If noise is your signal, what is your noise?
There is a derivation over on the math.SE site that might be useful.

Provided your data is Gaussian with variance $\sigma^2$.

If you use the unbiased estimator of the population variance:
$$
S^2_{N-1} = \frac{1}{N-1} \sum_{n=0}^{N-1} (x[n]- \bar{x})^2
$$
where $N$ is the number of points in your batch and $\bar{x}$ is the sample mean, then the variance of this estimate will be:
$$
\frac{2 \sigma^4}{N-1}
$$