# Power of Two Sample Variance Testing

Let $s_1^2$ and $s_2^2$ represent the variances of two independent samples of size n1 and n2. Let x+crit be the right critical value (based on the null hypothesis with significance level α/2) and x-crit be the left critical value, i.e.

Let δ = $s_1^2$/$s_2^2$. Then the beta value for the two-tailed test is given by

For the one-tailed test H0: $\sigma^2$ ≤ $\sigma_0^2$, we use

For the one-tailed test H0: $\sigma^2$ ≥ $\sigma_0^2$, we use

Example 1: Calculate the power for the one-tailed two-sample variance test where α = .05, the sizes of the two samples are 50 and 60 and the corresponding variances are 2.25 and 1.75.

The power of the test is 22.8% as shown in Figure 1.

Figure 1 – Power of two-sample variance testing

Example 2: Calculate the power for the two-tailed two-sample variance test where α = .05, the sizes of the two samples are 50 and 60 and the corresponding standard deviations are 1.5 and 1.2.

Note that this time we are given the standard deviations, and so we must square these values to find the ratio of the variances. The power of this test is 35.9% as shown in Figure 2.

Figure 2 – Power of two-sample standard deviation testing

Real Statistics Functions: The following functions are provided in the Real Statistics Resource Pack:

VAR2_POWER(ratio, n1, n2, tails, α) = the power of a one sample variance test where ratio =$\sigma_1^2$/$\sigma_2^2$ (effect size), n1 = size of sample 1, n2 = size of sample 2, tails = # of tails: 1 or 2 (default) and α = alpha (default = .05).

VAR2_SIZE(ratio, 1−β, tails, α, nratio) = the minimum sample size required to achieve power of 1−β (default .80) in a one sample variance test where ratio = $\sigma_1^2$/$\sigma_2^2$ (effect size), tails = # of tails: 1 or 2 (default) and α = alpha (default = .05).

For VAR2_SIZE, only the size of the first sample is returned. If the two samples don’t have the same size, you need to specify the size of the second sample in terms of the size of first sample using the nratio argument. E.g. if the size of the second sample is half of the first, then set nratio = .5.

If you set nratio to be a negative number then the absolute value of this number will be used as the sample size of the second sample. E.g. if nratio = -50, then the VAR2_SIZE function will find the size of the first sample assuming that the second sample has 50 elements.

For Example 1, =VAR2_POWER(E7,E8,E9,1,E12) = 0.228036, which is the same as the results shown in Figure 1. For Example 2 =VAR2_POWER(J7,J8,J9) = 0.35862, which is the same as the results shown in Figure 2.

Example 3: Calculate the sample size required to obtain a power of 80% for the test in Example 2, assuming both samples have the same size.

The sample size is 160, as calculated by the formula =VAR2_SIZE(J7).