In the definition of Wigner Matrix (a certain type of random Matrices) we take to independent family of i.i.d zero mean distributions $\{Z_{i,j}\}_{1<i<j}$ and $\{Y_{i}\}_{1\leq i}$ and then the entries of Wigner Matrix $X$ define as below:

$$\forall i,j; 1\leq i<j\leq N : X_N(i,j) = Z_{i,j}/\sqrt{n}$$ $$\forall i,j; 1\leq j<i\leq N : X_N(i,j) = Z_{j,i}/\sqrt{n}$$ $$\forall i; 1\leq i \leq N : : X_N(i,i) = Y_{i}/\sqrt{n}$$

After this task we will have a self-adjoint matrix that its eigenvalues are all real and we have Wigner's Semicircle Law about it.

My question is this: Is there any intuition or short proof that we should normalize the entries with $\frac{1}{\sqrt{N}}$ for controlling eigenvalues from being large.