What does R mean in the two-variance test?

R = σ1/σ2 is the ratio between the true standard deviations of the two groups under the alternative hypothesis. R=1.5 indicates that group 1 has 50% more spread than group 2.

What distribution does the F-test of variances use?

The statistic F = S1²/S2² follows an F(n-1, n-1) distribution under H0. Under the alternative R≠1, F/R² follows a central F(n-1, n-1), which allows the exact power to be computed without noncentral distributions.

Why does it assume the same size in both groups?

The calculator assumes equal groups by symmetry (n1=n2=n), which is the most efficient design for this two-tailed test. The result n is the size of each group.

What assumptions does the F-test of variances require?

Normality in both populations and independence of the samples. The F-test of variances is especially sensitive to non-normality; even small deviations can inflate the type I error rate.

Sample size for comparing two variances calculator

Calculator

Enter the ratio R = σ₁/σ₂, the significance level and the desired power. The same size is assumed in both groups (n₁ = n₂ = n).

Ratio R = σ₁/σ₂

Alpha (α)

Power (1-β)

Result pending…

Explanation

The Snedecor-Fisher F-test tests the equality of variances of two independent normal populations using the statistic \(F = S_1^2/S_2^2\), which under \(H_0\) follows an \(F(n_1-1, n_2-1)\) distribution.

Under the alternative \(\sigma_1 = R \cdot \sigma_2\) with \(R \neq 1\), we have \(F/R^2 \sim F(n_1-1, n_2-1)\), which allows the exact power to be computed without needing noncentral F distributions. The calculator searches for the smallest \(n\) (per group) so that the exact two-tailed power reaches the target.

Exact power formula

For the two-tailed test \(H_a: \sigma_1 \neq \sigma_2\) with equal-sized groups \(n\):

\( \text{Power} = F_{F(df,df)}\!\left(\frac{F_{\alpha/2,\,df,\,df}}{R^2}\right) + 1 - F_{F(df,df)}\!\left(\frac{F_{1-\alpha/2,\,df,\,df}}{R^2}\right) \)

where \(df = n-1\) and \(F_{p,df,df}\) is the \(p\) quantile of the \(F(df, df)\) distribution. The calculator searches for the smallest \(n\) that satisfies the target power.

R > 1: σ₁ > σ₂ (group 1 is more variable).
R < 1: equivalent to R'=1/R with the groups swapped; the power is the same by symmetry.

Quick settings

R: always use R > 1 (or its equivalent inverse by symmetry). R=1.5 is a moderate effect; R=2 is already large.
α: 0.05 is the standard; 0.01 for critical decisions about variability.
Power: 0.80 as a minimum; 0.90 for validation studies.
Normality: the F-test of variances is very sensitive to non-normality. If the data are not normal, consider the Levene or Bartlett test.

Worked example

A study compares the variability of reaction times between a group with and without pharmacological treatment. The researcher considers it relevant to detect differences of 50% or more in the standard deviation (R = 1.5), with \(\alpha = 0.05\) and 80% power.

The calculator returns approximately n = 80 per group (160 in total). The F-test with 79 degrees of freedom in each group has an 80% probability of rejecting \(H_0\) when the true ratio of standard deviations is ≥ 1.5.

Sensitivity analysis: with R = 2 (fourfold variance) the sample shrinks to n ≈ 22 per group. For R = 1.25, n ≈ 283 per group is needed. The sample size is highly sensitive to R in the region R ∈ (1, 2).

Model assumptions

Two random samples, independent of each other.
Both populations are normal: the \(F\) test is sensitive to departures from normality.
Observations within each group are independent.
The sample size is derived from the exact \(F\) distribution for the variance ratio.

How to interpret the result

The value \(n\) is the minimum size of each group in a balanced design to test whether two population variances differ using the Snedecor F-test. The total study size is \(2n\), and the numerator and denominator degrees of freedom are each \(n - 1\). Always round up; one fewer observation can reduce the degrees of freedom right at the critical margin. If you anticipate dropout or rejections, divide each \(n\) by \((1 - \text{dropout rate})\) to get the actual recruitment target per group.

The effect of interest is the variance ratio \(\lambda = \sigma_1^2 / \sigma_2^2\) (or its inverse if \(\lambda < 1\)). The design's sensitivity to the specification of \(\lambda\) is high: if the true ratio turns out to be closer to 1 than assumed, the test will have less power than planned. Run a sensitivity analysis by computing \(n\) for different values of \(\lambda\) within a reasonable range. It is important to keep in mind that the F-test is extraordinarily sensitive to non-normality: even moderate deviations from normality can seriously inflate the type I error rate, making the p-value unreliable. If the data are not clearly normal, consider alternative tests such as Levene or Bartlett (more robust to non-normality) and adjust \(n\) accordingly.

When the calculated \(n\) is very large, it usually indicates that the specified effect size (\(\lambda\) close to 1) is too small to be detected efficiently with a variance-ratio test; consider whether the difference between variances has practical relevance. If the goal is to estimate the variance ratio with a given precision rather than test it, use the sample size calculator for the variance CI. Once the data has been collected, analyze the results with the F-test for two variances.

References

Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.
Montgomery, D. C. & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.