F-statistic Calculator
Calculate F-statistics for variance comparison and regression analysis with ANOVA
F-statistic Calculation
Compare variances of two populations
Variance of the first population sample
Variance of the second population sample
Number of observations in first sample
Number of observations in second sample
Significance level for hypothesis testing
Example: Variance Comparison
Quality Control Study
Two production lines produce widgets. We want to test if they have equal variance in quality.
Line 1: Sample variance = 10.5, n₁ = 25
Line 2: Sample variance = 5.2, n₂ = 30
Calculation
F = S₁² / S₂² = 10.5 / 5.2 = 2.019
df₁ = n₁ - 1 = 24, df₂ = n₂ - 1 = 29
Critical value (α = 0.05) ≈ 1.90
Decision: F > F_critical, reject H₀
F-test vs T-test
F-test
- • Tests multiple coefficients jointly
- • Compares variances of populations
- • Based on F-distribution
- • Always positive values
T-test
- • Tests individual coefficients
- • Compares means of populations
- • Based on t-distribution
- • Can have negative values
F-distribution Properties
Always non-negative (F ≥ 0)
Right-skewed distribution
Defined by two degrees of freedom parameters
As df increases, approaches normal distribution
Used in ANOVA and regression analysis
Understanding F-statistic
What is F-statistic?
The F-statistic is a test statistic used to compare variances between groups or test the joint significance of multiple regression coefficients. It follows an F-distribution under the null hypothesis.
Applications
- •Testing equality of population variances
- •ANOVA (Analysis of Variance)
- •Overall significance in regression models
- •Testing nested model restrictions
Formulas
Basic F-test:
F = S₁² / S₂²
Where S₁² and S₂² are sample variances
Regression F-test:
F = [(SSRr - SSRf) / J] / [SSRf / (N - K)]
SSRr/SSRf: restricted/full model sum of squared residuals
J: number of restrictions, K: total coefficients, N: sample size
Degrees of Freedom
- Basic: df₁ = n₁-1, df₂ = n₂-1
- Regression: df₁ = J, df₂ = N-K
Interpretation Guidelines
Large F-statistic suggests:
- • Significant difference in variances
- • Joint significance of restricted variables
- • Evidence against null hypothesis
- • Model improvement with additional variables
Small F-statistic suggests:
- • No significant difference in variances
- • Restricted variables not jointly significant
- • Insufficient evidence against H₀
- • Simpler model may be preferable