounded to one decimal by the tool, which is why the Storey-style row shows 1.8 and 18.3 rather than exact halves.

What the study-level estimate assumes, and where it breaks

The base estimate assumes every null hypothesis is true. Expected false positives are then tests × alpha, the most false positives chance alone would produce on average, so the BH row is best read as an upper-bound style figure: real effects in the family pull the true FDR below it. That is a deliberate conservative choice for a planning tool, and it is the same simplification behind the worked example in most textbooks.

Three corners of the study tab are handled the textbook way, and they are worth stating plainly. First, every estimate is capped at 100%: the Benjamini-Yekutieli figure is the base estimate times the harmonic factor, capped at 1, so a 100-test study with 5 significant results at alpha 0.05 returns an estimated FDR of 100.00%, with 5.0 expected false discoveries and 0.0 expected true discoveries. A share cannot exceed 100% and expected true discoveries never go below zero: the uncapped product (518.74%) is named here only as the arithmetic the cap corrects. Second, with zero significant results there are no discoveries that could be false, so every method reports an FDR of 0.00% with 0 expected false and 0 expected true discoveries. Third, the Storey-style adjusted threshold is also capped at 1, the largest value a p-value threshold can take, so a very small assumed pi0 can no longer push it past 1. None of these change the BH p-value tab, which is the exact procedure; they are reasons to read the study tab as the estimate it is.

The power figure needs the same care. The tool computes it from the assumed share of real effects (1 minus your assumed pi0) and the share of tests called significant, power = (1 − pi0) × (significant ÷ tests) ÷ ((1 − pi0) + pi0 × alpha). That is a posterior-style estimate tied to the numbers you entered, not the design power of a study computed from effect size and sample size. For a design-stage power figure, run a proper power analysis alongside the statistical significance calculator for the per-test side of the plan, and use the number here only as a rough plausibility check on a finished study.

Choosing a method and a q level

Pick the method by the dependence you can defend, and q by what a false discovery costs you. BH is the default for independent tests and for the positive dependence common in genomics screens. Benjamini-Yekutieli is the safe pick when tests are correlated in ways you cannot characterise, for example expression measurements on the same samples, where the correlation coefficient calculator shows the kind of dependence involved: you pay for that safety in discoveries, as the four-method table shows. Bonferroni, which controls the family-wise error rate at alpha ÷ tests, suits a small family where a single false claim is expensive, such as confirmatory clinical endpoints. The Storey-style option suits large screens where you have a defensible assumption for the share of true nulls: it scales the estimate by that assumed pi0 and buys back power when the assumption is low. Storey’s full method, the approach Storey and Tibshirani set out for genomewide studies, estimates pi0 from the p-values themselves, so use the p-value tab or a q-value pipeline when you need the estimate made from data rather than assumed.

For q, 0.05 is the convention in most fields and 0.10 is common in exploratory screens that will be followed by validation. Above 0.10 you are no longer really controlling anything a reader would recognise. Whatever you choose, report it with the result: “significant at FDR q = 0.05 under Benjamini-Hochberg” is a complete statement, while a bare “p < 0.05” across hundreds of tests is not. In code, the same procedures live in R’s p.adjust with method “fdr” and in Python’s statsmodels multipletests, and Columbia University’s False Discovery Rate methods page walks through why estimating the share of true nulls is what separates Storey’s approach from plain BH.

What competing calculators miss

For this exact keyword, the competing tools each cover part of the job and stop.

• Brave Calculator: the article teaches the BH procedure step by step, but the tool itself is a bare two-input division (false positives over total discoveries). You cannot paste p-values into it, and the procedure it explains is never performed.
• Savvy Calculator: two different tools circulate under this name. One checks a single test at a time (m, rank, p-value, q in, threshold and yes/no out), which cannot reproduce the step-up behaviour where a later rank carries an earlier one. The other takes aggregate counts with a simple or conservative toggle, with no Benjamini-Yekutieli, no Storey option, no family-wise error rate and no power read.
• Code libraries (R p.adjust, statsmodels, SciPy): exact adjusted p-values and q-values, and the right choice for a pipeline, but they return arrays, not an explanation: no decision table a reviewer can read at a glance, and nothing for a reader who does not write code.
• The gap this page fills: the actual BH decision table from your own p-values, plus a study-level estimate with four named methods, the uncorrected FWER for scale, guidance matched to eight study types, and the estimate’s caps and limits written on the page instead of left for you to find.

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