Estimation the standard error of correlated (binomial) variables Say we have N voters from some group (N binomial variables).  Each voter has a preference on voting about some issue (probability of "success" p), which depends on the preferences of other voters.
Given a sample of votes for each voter (an estimation of their preference on that issue), and an estimation of the correlation matrix between each two voters.
Can we give a better estimation of the standard error of the average of the proportions (than the one offered by the standard $\sqrt{\frac{p(1-p)}{n}}$?
 A: I wouldn't use the term better, I would say correct the formula $p(1-p)/n$ applies under independence and is not correct if the sequence of observations is dependent.  The title of the question makes it sound like you want the formula for the variance when the observations have a given correlation $r$ between successive observations. But the scenario you pose is a problem where you haven't specified that each voter has the same p and the same correlation $r$ with every other voter.  So the formula could be more complicated if you have more than two parameters.  But let us take the simplest case which is probably what you intended.
The formula for the variance of the estimate (number of yes votes/number of voters) where $X_i$ is the $i$th voters vote) 1 for yes and 0 for no) is
\begin{align*}
&\sum_i \text{Var}(X_i)/n^2 +  2\sum\sum_{i\leq j} \text{Cov}(X_i, X_j)/n^2\\ 
&=p(1-p)/n  + 2 r [n(n+1)/2] p (1-p)/n^2\\
& = p(1-p)/n  + [r (n+1)/n ]p(1-p)\\
& = [p(1-p)/n] [1+r(n+1)].
\end{align*}
