Inverse F Calculator
Find F-value from percentile.
Probability must be between 0 and 1 (exclusive)
Enter probability and degrees of freedom
What Is the Inverse F Calculator?
The inverse F function finds the F-value corresponding to a given probability. While the F-Distribution Calculator finds P(F < x), this inverse asks "What F value has this probability below it?"
Given: Probability (cumulative), df₁ (numerator), df₂ (denominator) Returns: The F-value at that percentile
Common Critical Values
For right-tail probability α = 0.05 (cumulative p = 0.95):
| df₁ | df₂ = 10 | df₂ = 20 | df₂ = 30 |
|---|---|---|---|
| 1 | 4.96 | 4.35 | 4.17 |
| 2 | 4.10 | 3.49 | 3.32 |
| 3 | 3.71 | 3.10 | 2.92 |
| 5 | 3.33 | 2.71 | 2.53 |
| 10 | 2.98 | 2.35 | 2.16 |
| 20 | 2.77 | 2.12 | 1.93 |
For α = 0.01:
| df₁ | df₂ = 10 | df₂ = 20 | df₂ = 30 |
|---|---|---|---|
| 1 | 10.04 | 8.10 | 7.56 |
| 3 | 6.55 | 4.94 | 4.51 |
| 10 | 4.85 | 3.37 | 2.98 |
Applications
ANOVA Critical Values
"For one-way ANOVA with 4 groups and 50 total observations, what's the critical F at α = 0.05?"
df₁ = 4 - 1 = 3 df₂ = 50 - 4 = 46 F₀.₉₅(3, 46) ≈ 2.81
Reject H₀ if observed F > 2.81
Regression Model Testing
"Testing a regression with 5 predictors and n = 60, find critical F at α = 0.01."
df₁ = 5 df₂ = 60 - 5 - 1 = 54 F₀.₉₉(5, 54) ≈ 3.38
Comparing Variances
"Two samples of size 15 each. What's the critical F for testing equal variances at α = 0.05 (two-tailed)?"
df₁ = df₂ = 14 For two-tailed test: use α/2 = 0.025 in each tail F₀.₉₇₅(14, 14) ≈ 2.98
Reject if F > 2.98 or F < 1/2.98 = 0.34
Nested Model Comparison
Comparing full and reduced regression models requires F critical values based on the number of added terms.
One-Tailed vs. Two-Tailed
One-Tailed (most F-tests)
ANOVA and regression tests use the upper tail only:
- Compute F₀.₉₅ or F₀.₉₉
- Reject if F > critical value
Two-Tailed (variance comparison)
Testing σ₁² ≠ σ₂² uses both tails:
- Upper: F₁₋α/₂
- Lower: Fα/₂ = 1/F₁₋α/₂(df₂, df₁)
The reciprocal relationship: F₀.₀₅(df₁, df₂) = 1/F₀.₉₅(df₂, df₁)
Properties of F Critical Values
As df increases:
- F critical values decrease
- Distribution becomes less skewed
Symmetry: F₁₋α(df₁, df₂) = 1/Fα(df₂, df₁)
This allows computing lower percentiles from upper percentiles.
Historical Context
Ronald Fisher developed the F-distribution in the 1920s.
George Snedecor published extensive F-tables, making ANOVA practical for researchers.
Before computers, F-tables were essential—separate tables for different significance levels and degrees of freedom combinations.
Computational Methods
The inverse F has no closed form. Approaches include:
Newton-Raphson: Iterate to solve CDF(F) - p = 0
Beta function relationship: F relates to the incomplete beta function: P(F < f) = I(df₁×f/(df₁×f + df₂))
Inversion uses inverse beta algorithms.
Initial approximation: For large df, F ≈ 1 + z×√(2/df₂)
F-Distribution Properties
Mean: df₂/(df₂-2) for df₂ > 2
Mode: ((df₁-2)/df₁) × (df₂/(df₂+2)) for df₁ > 2
As both df → ∞: F → 1
| Percentile | Shape | Approximate F |
|---|---|---|
| 50th | — | ≈ 1 for large df |
| 95th | Upper tail | Depends on df |
| 99th | Far upper tail | Much larger |
Quantile Tables
For various cumulative probabilities:
| p | F(3,20) | F(5,30) | F(10,60) |
|---|---|---|---|
| 0.50 | 0.85 | 0.83 | 0.88 |
| 0.90 | 2.38 | 2.05 | 1.71 |
| 0.95 | 3.10 | 2.53 | 2.04 |
| 0.99 | 4.94 | 3.70 | 2.80 |
ANOVA and Regression Context
One-Way ANOVA
df₁ = k - 1 (groups minus 1) df₂ = n - k (total minus groups)
Two-Way ANOVA
Multiple F-tests:
- Factor A: df₁ = a - 1
- Factor B: df₁ = b - 1
- Interaction: df₁ = (a-1)(b-1)
- Error: df₂ = n - a×b
Regression
Overall F: df₁ = p (predictors), df₂ = n - p - 1 Nested F: df₁ = q (added predictors), df₂ = n - k - 1
Related Calculations
- F-Distribution Calculator — Forward calculation
- One-Way ANOVA Calculator — Uses F critical values
- Chi-Square Distribution Calculator — Related distribution
- T-Distribution Calculator — T² ~ F(1, df)
- Degrees of Freedom Calculator — Calculate df
Summary
The inverse F calculator converts cumulative probabilities to F-values for any combination of numerator and denominator degrees of freedom. Critical for determining ANOVA rejection regions, testing regression significance, and comparing variances, the inverse F function complements the forward CDF. Both degrees of freedom affect critical values, with larger df producing smaller critical values (closer to 1). For variance comparison tests, the reciprocal relationship allows computing lower percentiles from upper percentiles.