# Propagation of a Standard Deviation

I've got a simulation that tries to calculate the highest energy eigenstate of a Hamiltonian and obtains a value $$\langle A\rangle$$ as its best effort. I do this once each time for a number of different problem instances, say $$i$$ many. For each of these instances, I have the true value $$A_{\max}$$ for each instance and so I can calculate $$r = \langle A \rangle / A_{\max}$$ for each case. I can then calculate the mean of $$r$$ to see how well the simulation does. My question is, how would I report the spread of the $$r_{\text{mean}}$$?

It doesn't represent repeated measurements of the same value, so the standard error $$\sigma/ \sqrt{i}$$ doesn't seem appropriate. But then, neither does just taking the standard deviation of $$r_{\text{mean}}$$. What is the best practise in this case ?