Let  represent the hypothetical variance and s² the observed variance. 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 (two-tailed test) , i.e.

x_-crit = CHIINV(1−α/2,n−1)               x_+crit = CHIINV(α/2,n−1)

Let δ = /s². Then the beta value for the two-tailed test is given by

β = CHIDIST(δ*x_-crit,n−1)−CHIDIST(δ*x_+crit,n−1)

For the one-tailed test H₀:  ≤ , we use

x_crit= CHIINV(α,n−1)

β = 1−CHIDIST(δ*x_crit,n−1)

For the one-tailed test H₀:  ≥ , we use

x_crit = CHIINV(α,n−1)

β = CHIDIST(x_crit/δ,n−1)

Example 1: Calculate the power for the one-tailed and two-tailed tests from Example 3 of One Sample Variance Testing based on a sample of 50 pipes.

The results are shown in Figure 1: 73.4% for the one-tailed test and 63.9% for the two-tailed test.

Figure 1 – Power of a one sample test of the variance

Real Statistics Functions: The following function is provided in the Real Statistics Resource  Pack:

VAR1_POWER(ratio, n, tails, α) = the power of a one sample chi-square variance test where ratio = /s² (effect size), n = the sample size, tails = # of tails: 1 or 2 (default) and α = alpha (default = .05).

VAR1_SIZE(ratio, 1−β, tails, α) = the minimum sample size required to achieve power of 1−β (default .80) in a one sample chi-square variance test where ratio = /s² (effect size), tails = # of tails: 1 or 2 (default) and α = alpha (default = .05).

For Example 1, VAR1_POWER(E7,E8,1,E10) = 0.733927 and VAR1_POWER(J7,J8,2,J10) = 0.638379, which is the same as the results shown in Figure 1.

Example 2: Calculate the sample size required to find an effect of size .64 with alpha = .05 and power of .80 for the one-tailed and two-tailed tests.

The one tailed test requires a sample size of VAR1_SIZE(.64, .80, 1, .05) = 67 and the two tailed test requires a sample of size VAR1_SIZE(.64) = VAR1_SIZE(.64, .80, 2, .05) = 86.