About

In statistical hypothesis testing, comparing the variability of two distinct populations is a fundamental task. The F-Test Calculator allows researchers, students, and quality control analysts to strictly evaluate whether two samples come from populations with equal variances (homoscedasticity).

This tool uses the Fisher-Snedecor distribution to calculate the F-statistic and the corresponding p-value. It is critical for checking assumptions before proceeding to t-tests or ANOVA, ensuring that the subsequent analysis is statistically valid. Whether you are analyzing manufacturing tolerances or biological diversity, knowing if variances differ significantly is the first step in rigorous data analysis.

Formulas

The F-test statistic is calculated by taking the ratio of the two sample variances. By convention, the larger variance is usually placed in the numerator to calculate a right-tailed probability.

F = s₁²s₂²

The Degrees of Freedom (df) for each sample are calculated as:

df₁ = n₁ − 1

df₂ = n₂ − 1

Reference Data

Scenario	Variance A (s²₁)	Variance B (s²₂)	F-Value	Interpretation
Identical Variability	10.5	10.5	1.00	Perfect equality. Null hypothesis likely accepted.
High Divergence	50.0	5.0	10.00	Significant difference. Null hypothesis likely rejected.
Sample Size Impact	12.0 (n=5)	4.0 (n=5)	3.00	Critical value is high due to low degrees of freedom.
Sample Size Impact	12.0 (n=100)	4.0 (n=100)	3.00	Critical value is low; result is highly significant.
Inverse Ratio	4.0	12.0	0.33	F-tests typically place larger variance in numerator.

Frequently Asked Questions

Use an F-Test when you want to compare the variances (spread) of two different populations to see if they are significantly different. It is often used as a prerequisite check before performing a two-sample t-test assuming equal variances.

Yes, significantly. Larger sample sizes increase the degrees of freedom, making the test more sensitive to small differences in variance. With small samples, the variances must differ drastically to trigger statistical significance.

The p-value indicates the probability of observing an F-ratio as extreme as the one calculated, assuming the null hypothesis (equal variances) is true. If the p-value is lower than your alpha level (e.g., 0.05), you reject the null hypothesis.

Yes, the F-test for equality of variances is extremely sensitive to non-normality. If your data is not normally distributed, consider using Levene's test or the Brown-Forsythe test instead.