# Error of the variance

I have a collection of $(x,y,z)$ data points. I want to compute the mean, $\mu$, and variance, $\sigma^2$, along each axis, as well as the errors on each. I know that the standard error of the mean is $\Delta \mu = \sigma / \sqrt{N}$, but I'm not familiar with any methods to estimate the error of the variance, $\Delta (\sigma^2)$. Is there a method to estimate the error of the variance?

In case it makes a difference: I expect the data to be a 3D Gaussian, but I expect the variance to be different along different axes (i.e. $\sigma_x^2 \ne \sigma_y^2 \ne \sigma_z^2$).

The reason I'm looking for this is that the coordinates are the positions of particles that are expected to obey an advection-diffusion law, the means are related to the advection velocity, and the variances are related to the diffusion coefficients.