How do error bars change for variables squared?

Question

Let's say I have some variable $\mu$ with an uncertainty estimate:

$$\mu = 2 \pm .5$$

Let's say I have another variable $\nu = \mu^2$. Is the uncertainty estimate in $\nu$ equal to the the uncertainty in $\mu$ squared, such that

$$\nu = 4 \pm .25$$

This does not seem to be right to me. What would be the appropriate way of getting the uncertainty in $\nu$?

Answer 1

Uncertainty can be written for a quantity :

$$ X=Y \pm u(Y) $$

Then taking the square you have

$$ (Y\pm u(Y))^2=Y^2\pm 2Yu(Y)+u(Y)^2$$

Or a the first order for maybe more sense

$$ (Y\pm u(Y))^2=Y^2\pm 2Yu(Y)$$ It is ok for you ?