F-Distribution Formula
The F-distribution is the ratio of two chi-squared distributions.
Used in ANOVA, comparing variances, and testing overall regression model significance.
The Formula
Under H&sub0; (equal population variances): F = s&sub1;² / s&sub2;²
The F-distribution arises whenever you compare two variances or test whether multiple group means differ simultaneously. It is named after Ronald Fisher, who introduced it in 1924. The distribution is parameterized by two degrees of freedom: d&sub1; (numerator) and d&sub2; (denominator). It is always right-skewed and takes only positive values.
Variables
| Symbol | Meaning |
|---|---|
| F | F-statistic (ratio of two variances) |
| s&sub1;², s&sub2;² | Sample variances from groups 1 and 2 |
| σ&sub1;², σ&sub2;² | Population variances (assumed equal under H&sub0;) |
| d&sub1; | Numerator degrees of freedom = n&sub1; − 1 |
| d&sub2; | Denominator degrees of freedom = n&sub2; − 1 |
Critical F values (approximate, α = 0.05):
- F(1, 10): 4.96 — F(5, 10): 3.33 — F(10, 10): 2.98
- F(1, 30): 4.17 — F(5, 30): 2.53 — F(10, 30): 2.16
- Larger F means stronger evidence against equal variances (or equal means in ANOVA)
Example 1 — Comparing Two Variances
Two manufacturing lines produce bolts. Line 1 (n=21): s&sub1;² = 0.48 mm². Line 2 (n=16): s&sub2;² = 0.24 mm². Are the variances significantly different? (α = 0.05)
F = s&sub1;² / s&sub2;² = 0.48 / 0.24 = 2.0
d&sub1; = 20, d&sub2; = 15
Critical value F(20, 15) at α/2 = 0.025 ≈ 2.86
F = 2.0 < 2.86 critical value → fail to reject H&sub0;. Insufficient evidence that the variances differ.
Example 2 — ANOVA F-Test
ANOVA with 3 groups (k = 3) and 30 total observations. Mean square between groups MSB = 120, mean square within groups MSW = 40.
F = MSB / MSW = 120 / 40 = 3.0
d&sub1; = k − 1 = 2, d&sub2; = N − k = 27
Critical F(2, 27) at α = 0.05 ≈ 3.35
F = 3.0 < 3.35 → fail to reject H&sub0; at 5% level. The group means are not significantly different. (P-value ≈ 0.067)
When to Use It
Use the F-distribution when:
- Testing equality of two population variances (two-sample F-test)
- One-way or multi-way ANOVA — testing if group means differ significantly
- Multiple linear regression — testing whether the overall model is significant
- Comparing fit of two nested regression models (likelihood ratio test)
- Quality control — testing whether production line variability has changed