The CLT(central limit theorem) states $$ \lim_{n\rightarrow\infty}\frac{\bar{x}-\mu}{\sigma/\sqrt{n}} $$ conforms standard normal distribution $N(0,1)$, i.e. $\bar{x}$ coincides with $N(\mu,\sigma/\sqrt{n})$ as $n\rightarrow\infty$.
Why I could not say that $n\bar{x}$ will have maximum probability at $n\mu$ which equals to $$NormPDF_{(n\mu, \sigma/\sqrt{n})}(n\mu)$$
(NormPDF is normal distribution probability density function)
Since $$\sum_{k=1}^{n}x_k =n\bar{x} $$But actually $n\bar{x}$ conforms $N(\mu,\sqrt{n}\sigma)$ which is a more flat bell curve and $$NormPDF_{(n\mu, \sqrt{n}\sigma)}(n\mu)< NormPDF_{(n\mu, \sigma/\sqrt{n})}(n\mu)$$So why just scaling the random variable will have effect to its probability or probability density function?