Probability Distribution vs Sampling Distribution

A probability distribution shows how the total probability of $1$ is shared among a random variable's possible values. For example, rolling a fair six-sided die gives each value from 1 to 6 a probability of $\frac{1}{6}$. For a discrete variable the probabilities must satisfy $0 \le p_i \le 1$ and $\sum p_i = 1$ (each probability is between 0 and 1, and they add to 1). Continuous variables use a density curve instead, where probability is the area under the curve.

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.