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

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

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But the textbook says (without any proving) that 

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

Where i am wrong?