The wiki article:
Correlation and dependence - Wikipedia, the free encyclopedia
gives a definition in terms of covariance and individual random variable variance.
Now if you have these terms you need to include the covariance terms when you are finding the variances of correlated random variables. There is an inductive definition of this, but basically it involves calculation covariance terms for all the appropriate pairs of covariance terms.
Since you are able to get a definition for the covariance of any two random variables for a given rho (that greek p), if you know the standard deviations of each random variable you can get the covariance term from these two and then calculate the variance of the sample mean of all observations taking into account all covariance terms to get the variance of the sum of all the random variables, and then divide by n^2 to get the variance of the sample mean.