# 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.

• what is the underlying equation describing your phenomenon? May 11, 2015 at 20:33
• That's what I'm trying to find out. I believe I will have an advection-diffusion equation: df/dt = d/dx(v f) + d/dx(D df/dx). However, I generate my coordinates using something like a simplified Monte Carlo method subject to Lorentz forces. The theory is that this will yield an advection-diffusion equation under certain conditions. That being said: Analyzing the distribution to see if my data matches the expected behavior is a broader question. What I'm hoping to learn here is how to estimate the error of the variance (assuming a 3D Gaussian distribution if necessary). May 11, 2015 at 20:38
• There are some related threads on the site. I don't know if any of these will answer your question, but you may want to take a look: Variance of sample Variance; SD of SD; Why is chi square used to a CI for the variance?; Calculating required N, precision of variance estimate; & What is the distribution of the variance of a sample from an unknown distribution? May 11, 2015 at 21:12