Central Limit Theorem (Sum Version)
Let $X$ be a population random variable with finite mean $\mu_X$ and finite variance $\sigma^2_X$ and let $(X)_{i=1}^n$ be a random sample of $X$. Let $S= \displaystyle\sum_{i=1}^n X_i.$ Then for large sample sizes $n$ the distribution of $S$ is approximately normally distributed with mean $\mu_S = n \cdot \mu_X$ and variance $\sigma_S^2= n \cdot \sigma_X^2.$
What this part of the theorem says is this, if we take $n$ sample sizes from the population, the distribution is approximately normally distributed. Another words as $n$ grows larger, it converges to the normal distribution. The theorem is best used when you are taking $n>30$ sample size from the population. Since the sample size is approximately normally distributed, we can standardize the random variables so that we can estimate what percentage of values of the sample mean fall within some given interval. For example, if I have a population of 100,000 and I take a survey of 30 people to see what they think about math. Than we have $(X_i)_{i=1}^{30}$ where each $X_1$ represents the outcome of the 1st person, $X_2$ the second, $X_{30}$ the 30th person etc.
Central Limit Theorem (Sample Mean Version)
Let $X$ be a population random variable with finite mean $\mu_X$ and finite variance $\sigma_X^2$ and let $(X)_{i=1}^n$ be a random sample of $X$. Let $\bar{X}=\dfrac{1}{n}\displaystyle\sum_{i=1}^n X_i$ denote the sample mean. Then for large sample sizes $n$, the distribution of $\bar{X}$ is approximately normal with mean $\mu_{\bar{X}}=\mu_X$ and variance $\sigma^2_X= \dfrac{\sigma_X^2}{n}$.
This other part of the theorem is what is confusing me. I cant see the connection of the mean and variance like the other one. I think this one is when we want to consider the proportion of values rather the other one is the sum of values. But I don't understand why the variance and mean change. Also why is the mean for the 1st part of the theorem $n \cdot \mu_X$? You would think that the mean would simply be the sum of the outcomes divided by the number of observations which isn't the case in the 1st part of the theorem.