Suppose $\hat{m} = \sum_{i=1}^{N}(X_i)$ where $X_i \sim N(m,\sigma)$.

Are 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

$Var\left\{(\hat{m}-m)^2\right\} = \frac{1}{N^4}\sigma^2$

However the textbook says (without any proving) that

$Var\left\{(\hat{m}-m)^2\right\} \tilde{} \frac{1}{N^2} $

Where am I going wrong?