is close to one whenever N is large in comparison to n . (You don’t have to know this formula for the AP
exam. However, you do need to understand that when our sample size is larger than 10% of the
population, an adjustment must be made to the standard deviation formula.)
example: A large population is known to have a mean of 23 and a standard deviation of 2.5. What
are the mean and standard deviation of the sampling distribution of means of samples of size 20
drawn from this population?
solution:Central Limit Theorem
The discussion above gives us measures of center and spread for the sampling distribution of but tells
us nothing about the shape of the sampling distribution. It turns out that the shape of the sampling
distribution is determined by (a) the shape of the original population and (b) n , the sample size. If the
original population is normal, then it’s easy: the shape of the sampling distribution will be normal if the
population is normal.
If the shape of the original population is not normal, or unknown, and the sample size is small, then the
shape of the sampling distribution will be similar to that of the original population. For example, if a
population is skewed to the right, we would expect the sampling distribution of the mean for small
samples also to be somewhat skewed to the right, although not as much as the original population.
When the sample size is large, we have the following result, known as the central limit theorem : For
large n , the sampling distribution of will be approximately normal. The larger is n , the more normal
will be the shape of the sampling distribution.
A rough rule of thumb for using the central limit theorem is that n should be at least 30, although the
sampling distribution may be approximately normal for much smaller values of n if the population doesn’t
depart much from normal. The central limit theorem allows us to use normal calculations to do problems
involving sampling distributions without having to have knowledge of the original population. Note that
calculations involving z -procedures require that you know the value of s, the population standard
deviation. Since you will rarely know s, the large sample size essentially says that the sampling
distribution is approximately , but not exactly, normal. That is, technically you should not be using z -
procedures unless you know s but, as a practical matter, z -procedures are numerically close to correct for
large n . Given that the population size (N ) is large in relation to the sample size (n ), the information
presented in this section can be summarized in the following table: