The central limit theorem can be stated as follow:
$$\overline{X}\overset{d}{\rightarrow}N(\mu,\frac{\sigma^2}{n})\quad\text{as}\quad n\to\infty\\ Z=\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\overset{d}{\rightarrow}N(0,1)\quad\text{as}\quad n\to\infty $$
However, I also spot a statement from the text
Arak M. Mathai, Hans J. Haubold-Probability and Statistics:
The importance of this limit theorem is that whatever be the population, whether discrete or continuous, the standardized sample mean will go to a standard normal when $n\to\infty$ and when the population variance is finite. Thus the normal or Gaussian distribution becomes a very important distribution in statistical analysis. Misuses come from interpreting this limit theorem in terms of $\bar{x}-\mu$ or $\bar{x}$. This limit theorem does not imply that $\bar{x}-\mu\sim N(0,\sigma^2/n)$ for large n. It does not imply that $\bar{x}\sim N(\mu,\sigma^2/n)$ for large n
The textbook only admits the standardized form
I am wondering which one is more accurate, and why does the second graph say so?
