Aggregate Mean and Std Dev

The aggregate mean, E[X], is simple: dot product of sample mean and sample size arrays divided by sum of sample size array.

Formula:

=SUMPRODUCT(B2:B26,C2:C26)/B28

Aggregate standard deviation is more involved. Assuming each sample's variance were based on dividing by the sample's N-1 rather than N, I believe it should be

SD[X_s]² = Var[X_s] = sum((X_si - E[X_s])²) / (N_s - 1)

(N_s - 1) SD[X_s]² = sum((X_si - E[X_s])²) = sum(X_si²) - N_s E[X_s]²

sum(X_si²) = (N_s - 1) SD[X_s]² + N_s E[X_s]²

Var[X] = sum_s(sum(X_si²)) / sum_s(N_s) - E[X]²

Formula:

=SQRT((SUMPRODUCT(B2:B26-1,D2:D26^2)+SUMPRODUCT(B2:B26,C2:C26^2))/B28-C28^2)