Estimate variance of sub-sets from overall variance

Question

I am looking for a way to estimate the variance of a summed sub-set based on the variance of those sums.

Si = sum( Ai )
S = { S0...Sn }
V = variance( S )

That is, each sample value Si is actually the sum of a set Ai. V is then the normal variance of the set S.

Is there any way to estimate the variance of the values in sets Ai using the final variance, or some result from the above?

Currently I do variance / (len(Ai)^2) but the value is then an averaged value of the variance. I'm wondering if there is a way to improve upon this.

---

Background: The reason the I don't calculate variance on individual samples from the population is that the measurement is then too small. I am measuring elapsed time, and so individual samples the elapses time is too small for the timing device to measure, so instead I do a series of operations and calculation the time for the whole group.

Note: In the application I have no ability to store samples and am using an online algorithm (Knuth).

Answer 1

Making the strong assumptions that all the various $A_{i,j}$ are identically and independently distributed, and if there are the same number in each sum $S_j$, then since $$Var(S_j) = Var\left(\sum_i A_{i,j}\right) = \sum_i Var(A_{i,j})$$ your method is reasonable.

But this does depend on what is really happening: if the $A_{i,j}$s are correlated, or if their population means or variances vary with $j$, or if there are different numbers in each sum, then you will need another approach.