How to test statistical significance of standard deviation? I have two lists of values A and B. I calculated the standard deviation of A and the standard deviation of B. The standard deviation of A is bigger than that of B. How can I test if the difference in the standard deviation is significant? I understand that t-tests are used to test differences in mean but how can we do the same for standard deviation or for variation coefficient? Can you give some examples of tests to check the significance of stdv differences between two samples?

The first test one learns for this is an F-test of variances, discussed in this nice video. While that explicitly tests for variance equality, variance equality is the same as standard deviation equality, so this will test for both.
A caveat is that, while the t-test is fairly robust to deviations from the assumption of normal distributions, the F-test is not. My question here mentions some alternatives to the F-test and also discusses some of their issues.
An additional point is that F-testing is more famous for its use in ANOVA, which is a comparisons of group means, not if group variances. The trick is that the F-test is such a case cleverly uses a comparison of two variances to say something about means.
