How do you run correlations on mean values? I have five random variables measured on units in four sites, and I'm interested in comparing between sites to determine if there are significant differences. I'm extremely interested in the correlations between these random variables. 
For simplicity, let's say that I took data from NYC, Minsk, San Francisco, and Paris. I polled ten women in those cities and I got data on their weight, height, and hair color (3 out of my 5 variables). I then polled ten men from those cities and got data on their weight, and asked them about their wife's weight. So I have 3 random variables that come from the same person (and thus can be correlated), and two other r.v.s that come from different sources.
I want to force these data into some form that can be correlated between cities.
Can I correlate the means for each city? I know this is frowned upon because you won't be representing the proper variability of the data, but do I have any other option?
 A: Yes, you can correlate the means of men's weights with the means of women's weights across four cities.  (With only four data, you will have very low power, though.)  The interpretation of the correlation will only apply to the means, however—it won't say anything about the rest of the distributions.  You can extend this by getting the means of all of your variables for each of the four cities and forming a correlation matrix.  You would run these correlations the same way you run any other correlation, you just need to get your data set up correctly: put each variable in a column, each city in a row, and the elements would be the means of those variables in that city.  For example:  
city    m.weight   w.weight   m.height   w.height   w.hair.color
NYC     (mean)     ...
Minsk   ...
SF
Paris

If you want to know if there are differences in the sites / cities, a correlation won't do that for you.  If the variables are normally distributed (you couldn't use hair color), you could try a MANOVA.  
