I'm doing a Monte Carlo calculation and use the standard deviation of the mean $\sigma_M$ as the error. To get an estimate of this from the regular standard deviation I use
$$\sigma_M=\dfrac\sigma {\sqrt{n-1}} \ ,$$
where $n$ is the number of samples.
But then to try to confirm the result I find the "actual" standard deviation of the mean, let's call it $\widetilde{\sigma}_M$, by doing the whole calculation a lot of times and taking the standard deviation of the results.
I find that my estimate is systematically too small, usually around $\sigma_M\sim0.6\ \widetilde{\sigma}_M$. I've tried using bootstrap estimation to find $\sigma_M$, in case the samples are not independent, with pretty much the same result.
One solution would be to just multiply my estimate with $2$ for example. The problem is I would have to double check this factor every time I change any parameter, so it is better to have a reliable estimate from the get-go.
Any ideas how to deal with this? Other sources of the discrepancy than dependent samples? Should I be happy with $0.6\ \widetilde{\sigma}_M$ and move on?