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I'm fairly new to stats, so please excuse me if this problem is hopelessly elementary or misinformed. Basically, I'm wondering if you can help me understand whether I'm using the F-Test for variance correctly.

I have two fairly small populations (n=15), and I'd like to employ a statistical test to determine their respective variances differ significantly. Group A is relatively normally distributed, and group B is skewed left. Originally, I used an F-Test for variance to test for a difference in variance, but then I learned that the F-test can yield false positives if a population isn't normally distributed (as is the case with group B). To remedy this, I created a distribution of sample means for each of the two groups and then performed the F-Test to compare the variance of those distributions. Those distributions of sample means are obviously normal. However, is it appropriate to do an F-Test for variance on those distributions? And how exactly should I interpret the test results?

Just in case it's important, the software I'm using is R and the F-test in R is accessed through var.test()

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    $\begingroup$ Presumably you have two small samples of much larger populations or processes, for otherwise applying a statistical test makes no sense. If one of those samples looks strongly skewed, and you believe that implies the population it represents must be skewed, too, then please pause to ponder why you are comparing variances in the first place. You already believe the population distributions differ (in shape) and the variance has a different interpretation for a skewed population than it does for a symmetrical one. What, then, would the point of testing the variances be? $\endgroup$
    – whuber
    Jul 9, 2014 at 19:22
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    $\begingroup$ How exactly did you create "a distribution of sample means for each of the two groups"? $\endgroup$ Jul 9, 2014 at 19:37

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You started out wanting a test of equality of variances of the raw data, which seems reasonable. Then you switched to considering variances of means. No motivation was provided for doing this, and I suspect this step is not logical.

To your original problem, if a distribution is skewed, one could argue that the variance is not an optimum dispersion measure. What was the motivation for wanting to use variance as the dispersion measure, and what root problem are you trying to solve? If you are ultimately just wanting to know of observations from population A are bigger than those in B, the Wilcoxon test may be in order.

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    $\begingroup$ What do you think would be an optimal measure of dispersion if a distribution is skewed? (I can ask this as a new question, if that would be better.) $\endgroup$ Jul 9, 2014 at 19:35
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    $\begingroup$ @Frank_Harrell Thanks for the help! To back up a bit and give some context for this test, I am analyzing how states' past experience with fossil fuel extraction is related to the stringency of their regulations for hydraulic fracturing (fracking). A scatterplot of this relationship shows that states with less experience have a wider range of regulatory stringency (some states have many regulations; others have very few), while experienced states all seem to have a "medium" number of regulations. I wanted to use the F-test to confirm that the pattern I'm eyeballing is statistically significant. $\endgroup$
    – Josh
    Jul 9, 2014 at 20:34
  • $\begingroup$ I see. You may want to create a customized statistical model for variance or log variance as a function of quantitative past experience. $\endgroup$ Jul 9, 2014 at 22:13

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