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
- A population; each unit in the population has a
quantitative value (the variable) associated with
- Parameters: The mean m
and standard deviation s
for the population of values are
- 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:
- How close to normal the population
distribution is. The closer, the
- 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.
From here link to worksheets.