Let's suppose we're running a simulation of a system and we are interested in the distribution of a quantity (say $M$). One would then run the simulation for a long time and construct a histogram of $M$. The counts in the histogram corresponding to a value of $M$ are $N(M)$. One can denote by $\langle N(M) \rangle$ the ensemble average of $N(M)$, over all simulations of a given length. The statement is that the error in the histogram goes like:
$\delta^2N(M) = (1+2 \tau) \langle N(M) \rangle$
Here $\tau$ is the correlation time of the system, or, of the quantity of interest. My first question is, is $\delta^2N(M)$ the variance or the standard deviation from the ensemble average? Second, how would you prove the above relation?
As an example, one could think of a Monte Carlo simulation of an Ising system, trying to obtain a distribution of the magnetization.