How to Calculate F-Distribution
What is F-Distribution?
The F-distribution arises in statistical tests that compare variances or mean squares, such as ANOVA and regression analysis. The F-statistic is the ratio of two chi-squared distributions divided by their degrees of freedom.
Formula
F = s₁²/s₂² where s₁² and s₂² are sample variances from two populations
- s₁²
- variance of first sample
- s₂²
- variance of second sample
- F
- F-statistic — ratio of variances
- df₁, df₂
- degrees of freedom — for numerator and denominator
Step-by-Step Guide
- 1F = (s₁²/σ₁²) / (s₂²/σ₂²)
- 2Degrees of freedom: df₁ (numerator) and df₂ (denominator)
- 3If F > critical value: reject H₀ at chosen α
- 4F-test used in ANOVA, regression significance
Worked Examples
Input
F=3.5, df₁=3, df₂=20, α=0.05
Result
Critical value ≈ 3.10; F > critical, reject H₀
Input
F=1.2, df₁=5, df₂=10
Result
Fail to reject H₀ — not significant
Frequently Asked Questions
When is the F-distribution used?
F-tests compare variances of two populations, and in ANOVA to test if multiple group means are equal.
Is the F-distribution symmetric?
No, it's right-skewed. F-values are always positive (ratios of squared quantities).
What does an F-value of 1 mean?
F=1 suggests equal variances in both samples. F>1 indicates first sample has larger variance.
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