$\operatorname{Var}(X^2)$, if $\operatorname{Var}(X)=\sigma^2$ What would be $\operatorname{Var}(X^2)$, if $\operatorname{Var}(X)=\sigma^2$?
 A: Error propagation via Taylor's rule (aka "delta" method) --
$$\operatorname{Var}(X^2) \approx 4\operatorname{\mathbb{E}}(X)^2 \operatorname{Var}(X)$$
A: As a simple example of the responses of @user2168 and @mpiktas:
The variance of the set of values 1,2,3 is 0.67, while the variance of its square is 10.89. On the other hand, the variance of 2,3,4 is also 0.67, but the variance of the squares is 24.22.
These are just variances for finite sets of data, but the idea extends to distributions.
A: It's easy to see that the relationship between then is not constant by taking $X'=X+c$. Shifting a distribution by a constant doesn't affect the variance, but $Var((X+c)^2)$ can be made arbitrarily large. $Var(X^2)$ is a fourth-order statistic (i.e. is a combination of moments of order four and smaller), and cannot be written in terms of lower order statistics such as variance and mean.
A: Error propagation via Taylor's rule (aka "delta" method) --
$$\operatorname{Var}(X^2) \approx  4\operatorname{\mathbb{E}}(X)^2 \operatorname{Var}(X) - \operatorname{Var}(X)^2 $$
Sorry i have expanded the taylor's rule in one extra order, because to just approximate the $\operatorname{Var}(X)$ linearly caused some problem with my algorithm, thought it would help other people to realize it's not linear...
