Sampling Distributions and the Central Limit Theorem

Three distributions must stay apart: the population (fixed shape, mean $\mu$, standard deviation $\sigma$), one sample (its histogram resembles the population), and the sampling distribution of a statistic (the values that statistic takes over all possible samples of size $n$). For the sample mean, $\mu_{\bar{x}} = \mu$, so $\bar{x}$ is unbiased, and $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, so quadrupling $n$ halves the spread. The Central Limit Theorem adds the shape: for large enough $n$ the sampling distribution of $\bar{x}$ is approximately normal whatever the population looks like, which is what makes a probability about $\bar{x}$ a normal calculation, $z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$.

The errors are all misplacements. The CLT is read as a claim that the population or the sample data become normal, when both keep the population's shape and only the statistic's distribution goes normal. Larger samples have less variability is applied to the sample's own standard deviation, which estimates $\sigma$ and does not shrink. And $\sigma$ is used where $\frac{\sigma}{\sqrt{n}}$ belongs: for bottles with $\mu = 12.0$ and $\sigma = 0.4$, the probability that one bottle exceeds 12.2 oz is about 0.309, while the probability that a sample of 16 averages more than 12.2 is about 0.023.