Suppose $\hat{m}=\sum_{i=1}^{N}(x_i)$ where $x_i$'s are samples from one-dimensional normal distribution $N(m,\sigma)$. Is the following steps correct? $Var\left\{(\hat{m}-m)^2\right\} = E\left\{(\hat{m}-m)^4\right\} - E^2\left\{(\hat{m}-m)^2\right\}$ $= 3E^2\left\{(\hat{m}-m)^2\right\} - E^2\left\{(\hat{m}-m)^2\right\}$ $= 2E^2\left\{(\hat{m}-m)^2\right\}$ and I know that $ E\left\{(\hat{m}-m)^2\right\} = \frac{1}{N^2}\sigma$. Then $Var\left\{(\hat{m}-m)^2\right\} = \frac{1}{N^4}\sigma^2$ ---- But the textbook says (without any proving) that $Var\left\{(\hat{m}-m)^2\right\} \tilde{} \frac{1}{N^2} $ Where i am wrong?