Sampling distributions

The sampling distribution of the sample mean describes how $\bar{x}$ varies across all possible samples of size $n$ from a population. Its center is the population mean $\mu$, and its spread is the standard error ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$. If the population is normal, or if $n$ is large enough for the Central Limit Theorem, $\bar{x}$ is approximately normally distributed. To find probabilities or percentiles, convert $\bar{x}$ to a $z$-score using $z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$, then use the standard normal table or technology.