I have measured a large data sample from an underlying Gaussian distribution and want to estimate the variance. However, the measured values are noisy with some Gaussian noise with a standard deviation that is approximately known. How can I estimate the error of the sample variance in this case? First I computed the mean squared error $$\frac{2}{n-1} \sigma^4$$
which does not take into account the noise and seems to be too small.

What I also do not understand is, how one can compute the `MSE` if one does not know the true distribution, thus don't know the true $ \sigma$.