Variance of mean of correlated variables

Question

Page 228 of THIS BOOK provides the formula for the variance of the mean of more than two correlated random variables:

where $m$ is the number of variables, $r$ is the correlation between the variables, and $V$ is the variance of each of the variables.

The same book, however, provides a different formula for the variance of the mean of two correlated variables:

The formula formula for more than two variables doesn't seem to be an extension of the formula for two variables. Specifically a $2$ is in the two-variable formula that is absent in the more-than-two-variable formula.

Is this by design?

Answer 1

The formula for $m>2$ is a generalization of the other formula:

When $m=2$: $$ \left(\frac{1}{m}\right)^2 = \frac{1}{4}, $$

The sum of $V_i$ equals $V_1 + V_2$,

And for the last summation,
$$ r_{12} \cdot \sqrt{V_1} \cdot \sqrt{V_2} + r_{21} \cdot \sqrt{V_2} \cdot \sqrt{V_1} = 2r \cdot \sqrt{V_1} \cdot \sqrt{V_2} $$

Here's an R code for computing this sum:

myVariances <- c(0.25,0.5,0.75) # this is a vector of the variances

myCorrelations <- matrix(data = c(1,0.1,0.2,0.1,1,0.3,0.2,0.3,1), nrow = 3, ncol = 3) # this is the matrix of correlations

mySum <- 0 # initializes mySum to zero

for (i in 1:nrow(myCorrelations)) {
  for (j in 1:nrow(myCorrelations)) {
    mySum <- mySum + myCorrelations[i,j] * sqrt(myVariances[[i]]) * sqrt(myVariances[[j]])
  }
} # this loop computes the sum

(1/nrow(myCorrelations))^2 * mySum # this multiplies that sum by (1/m)^2

The above code assumes that your matrix of correlations includes 1's on the diagonal, to represent that the variables are perfectly correlated with themselves.