Central Limit Theorem vs Sampling Distribution

The central limit theorem is what lets you use normal-based methods for means even when the underlying data are not normal. For example, individual household incomes are strongly right-skewed, yet the distribution of mean income from samples of 50 households is close to a bell shape. As the sample size $n$ grows, the sampling distribution of $\bar{x}$ (x-bar, the sample mean) approaches a normal shape centered at $\mu$ (mu) with standard deviation $\frac{\sigma}{\sqrt{n}}$ (sigma over root n). A common guideline is that $n \ge 30$ is large enough for many populations.

A sampling distribution describes how a statistic, such as the sample mean, changes from one random sample to the next. For example, taking many samples of 40 students and recording each sample mean builds up the sampling distribution of $\bar{x}$ (x-bar, the sample mean). For the sample mean its center is the population mean $\mu$ (mu) and its spread, the standard error, is $\frac{\sigma}{\sqrt{n}}$ (sigma over the square root of n). This spread shrinks as the sample size grows, so larger samples give more consistent estimates.