# Comparing standard deviations of two dependent samples

As far as I know you can do a F-test ($F = s_1^2/s_2^2$) or a chi-squared test ($\chi^2 = (n-1)(s_1^2/s_2^2$) for testing if the standard deviations of two independent samples are different. But does this also hold for dependent samples?

• In some situations an F test or $\chi^2$ test will work as expected and in others they won't, depending on how the data are assumed to depart from independence. In what way, precisely, do you suppose your two samples are dependent?
– whuber
Feb 20, 2012 at 14:59
• Have you checked the Morgan-Pitman-Test? I rarely see it mentioned, and I have no information on its strength and weaknesses. Still, it seems to be a test for the equality of variances in two dependent groups. Feb 20, 2012 at 15:09

The Morgan-Pitman test is the clasisical way of testing for equal variance of two dependent groups.

For $n$ pairs of randomly sampled observations

$(X_{11}, X_{12}),...,(X_{n1},X_{n2})$

define

$U_i = X_{i1}-X_{i2}$

and

$V_i = X_{i1}+X_{i2}$

for $(i=1,...,n)$.

Then under

$H_0: \sigma_1^2=\sigma_2^2$

the correlation of $U$ and $V$ is zero.

If the distributions of the two variables differ in shape then you should use a robust method of testing the hypothesis of $\rho_{uv}=0$.

A good description is in Wilcox's Modern Statistics for the Social and Behavioral Sciences (Chapman & Hall 2012), including alternative ways of comparing robust measures of scale rather than just comparing the variance.

• This misses the important assumption of bivariate normality of $X_1$ and $X_2$. Dec 11, 2022 at 21:26