F-statistic Calculator
Free F-statistic calculator. ANOVA, regression F-test, variance ratio. F-distribution, p-value, effe
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
F-Statistic — ANOVA, Regression, Variance Ratio
F-statistic, p-value, F-distribution, effect size (η², ω²). Step-by-step breakdown with interactive visualization.
Real-World Scenarios — Click to Load
Mode
F-Distribution (df1=2, df2=12) — Rejection Region
Observed F = 4.5000. Red = rejection region (α = 0.05).
Variance Comparison (MS Between vs MS Within)
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • F-statistic: F = (explained variance) / (unexplained variance). Ratio of mean squares.
- • ANOVA: F = MS_between / MS_within. MS = SS/df. Reject H₀ when F > F_critical.
- • Regression: F = MS_regression / MS_residual. From R²: F = (R²/p) / ((1−R²)/(n−p−1)).
- • Variance test: F = s₁²/s₂² (larger/smaller). df1 = n₁−1, df2 = n₂−1.
- • p-value: P(F_df1,df2 ≥ F_observed). Uses F-distribution (ratio of chi-squares).
- • Effect size: η² = SS_between/SS_total. ω² = (SS_between − (k−1)MS_within)/(SS_total + MS_within).
Did You Know?
Expert Tips
ANOVA vs Regression F
ANOVA: comparing means across groups. Regression F-test: testing if the regression model (all predictors) explains significant variance. Both use the same F-distribution.
Effect Size Matters
A significant F-test does not mean a large effect. Always report η² or ω². Cohen: 0.01=small, 0.06=medium, 0.14=large for η².
Post-Hoc Tests
A significant ANOVA F-test only tells you that at least one group differs. Use Tukey, Bonferroni, or Scheffé for pairwise comparisons. See our Bonferroni Correction Calculator.
Variance Ratio Convention
For the F-test of variances, always put the larger variance in the numerator so F ≥ 1. This makes it a one-tailed test in the upper tail.
Formulas
ANOVA: F = MS_between / MS_within
MS = SS/df
From R²: F = (R²/p) / ((1−R²)/(n−p−1))
p = number of predictors
Variance ratio: F = s₁²/s₂² (larger/smaller)
df1 = n₁−1, df2 = n₂−1
η² = SS_between / SS_total
Eta-squared effect size
Modes Explained
1. ANOVA summary
Enter SS_between, df_between, SS_within, df_within from ANOVA table. F = MS_between / MS_within.
2. From R² and n, p
Enter R², sample size n, and number of predictors p. Computes regression F-test.
3. Two variances
Enter s₁², n₁, s₂², n₂. F = larger variance / smaller. Tests H₀: σ₁² = σ₂².
4. Raw group data
Paste groups (one per line, comma-separated values). Computes one-way ANOVA from raw data.
Frequently Asked Questions
When do I use ANOVA vs regression F-test?
ANOVA: comparing means across groups. Regression F-test: testing if the regression model (all predictors) explains significant variance.
What does a significant F mean?
Reject H₀. For ANOVA: at least one group mean differs. For regression: model is useful. For variance test: variances differ.
What is the F-distribution?
Distribution of (χ²_df1/df1) / (χ²_df2/df2). Right-skewed. Critical value depends on df1, df2, and α.
How do I interpret η²?
Proportion of variance explained. 0.01=small, 0.06=medium, 0.14=large (Cohen).
What if F < 1?
For ANOVA/regression, F < 1 means within-group variance exceeds between-group — no evidence of group differences.
What are the assumptions?
ANOVA: independence, normality within groups, homogeneity of variance. Regression: normal residuals, homoscedasticity. Variance test: both samples from normal populations.
Effect Size Interpretation
| η² | Interpretation |
|---|---|
| 0.01 | Small |
| 0.06 | Medium |
| 0.14 | Large |
Applications
Clinical Trials
Compare treatment arms (ANOVA).
Regression
Overall model significance.
Quality Control
Compare process variances.
A/B/C Testing
Multiple variants.
Chart Interpretation
The F-distribution chart shows the null distribution. The red shaded area is the rejection region (F ≥ F_critical). If your observed F falls in the red region, reject H₀. The variance comparison bar shows group means (or MS_between vs MS_within when using summary stats).
Official Data Sources
Disclaimer: This calculator provides statistical guidance. Verify assumptions (normality, homogeneity of variance) before interpreting results. For publishable research, verify with R, Python scipy, SAS, or SPSS.
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