# Power of One Sample Variance Testing

Let $\sigma_0^2$ 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 = $\sigma_0^2$/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: $\sigma^2$ ≤ $\sigma_0^2$, we use

xcrit = CHIINV(α,n−1)

β = 1−CHIDIST(δ*xcrit,n−1)

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

xcrit = CHIINV(α,n−1)

β = CHIDIST(xcrit/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 = $\sigma_0^2$/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 = $\sigma_0^2$/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.

### 12 Responses to Power of One Sample Variance Testing

1. Jonathan Bechtel says:

Hi Charles,

1). The sample size you use for this problem is 50, but the sample size from the problem you reference is 25.

2). At the top of the page you say X-crit = CHIINV(a/2, n-1) and X+crit = CHIINV(1-a/2, n-1) but in your two tailed problem you use the inverse.

3). When you talk about the answers given by var1_power you use var1_power(E7, E8, 1, E10) for the 2 tailed case. I’m guessing you mean var1_power(J7, J8, 2, J10), which does give the correct answer.

4). When I plug in VAR1_SIZE(0.64, 0.80, 1, .05) and VAR1_SIZE(0.64, .80, 2, .05) I get 67 and 86. I’m using Excel 2013 on a Windows.

• Charles says:

Jonathan,
Thanks for carefully looking at this webpage and finding the mistakes identified in your comment. I have now made the necessary corrections on the webpage. I really appreciate your diligence and help in making the website more accurate, and therefore easier to understand.
Charles

• Jonathan Bechtel says:

No problem.

I LOVE this website.

Studying its contents has now become my sacred daily ritual and acquiring a higher understanding of mathematics is incredibly gratifying.

There is no way I’d ever be able to learn what I’m learning without this site.

• Charles says:

Jonathan,
I am please to hear this. I am glad that you are learning a lot from the site and I appreciate your diligence in finding errors to help improve the site.
Charles

2. Alex Bartic says:

Hi,

Why for the second one-tailed test, at the call of CHIDIST, did you divided by delta instead of multiply?

• Charles says:

Alex,
It is just the way the algebra works out.
Charles

3. Javier says:

Hi. Already using the RealStats pack to do some underrated but quite interesting statistical power calculations.

When I plug VAR1_POWER(0.64,50,1,0.05) I get 0,677983 and when plugging VAR1_POWER(0.64,50,2,0.05) I get 0,539065. The VAR1_SIZE values I get are the same than in the example. I’m using Excel 2016.

Could this be a matter of some iteration or precision settings somewhere in Excel I need to adjust?

• Charles says:

Javier,
Sorry, but I don’t understand what you mean “the VAR1_SIZE values I get are the same than in the example”. Please explain.
Charles

• Daniel says:

Hi Charles,

I love the website and the add in. I tend to recommend it to people whenever the opportunity arises.

I had exactly the same problem that Javier had, and found that I could get the correct outputs when I used 1/δ as my first argument. So it seems like the issue is based around which variance is taken to be the hypothetical. However, Example 1 as illustrated, is still producing 0.677983 and 0.539065 for one and two-tailed tests respectively (at least in my experience).

Thanks.

• Charles says:

Daniel,
I thought that this was already fixed in Rel 4.14, but if not I will make sure that it is fixed in the next release (Rel 4.15). I am finishing up the testing for this release and it should be available soon.
Charles

• Daniel says:

I just double checked the example using a fresh copy of your add-in. I can confirm that my results are still as described above.

As always, thank you for your generous work.

• Charles says:

Daniel,
I will make sure that this change is made in Rel 4.15.
Charles