F-Statistic Calculator
Calculate the F-ratio to test equality of two population variances. Enter sample variances and sizes to get the F-statistic, degrees of freedom, and p-value. Related tools: ANOVA Calculator, T-Test Calculator, and Variance Calculator.
How to Use the F-Statistic Calculator
The F-test compares two population variances to determine if they are significantly different. It is commonly used as a preliminary test before a two-sample t-test (to check the equal variance assumption) and in ANOVA to compare variability between groups to variability within groups.
Enter the sample variance (s²) and sample size (n) for each group. The larger variance should typically be placed in the numerator (group 1) for a right-tailed test. Select your significance level and tail type, then click Calculate. The calculator returns the F-ratio, degrees of freedom, and p-value.
The F-test assumes both populations are normally distributed. It is sensitive to departures from normality — more so than the t-test. For non-normal data, consider Levene's test or Bartlett's test as alternatives for testing equality of variances.
Formula
F-Statistic:
F = s₁² / s₂² (larger variance in numerator)
Degrees of Freedom:
df₁ = n₁ - 1 (numerator)
df₂ = n₂ - 1 (denominator)
Hypotheses (two-tailed):
H₀: σ₁² = σ₂²
H₁: σ₁² ≠ σ₂²
Decision Rule:
Reject H₀ if F > F(α/2, df₁, df₂) or F < F(1-α/2, df₁, df₂)
Example Calculation
Test whether two manufacturing processes have equal variance:
Given: s₁² = 25, n₁ = 21, s₂² = 16, n₂ = 16
F = 25 / 16 = 1.5625
df₁ = 21 - 1 = 20, df₂ = 16 - 1 = 15
p-value (two-tailed) ≈ 0.3842
At α = 0.05: Fail to reject H₀
Conclusion: No significant difference in variances.
The equal variance assumption is reasonable for a t-test.
F Critical Values Reference Table
| df₁ | df₂ | F₀.₁₀ | F₀.₀₅ | F₀.₀₁ |
|---|---|---|---|---|
| 1 | 10 | 3.285 | 4.965 | 10.044 |
| 2 | 10 | 2.924 | 4.103 | 7.559 |
| 5 | 10 | 2.522 | 3.326 | 5.636 |
| 5 | 20 | 2.158 | 2.711 | 4.103 |
| 10 | 10 | 2.323 | 2.978 | 4.849 |
| 10 | 20 | 1.937 | 2.348 | 3.368 |
| 15 | 15 | 2.014 | 2.403 | 3.522 |
| 20 | 20 | 1.794 | 2.124 | 2.938 |
| 30 | 30 | 1.606 | 1.841 | 2.386 |
| 60 | 60 | 1.404 | 1.534 | 1.836 |
Frequently Asked Questions
What is the F-test?
The F-test is a statistical test that compares two variances to determine if they are significantly different. It uses the F-distribution, which is the ratio of two chi-square distributions divided by their respective degrees of freedom. The test is named after Sir Ronald Fisher and is fundamental to ANOVA and regression analysis. The F-statistic is always positive since it is a ratio of variances.
When should I use the F-test?
Use the F-test to: (1) test equality of variances before performing a two-sample t-test, (2) compare variability between groups in ANOVA, (3) test overall significance of a regression model, or (4) compare nested regression models. For testing variance equality specifically, the F-test requires normally distributed data. If normality is questionable, use Levene's test instead.
How do I interpret the F-ratio?
An F-ratio of 1 means the variances are equal. F > 1 means the numerator variance is larger. The further F is from 1, the stronger the evidence that variances differ. However, you must compare F to the critical value (or use the p-value) to determine statistical significance. A large F with a small p-value indicates the variances are significantly different.
What is the relationship between F-test and ANOVA?
ANOVA uses the F-test to compare means of three or more groups. The ANOVA F-statistic is the ratio of between-group variance to within-group variance: F = MS_between / MS_within. A large F indicates that group means differ more than expected by chance. The two-sample F-test for variances is a special case comparing just two groups' variability.
Why should the larger variance be in the numerator?
For a right-tailed test (H₁: σ₁² > σ₂²), placing the larger variance in the numerator ensures F ≥ 1, simplifying the comparison with critical values from standard F-tables (which only list right-tail values). For a two-tailed test, you can place either variance in the numerator and double the one-tailed p-value, or always use the larger variance and compare to F(α/2).
What are the assumptions of the F-test for variances?
The F-test assumes: (1) both populations are normally distributed — this is critical as the test is very sensitive to non-normality, (2) samples are independent, (3) data are continuous. Unlike the t-test, the F-test for variances does not become robust with large samples. For non-normal data, alternatives include Levene's test (uses absolute deviations from median) or the Brown-Forsythe test.