F Test for Two Variances Calculator

Enter two arrays of numeric observations and the calculator returns the F statistic, defined as the first sample variance divided by the second, together with a two-sided p-value. The calculation uses unbiased sample variances and the exact F distribution for the corresponding sample sizes. It is deterministic, uses no simulation or network request, and requires at least two observations in each group.

What the two-variance F test measures

The test starts with the null hypothesis that the two underlying population variances are equal. It computes an unbiased sample variance for each group, using a denominator of sample size minus one, and divides the first variance by the second. That ratio is the F statistic. A value near one means the observed spreads are similar, while a value far above one means the first group is more variable and a value far below one means the second is more variable. The ordering matters for the reported statistic: swapping the arrays produces the reciprocal ratio and exchanges the two degrees of freedom. The two-sided p-value, however, evaluates unusually large discrepancies in either direction. It is twice the smaller tail probability under the F distribution, capped at one. A small p-value indicates that a variance ratio at least this extreme would be unusual if both populations truly had the same variance. It does not state the probability that their variances are equal, and it does not describe how important the difference is in practical terms.

How to enter data and interpret the result

Provide each sample as an array containing at least two finite numbers. The arrays may have different lengths, and their observation order does not affect the calculation. Each group is summarized independently with a numerically stable one-pass variance algorithm. The returned F statistic is always the variance of group1 divided by the variance of group2, so retain a consistent group order when comparing results across reports. The p-value is two-sided and already accounts for a discrepancy in either direction; do not double it again. Before examining the data, choose a significance level appropriate to the analysis, such as a threshold specified by a study plan or quality procedure. A p-value below that threshold is evidence against equal population variances under the model assumptions. A larger value means the data do not provide strong evidence of unequal variances, but it does not prove equality. Report sample sizes, sample variances, the group ordering, the F statistic, and the p-value when documenting the analysis, and consider a confidence interval or a robust spread comparison when practical magnitude matters.

Assumptions and important limitations

The classical F test is sensitive to its assumptions. Observations should be independent within and between groups, the samples should represent the populations to which the conclusion is applied, and both underlying populations should be approximately normal. Skewness, heavy tails, or outliers can substantially change sample variances and inflate or suppress the test statistic, especially with small samples. Inspect the data distribution rather than treating the p-value as a standalone verdict. For clearly non-normal measurements, consider a more robust procedure such as Levene's test or the Brown–Forsythe test. The calculator rejects a group with fewer than two values because a sample variance cannot then be estimated. It also rejects nonnumeric or non-finite observations, arrays above the declared size limit, and groups with zero or non-finite sample variance because the ordinary continuous F ratio and its p-value are not well defined for those inputs. Statistical significance does not establish causation, measurement quality, or business importance. Use the result as one diagnostic within a documented analysis, not as an automatic decision detached from sampling design and subject-matter context.