• Say that we have a set of samples from a Monte Carlo experiment
• Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean \( \langle \mathbf{M}\rangle \) is given by

$$ \sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} $$

• If the samples are correlated we can rewrite our results to show that

$$ \sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} $$ where \( \tau \) is the correlation time (the time between a sample and the next uncorrelated sample) and \( \Delta t \) is time between each sample