Calculating the average variance of several normal distributed variables


I have the following problem. I'm trying to average several normal distributed variables. To make it more easy, I'm using z-scores.

The mean of the average is (Mx1 + Mx2 + Mx3)/3, in this case 0.

But how do you calculate the variance?

If I assume they are independent, am I correct when I think it is: Var[y] = (1/3)^2 * (Var[x1] + Var[x2] + Var[x3])

And when they are correlated, the formula is: Var[y] = (1/3)^2 * (Var[x1] + Var[x2] + Var[x3] + 2Cov[x1,x2] + 2Cov[x1,x3] + 2Cov[x2,x3])