Central Limit Theorem Equation

I was wondering what the difference is between $Z_n=\frac{S_n-n\mu}{\sigma \sqrt{n}}=\frac{X_1+\ldots+X_n -n\mu}{\sigma \sqrt{n}}$ and $Z_n=\frac{X-n\mu}{\sigma/\sqrt{n}}$?

What is the benefit of one over the other?

• I am not sure you have written a true equation? \begin{align} Z_n &=\frac{X_1+\ldots+X_n -n\mu}{\sigma \sqrt{n}}\\ & =\frac{n(\frac{X_1+\ldots+X_n}{n}-\mu)}{\sigma \sqrt{n}}\\ &= \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \end{align}. If you just have one X then why you need to normalize under $n\mu$? – TPArrow Jul 29 '15 at 6:25
• Do you mean $\bar{X}$ rather than $X$ in your second one? – Glen_b -Reinstate Monica Jul 29 '15 at 10:01

$$Z=\frac{\sum_{i=1}^n X_i-n\mu}{\sqrt{n}\sigma}=\sqrt{n}\frac{\bar{X}-\mu}{\sigma} \xrightarrow{D} \mathcal{N}\left(0,1 \right)$$