STATISTICSInference & TestsStatistics Calculator
📊

Bonferroni Correction Calculator

Free Bonferroni correction calculator. Compare Bonferroni, Šidák, Holm, Benjamini-Hochberg. Multiple

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

α
STATISTICSInference & Tests

Bonferroni Correction — Multiple Comparison Adjustment

Bonferroni, Šidák, Holm, Benjamini-Hochberg. Control FWER or FDR. Step-by-step breakdown with p-value ladder.

ANOVA Post-hoc →Hypothesis Tests →

Real-World Scenarios — Click to Load

Configuration

Total number of comparisons
Usually 0.05

Comma or space separated. Leave empty for adjusted α only.

bonferroni_correction.sh
CALCULATED
$ bonferroni --m=10 --alpha=0.05
Bonferroni α
5.000000e-3
Šidák α
5.116197e-3
Original α
0.05
Tests (m)
10
Share:
Bonferroni Correction Result
m = 10 tests, α = 0.05
α_adjusted = 5.0000e-3
Šidák: 5.1162e-3
numbervibe.com/calculators/statistics/bonferroni-correction-calculator

Corrected vs Uncorrected α

Calculation Breakdown

INPUT
Number of tests (m)
10
ext{Total} ext{comparisons}
INPUT
Significance level (α)
0.05
ext{Usually} 0.05
BONFERRONI
Bonferroni α_adjusted
5.000000e-3
α/m = 0.05/10
ŠIDÁK
Šidák α_adjusted
5.116197e-3
1 − (1−α)^(1/m) = 1 − (1−0.05)^(1/10)

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • Bonferroni: α_adjusted = α/m. Simple, conservative. Controls FWER.
  • Šidák: α_adjusted = 1 − (1−α)^(1/m). Slightly less conservative than Bonferroni.
  • Holm-Bonferroni: Step-down procedure. More powerful than Bonferroni while controlling FWER.
  • Benjamini-Hochberg: Controls FDR (false discovery rate), not FWER. More powerful for exploratory analysis.
  • Power loss: As m increases, Bonferroni becomes very conservative; many true effects may be missed.

Bonferroni vs Holm vs Benjamini-Hochberg

MethodControlsPowerWhen to Use
BonferroniFWERLowestConfirmatory, few comparisons, strict control
ŠidákFWERSlightly higherSimilar to Bonferroni, independent tests
HolmFWERHigher than BonferroniConfirmatory, want more power than Bonferroni
Benjamini-HochbergFDRHighest (exploratory)Exploratory, many tests (genomics, screening)

FWER vs FDR

Family-Wise Error Rate (FWER)

Probability of making at least one false positive among all tests. Bonferroni and Holm control FWER.

False Discovery Rate (FDR)

Expected proportion of false positives among rejected hypotheses. Benjamini-Hochberg controls FDR.

Formulas Reference

Bonferroni: α_adjusted = α / m

Šidák: α_adjusted = 1 − (1−α)^(1/m)

Holm: Reject p_(i) if p_(i) < α/(m−i+1) for sorted p-values

Benjamini-Hochberg: Reject p_(i) if p_(i) < (i/m)×α

Step-by-Step: Holm-Bonferroni

Step 1: Sort p-values: p_(1) ≤ p_(2) ≤ … ≤ p_(m).

Step 2: Compare p_(1) to α/m. If p_(1) ≥ α/m, reject none and stop.

Step 3: If p_(1) < α/m, reject H_(1). Compare p_(2) to α/(m−1).

Step 4: Continue until p_(i) ≥ α/(m−i+1). Reject all hypotheses before that point.

Frequently Asked Questions

Why do we need multiple comparison correction?

When you run many tests, the chance of at least one false positive increases. Without correction, 5% of tests will be false positives by chance when α=0.05.

Is Bonferroni too conservative?

Yes, for many tests. With m=100, Bonferroni gives α=0.0005. Holm is less conservative; BH (FDR) is even less so when you accept some false discoveries.

What is the difference between Holm and Bonferroni?

Holm is a step-down procedure: it rejects the smallest p-value if < α/m, then the next if < α/(m−1), etc. It always rejects at least as many as Bonferroni and often more.

When should I use FDR instead of FWER?

Use FDR (e.g., BH) when doing exploratory screening (e.g., gene expression) and you can tolerate some false positives. Use FWER for confirmatory studies.

Did You Know?

📊The Bonferroni correction is named after Carlo Emilio Bonferroni (1892–1960), an Italian mathematician.
🧬Genome-wide association studies often use p < 5×10⁻⁸ as the significance threshold for ~1M SNPs.
📈Benjamini-Hochberg (1995) revolutionized multiple testing by introducing FDR control.
🔬Holm (1979) improved on Bonferroni by using a step-down procedure that uniformly dominates it.

Disclaimer: This calculator is for educational purposes. For research or publication, consult a statistician and use established software (R, Python, etc.).

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

P-Value Calculator

Compute p-values from test statistics for z, t, chi-square, and F distributions. One-tailed and two-tailed tests. Significance stars and decision rule.

Statistics

Chi-Square Calculator

Perform chi-square goodness-of-fit and independence tests. χ² statistic, p-value, degrees of freedom, and Cramér's V.

Statistics

F-statistic Calculator

Computes F-statistic and p-value for ANOVA, regression F-test, and variance ratio tests. Includes F-distribution visualization.

Statistics

Z-test Calculator

Performs z-tests: one-sample (σ known), two-sample, one-proportion, two-proportion. CI, effect size, power.

Statistics

ANOVA Calculator

One-way analysis of variance (ANOVA) calculator. Compare means of multiple groups, compute F-statistic, p-value, eta squared, and ANOVA table with...

Statistics

Hypothesis Testing Calculator

Comprehensive hypothesis testing: one-sample z/t, two-sample t, paired t, one/two-proportion z. Test statistic, p-value, confidence interval, decision...

Statistics