Error in average of $x^2$ from error in average of $x$?

Question

Is there an easy way to obtain the error in $\langle{x^2}\rangle$ from $\langle{x}\rangle$ or are they independent?

The values of x are from a molecular simulation application, I obtained a set of values of x but I've used block averaging to make them uncorrelated, giving me a standard error in the mean of x.

Awaiting any advice

Thanks

Answer 1

From the fact that if $z=xy$ then the errors add in quadrature: $$\left(\frac{\sigma_{z}}{\bar{z}}\right)^2=\left(\frac{\sigma_{x}}{\bar{x}}\right)^2+\left(\frac{\sigma_{y}}{\bar{y}}\right)^2$$

Now if $z=x^2$$$\sigma_{x^2}=\sqrt{2\frac{\bar{x}^2}{\sigma_{x}^2}}$$