Sampling For Correlation Suppose we have population P that is split into large number of close size sub-populations (e.g. country is split to zips).
For each sub-population a small sample was drawn and average of a variable X was calculated. 
The researcher has only access to the average values and he/she needs to study a correlation between X and a random variable Y that researcher has the full access to the population data.
What is the correct way to organize the correlation calculation?
Concern #1 is that the researcher doesn't have access to the individual instances used for the sampling of Y. So if he/she take a ratio between the mean of X and mean of Y it might not be the same as the mean of the ratio X/Y. 
Concern #2 is the small size of the X samples. However, since the number of sub-populations is large the same forces that lead to the CLT will be applied to the correlation.
 A: This isn't an answer, but a proof that, as @whuber says, the effort is doomed without more information.
The correlation is defined as:
\begin{equation}
 corr(X,Y) = \frac{cov(X,Y)}{\sqrt{var(X)var(Y)}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (1)
\end{equation}
$var(Y)$ is known, so it's only necessary to estimate $var(X)$ and $cov(X,Y)$. 
Let $Z$ index sub-populations. The following equalities are proved in introductory stat theory classes.
\begin{equation}
 var(X) = E_Z(var(X|Z)) + var_Z(E(X|Z)  \quad\quad\quad\quad\quad\quad\quad\quad  (2)
\end{equation}
\begin{equation}
cov(X,Y) = E_Z(cov(X,Y)|Z) + cov_Z(E(X|Z), E(Y|Z))  \quad\quad\quad (3)
\end{equation}
Here $E_Z$ means that expectation is taken over the distribution of sub-populations, and a similar definition applies to $var_Z$ and $cov_Z$.  The $|$ indicates a within-sub-population distribution; for example,  $E(X|Z)$ is the mean of X in sub=population Z.
The second term in Eqn 2 and in Eqn 3 depends on the sub-population means of $X$; so both terms can be estimated. However the first term in each equation depends on the within-sub-population distribution of X (Eqn 2) and (X,Y) (Eqn 3), so neither term can be estimated from the available data.
