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 |
Step-by-Step Decision Process
- Identify the purpose: F-tests compare variances (two populations) or test overall significance in ANOVA/regression.
- State hypotheses for variance comparison: H0: sigma1^2 = sigma2^2. H1: sigma1^2 ≠ sigma2^2 (or one-tailed).
- Calculate F-ratio: F = s1^2 / s2^2 (larger variance in numerator for right-tailed test).
- Determine degrees of freedom: df1 = n1-1 (numerator), df2 = n2-1 (denominator).
- Find critical value or p-value: The F-distribution is right-skewed and always positive.
- Make decision: If F exceeds the critical value, reject H0 (variances are significantly different).
Additional Solved Examples
Example: Comparing Variability of Two Machines
Machine A (n=16): variance = 2.4. Machine B (n=21): variance = 5.8. Do the machines have different precision at alpha = 0.05?
H0: sigma_A^2 = sigma_B^2
H1: sigma_A^2 ≠ sigma_B^2 (two-tailed)
F = larger/smaller = 5.8/2.4 = 2.417
df1 = 21-1 = 20 (Machine B), df2 = 16-1 = 15 (Machine A)
F-critical (0.025, 20, 15) = 2.76 (upper tail for two-sided)
Since 2.417 < 2.76, fail to reject H0
Answer: There is insufficient evidence to conclude the machines have different variability (F(20,15) = 2.42, p > 0.05). We cannot reject the assumption of equal variances.
Example: F-Test in Regression
A multiple regression with 3 predictors and n=50 observations yields: SSR = 450, SSE = 300. Test overall model significance.
df_regression = k = 3
df_residual = n - k - 1 = 50 - 3 - 1 = 46
MSR = 450/3 = 150
MSE = 300/46 = 6.522
F = MSR/MSE = 150/6.522 = 23.0
F-critical (0.05, 3, 46) = 2.81
Since 23.0 >> 2.81, reject H0 (p < 0.001)
R-squared = SSR/SST = 450/750 = 0.60
Answer: The regression model is highly significant (F(3,46) = 23.0, p < 0.001). The three predictors collectively explain 60% of the variance in the response variable.
Interpreting Results
Properties of the F-Distribution
- The F-distribution is always non-negative (ratio of two positive quantities)
- It is right-skewed, becoming more symmetric as df increase
- F = 1 means the two variances are equal (no difference between groups in ANOVA)
- The F-distribution has two df parameters: df1 (numerator) and df2 (denominator) - order matters
- F(df1, df2) = 1/F(df2, df1) - swapping numerator and denominator inverts the statistic
Key Takeaways
- The F-statistic is the ratio of two variances (or mean squares). F near 1 means no difference; F much greater than 1 suggests significance.
- In ANOVA, F = MS_between / MS_within tests whether group means differ beyond random variation.
- In regression, the overall F-test determines whether the model as a whole has predictive value.
- The F-test for equal variances is sensitive to non-normality - Levene's test is a more robust alternative.
- The F-distribution is the square of the t-distribution for single-parameter tests: F(1, df) = t(df)^2.
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.