False Discovery Rate Calculator
Solve false discovery rate problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Calculator
Adjust values & calculateFDR-Adjusted Results
| Rank | Test # | P-value | Threshold | Adj. P (q) | Result |
|---|---|---|---|---|---|
| 1 | #1 | 0.001000 | 0.005000 | 0.010000 | Sig. |
| 2 | #2 | 0.008000 | 0.010000 | 0.040000 | Sig. |
| 3 | #3 | 0.039000 | 0.015000 | 0.084000 | N.S. |
| 4 | #4 | 0.041000 | 0.020000 | 0.084000 | N.S. |
| 5 | #5 | 0.042000 | 0.025000 | 0.084000 | N.S. |
| 6 | #6 | 0.060000 | 0.030000 | 0.100000 | N.S. |
| 7 | #7 | 0.100000 | 0.035000 | 0.142857 | N.S. |
| 8 | #8 | 0.250000 | 0.040000 | 0.312500 | N.S. |
| 9 | #9 | 0.500000 | 0.045000 | 0.555556 | N.S. |
| 10 | #10 | 0.800000 | 0.050000 | 0.800000 | N.S. |
Formula
The Benjamini-Hochberg procedure sorts p-values, calculates rank-based thresholds, and finds the largest rank where the p-value falls below its threshold. All hypotheses up to that rank are rejected. Adjusted p-values (q-values) represent the minimum FDR for significance.
Last reviewed: December 2025
Worked Examples
Example 1: Genomics Study with 10 Tests
Example 2: A/B Testing Multiple Variants
Background & Theory
The False Discovery Rate Calculator applies the following established principles and formulas. Statistics and probability provide the mathematical framework for drawing conclusions from data under uncertainty. The measures of central tendency describe where data cluster. The mean is the arithmetic average, computed as the sum of all values divided by the count. The median is the middle value of an ordered dataset, robust to extreme outliers. The mode is the most frequent value. Spread is quantified by variance, the average squared deviation from the mean, and by its square root, the standard deviation. For a sample, variance uses n minus one in the denominator to correct for bias in estimation. The normal distribution, defined by its mean and standard deviation, is the cornerstone of parametric statistics. Its bell-shaped probability density follows the formula f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2). The empirical rule states that approximately 68 percent of observations fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. A z-score standardizes a data point by subtracting the mean and dividing by the standard deviation, expressing how many standard deviations an observation lies from the mean. In hypothesis testing, the p-value is the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. Confidence intervals express the range within which the true population parameter falls with a specified probability, typically 95 percent. Correlation measures linear association between two variables, with Pearson's r ranging from negative one to positive one. Correlation does not imply causation. Linear regression fits a line of the form y = a + bx to minimize the sum of squared residuals. Bayes' theorem relates conditional probabilities: P(A|B) = P(B|A) * P(A) / P(B), allowing prior beliefs to be updated on new evidence. The law of large numbers guarantees that the sample mean converges to the population mean as sample size grows. The central limit theorem states that the distribution of sample means approaches normality regardless of the population distribution, provided the sample size is sufficiently large, typically 30 or more.
History
The history behind the False Discovery Rate Calculator traces back through the following developments. The mathematical study of probability emerged in the 17th century from correspondence between Blaise Pascal and Pierre de Fermat in 1654. Their exchange, prompted by a gambling problem posed by the Chevalier de Mere, established the foundations of probability theory by calculating expected outcomes through systematic enumeration of cases. Jacob Bernoulli formalized the law of large numbers in his posthumously published Ars Conjectandi of 1713, proving rigorously that empirical frequencies converge to theoretical probabilities with increasing observations. His work laid the groundwork for inferential statistics by connecting mathematical probability to observed data. Carl Friedrich Gauss developed the method of least squares around 1795 while adjusting astronomical observations, and he recognized the bell-shaped error distribution that now bears his name. Pierre-Simon Laplace independently worked on the normal distribution and proved an early version of the central limit theorem around 1810, demonstrating why errors in measurement tend toward normality. The late 19th century saw statistics emerge as a distinct scientific discipline. Francis Galton introduced regression and correlation in the 1880s while studying heredity. Karl Pearson formalized these concepts, developed the chi-squared test, and founded the journal Biometrika in 1901, establishing statistics as a rigorous academic field. Ronald Fisher transformed statistical practice in the early 20th century. His 1925 book Statistical Methods for Research Workers introduced significance testing, analysis of variance, and the concept of the p-value as a decision threshold, establishing the framework still used in scientific research. Fisher and Jerzy Neyman engaged in a prolonged methodological dispute over the interpretation of hypothesis tests. The Bayesian approach, rooted in the 18th century work of Thomas Bayes and Laplace, was largely eclipsed by frequentist methods through much of the 20th century but experienced a revival after World War II and accelerated with computational advances. The late 20th and early 21st centuries brought statistics into every domain through big data, machine learning, and the routine availability of software capable of processing millions of observations.
