# Sampling Distributions for Means

The "Central Limit Theorem." This theorem -- which involves averages computed from random samples of data -- is described below. The basic setting is as follows:

• A population; each unit in the population has a quantitative value (the variable) associated with it.
• Parameters: The mean m and standard deviation s for the population of values are parameters.
• A simple random sample. n units are randomly selected from the population in such a way that all possible samples are equally likely to be the selected sample.
• Statistics: The mean x-bar and standard deviation s for the sample are statistics. They are used as estimates of the parameters. Statistics are variables.

The sample mean x-bar is the focus here. It is a variable (each random sample results in a different sample mean); as such it has a distribution.

• The mean of this distribution is m.
• The standard deviation of this distribution is s/sqrt(n) (where sqrt(n) means "square root of n").
• The central limit theorem describes the pattern of variability. The distribution of x-bar is approximately normal. The quality of the approximation depends on two factors:
1. How close to normal the population distribution is. The closer, the better.
2. How large the sample size is. The larger, the better.

Avoid using this result for situations in which the combination of both nonnormal data and small sample size are present.