Statistical significance test for averages of correlation coefficients My setting looks like that I have several subjects $i$ (let's say 5) and for each subject I measure the lists/vectors $A_i$, $B_i$, $C_i$ and $D_i$. Then, for each subject $i$, I calculate the correlation between $A_i$ and $B_i$ (let me call this correlation $AB_i$) as well as between $C_i$ and $D_i$ ($CD_i$). Now, I have for each subject $i$ one correlation coefficient $AB$ and $CD$. Thus, for e.g., five subjects, I have five $AB_i$ correlation coefficients such as $AB'=[AB_1=-0.1, AB_2=0.2, AB_3=0.25, AB_4=0.3, AB_5=0.2]$ and five $CD_i$ coefficients such as $CD'=[CD_1 = 0.8, CD_2=0.8, CD_3=0.75, CD_4=0.9, CD_5=0.7]$.
Now, I want to test the null hypothesis that the correlation coefficients $AB'$ are similar to $CD'$. So basically, whether the mean of $AB'$ is the same as the mean of $CD'$.
Note that $A_i$, $B_i$, $C_i$ and $D_i$ are not repeated measures and focus on different things. Actually, $A_i$ and $C_i$ as well as $B_i$ and $D_i$ correspond to the same variable being measured, but for different sub-groups of my subjects. So, a subject can correspond to a specific event where I can distinguish between sub-groups. E.g., suppose that you study five different school classes (subjects) and for each class you distinguish between males and females (sub-groups). Then you measure two variables (vectors) for both males ($A_i$, $B_i$) and females ($C_i$, $D_i$). Now, I want to know whether $A_i$ correlates similarly to $B_i$ as $C_i$ correlates to $D_i$ across all subjects (school classes).
I know how to calculate a statistical significance test regarding the differences between single correlation coefficients for individual subjects. There are several approaches available; one is to do a Fisher z transformation and then determine the z statistic with known standard deviation. However, how can I do this across subjects (for the mean) in one step?
Normally, I would just pick a t-test. However, the issue I see here is that the correlation coefficients are not normally distributed which is why we could again work with the Fisher transformation somehow. I am unsure how though.
Maybe average the transformed correlations and then conduct a z-test? 
I have studied some similar questions withouth finding any answer to this problem. Hope someone has an idea of how to approach this. 
 A: Short answer: Yes, transform each correlation ($AB_i$ and $CD_i$) using the Fisher $r-to-z'$ transform:
$f(r)=\frac12 ln \frac{1+r}{1-r}$.
  Then, perform an independent samples t-test to test the null hypothesis of  $\mu_{f(AB)}=\mu_{f(CD)}$.
Rationale: You're right that approximate normality is important here.  With only a small number of subjects, you can't count on the central limit theorem to address the (often) non-normal sampling distribution of r.  The sampling distribution of r will only be approximately normal when $\rho$ is close to 0 or when n is very large (here, I refer to the n used to compute the correlation, not the n indexed by i in your question).  Based on the examples in your question, I'm guessing n is modest, and if you already knew the $\rho$'s, there'd be no point in asking your question. Bottom line: the Fisher $r-to-z'$ will probably help here. 
It sounds like you're comparing independent samples, especially based on this part of your description:

$A_i$ and $C_i$ as well as $B_i$ and $D_i$ correspond to the same variable being measured, but for different sub-groups of my subjects.

So, it makes sense to use an independent samples t-test.  