Frequently Asked Questions
Formula
BH threshold(i) = (i / m) ร ฮฑ | Adjusted p(i) = min(p(i) ร m / i, 1)
The Benjamini-Hochberg procedure sorts p-values, calculates rank-based thresholds, and finds the largest rank where the p-value falls below its threshold. All hypotheses up to that rank are rejected. Adjusted p-values (q-values) represent the minimum FDR for significance.
Worked Examples
Example 1: Genomics Study with 10 Tests
Problem: A researcher tests 10 genes for differential expression and obtains p-values: 0.001, 0.008, 0.039, 0.041, 0.042, 0.06, 0.10, 0.25, 0.50, 0.80. Apply BH correction at FDR = 0.05.
Solution: Sort p-values and calculate BH thresholds:\nRank 1: p=0.001, threshold=0.005 โ Significant\nRank 2: p=0.008, threshold=0.010 โ Significant\nRank 3: p=0.039, threshold=0.015 โ Not significant\nRank 4: p=0.041, threshold=0.020 โ Not significant\nLargest significant rank: 2\nBonferroni would require p < 0.005 (only 1 discovery)
Result: 2 discoveries at FDR 0.05 vs 1 with Bonferroni | BH is more powerful
Example 2: A/B Testing Multiple Variants
Problem: An A/B test platform runs 5 tests with p-values: 0.01, 0.03, 0.04, 0.15, 0.60. Determine which are significant at FDR = 0.10.
Solution: Sort and apply BH procedure at alpha=0.10:\nRank 1: p=0.01, threshold=0.02 โ Significant (q=0.05)\nRank 2: p=0.03, threshold=0.04 โ Significant (q=0.075)\nRank 3: p=0.04, threshold=0.06 โ Significant (q=0.067)\nRank 4: p=0.15, threshold=0.08 โ Not significant\nRank 5: p=0.60, threshold=0.10 โ Not significant
Result: 3 significant variants at FDR 0.10 | Expected false discoveries: 0.3
Frequently Asked Questions
What is the False Discovery Rate (FDR) and why is it important?
The False Discovery Rate is the expected proportion of false positives among all rejected hypotheses in multiple testing scenarios. When you perform many statistical tests simultaneously, like testing thousands of genes for differential expression or running hundreds of A/B test variants, the probability of obtaining false positives by chance alone increases dramatically. At a significance level of 0.05, testing 1,000 hypotheses would yield approximately 50 false positives even if none of the effects are real. FDR control provides a principled way to manage this multiple comparisons problem while retaining more statistical power than ultra-conservative methods like Bonferroni correction, making it essential in genomics, neuroimaging, and large-scale A/B testing.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
Why might my result differ from another tool or reference?
Differences typically arise from rounding conventions, the specific version of a formula (for example, simple vs compound interest), or unit inconsistencies between inputs. Check that both tools are using the same formula variant and the same units. The References section links to the authoritative source behind the formula used here.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
How do I verify False Discovery Rate Calculator's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
What inputs do I need to use False Discovery Rate Calculator accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting โ for example, a weight measurement in kilograms, a distance in metres, or a dollar amount โ and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy