Normal Distribution vs Sampling Distribution

The normal distribution is a bell-shaped curve centered at its mean $\mu$ (mu) and spread out by its standard deviation $\sigma$ (sigma). Many natural measurements, such as heights or measurement errors, are approximately normal. For example, adult heights cluster near an average, with fewer people far above or below it. The empirical rule says about 68, 95, and 99.7 percent of the data fall within 1, 2, and 3 standard deviations of the mean.

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.