Variance Sum Law

The variance sum law is an expression for the variance of the sum of two variables. If the variables are independent and therefore Pearson's r = 0, the following formula represents the variance of the sum and difference of the variables X and Y:

Note that you add the variances for both X + Y and X - Y.

If X and Y are correlated, then the follownig formula (which the former is a special case) should be used:

where ρ is the population value of the correlation.
In a sample r is used as an estimate of ρ.