Chapter 5 Probability Distributions 217Figure 5-27
Sampling
distribution
for average
from the
uniform
distribution
sample size = 1 sample size = 4
sample size = 9 sample size = 165 Try some of the other distributions in the list under various sample
sizes. Close the workbook when you’re fi nished. You do not have to
save your changes.A few fi nal points about the Central Limit Theorem should be considered.
First, the theorem applies only to probability distributions that have a fi nite
mean and standard deviation. Second, the sample size and the properties of
the original distribution govern the degree to which the sampling distribution
approximates the normal distribution. For large sample sizes, the approxi-
mation can be very good, whereas for smaller samples, the approximation
might not be good at all. If the probability distribution is extremely skewed,
a larger sample size will be necessary. If the distribution is symmetric, the
sample size usually need not be very large. How large is large? If the original
distribution is symmetric and already close in shape to a normal distribu-
tion, a sample size of 15 or 20 should be large enough. For a highly skewed
distribution, a sample size of 40 or 50 might be required. Usually the Central
Limit Theorem can be safely applied if the sample size is 30 or more.
The Central Limit Theorem is probably the most important theorem in
statistics. With this theorem, statisticians can make reasonable inferences
about the sample mean without having to know about the underlying prob-
ability distribution. You’ll learn how to make some of these inferences in
the next chapter.