Assume $x_i$s are identical and independent random variables. I have a function $$ g= 2 - \frac{\sum_i x_i^2/N}{\left ( \sum_i x_i/N \right)^2} $$ (where $N$ is the number of $x_i$s in the sum) from which I'd like to estimate an error. I define $$ \frac{\sum_i x_i}{N} = <x>$$ and $$ \frac{\sum_i x_i^2}{N} =<x^2>$$ Then the error propagation formula estimates $$\sigma_g^2 \approx \sum_i (\partial g/\partial x_k)^2_{x_i=<x>} \sigma_{x_i}^2 $$ where in the derivative the $x_i$s are evaluated at the estimate mean. However when calculating this I get zero. Also, I am not sure what $g$ would be if doing the same. If I replace all $x_i$s with estimate mean then $g = 2 - N^2<x>^2/(N<x>)^2 = 2 - 1 = 1$.
I am not sure how I should proceed, or if it is possible. Any help is appreciated.