1.5625
19
19
25
16
9
1.5625
56.25
%
1.5625
19
19
25
16
9
1.5625
56.25
%
The F-Test Calculator compares the variances of two independent populations to determine if they are significantly different. Also known as the variance ratio test, the F-test is fundamental in statistics — it serves as a preliminary check for homogeneity of variances before running a t-test and forms the basis of ANOVA.
Enter the sample variances and sample sizes to compute the F-statistic, which is the ratio of the larger variance to the smaller variance, along with the appropriate degrees of freedom.
The F-test for equality of variances compares two sample variances to test whether the corresponding population variances are equal. The test statistic is the ratio of the larger to the smaller sample variance:
$$F = \frac{s_1^2}{s_2^2} \quad \text{where } s_1^2 \geq s_2^2$$
By convention, the larger variance is placed in the numerator so that F ≥ 1. The degrees of freedom are:
$$df_1 = n_1 - 1 \quad (\text{numerator, larger variance})$$
$$df_2 = n_2 - 1 \quad (\text{denominator, smaller variance})$$
Under the null hypothesis \(H_0: \sigma_1^2 = \sigma_2^2\), the F-statistic follows an F-distribution with \(df_1\) and \(df_2\) degrees of freedom. The F-distribution is right-skewed and bounded below by 0, with a mean near 1 when the null hypothesis is true.
For a two-tailed test (testing whether variances differ in either direction), compare F to the critical value at α/2. For a one-tailed test (testing if one specific variance is larger), use α directly.
Interpreting the F-test results:
Inputs
Results
Two measurement methods with variances 25 and 16. F = 1.5625 < 2.53 (critical at α=0.05 two-tailed), so we cannot reject equal variances.
Inputs
Results
Machine A shows much higher variability than Machine B. F = 3.75 > 3.09 (critical at α=0.05 two-tailed, df 14,11), suggesting significantly different variances.
By convention, placing the larger variance in the numerator ensures F ≥ 1, simplifying the use of F-distribution tables (which typically only list right-tail critical values). This convention means you only need upper critical values. For a two-tailed test, you compare F to Fα/2 rather than checking both tails.
The F-test for equality of variances is highly sensitive to departures from normality — more so than the t-test or ANOVA F-test. Even moderate skewness or heavy tails can inflate the Type I error rate. For non-normal data, Levene's test or the Brown-Forsythe test are more robust alternatives for testing equality of variances.
ANOVA uses an F-test, but the F-statistics serve different purposes. The variance equality F-test compares two sample variances. ANOVA's F-test compares between-group variance to within-group variance. Both use the F-distribution, but they test different hypotheses. The variance F-test is a prerequisite check; ANOVA's F-test is the main analysis.
For comparing two means, the F-test is used as a preliminary test to check the equal variance assumption of the pooled (Student's) t-test. If the F-test indicates unequal variances, Welch's t-test should be used instead. Additionally, for two groups, F = t² — the ANOVA F-statistic equals the square of the two-sample t-statistic.
The two-sample F-test only compares two variances. For comparing variances across three or more groups, use Bartlett's test (assumes normality) or Levene's test (more robust). These generalize the variance equality concept to multiple groups simultaneously.
Power analysis for the F-test depends on the true variance ratio, significance level, and desired power. As a rough guide, each sample should have at least 10-20 observations for reasonable power to detect moderate variance differences (ratio of 2-3). Very unequal sample sizes reduce power and can amplify sensitivity to non-normality.
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