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Figure 10-1 compares the t- and standard normal (Z-) distributions in their
most general forms.
Standard normal
distribution (Z-distribution)
Figure 10-1:
t-distribution
Comparing
the standard
normal (Z-)
distribution
to a generic
t-distribution.
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The t-distribution is typically used to study the mean of a population, rather
than to study the individuals within a population. In particular, it is used
in many cases when you use data to estimate the population mean — for
example, to estimate the average price of all the new homes in California. Or
when you use data to test someone’s claim about the population mean — for
example, is it true that the mean price of all the new homes in California is
$500,000?
These procedures are called confidence intervals and hypothesis tests and are
discussed in Chapters 13 and 14, respectively.
The connection between the normal distribution and the t-distribution is that
the t-distribution is often used for analyzing the mean of a population if the
population has a normal distribution (or fairly close to it). Its role is espe-
cially important if your data set is small or if you don’t know the standard
deviation of the population (which is often the case).
When statisticians use the term t-distribution, they aren’t talking about just
one individual distribution. There is an entire family of specific t-distributions,
depending on what sample size is being used to study the population mean.
Each t-distribution is distinguished by what statisticians call its degrees of
freedom. In situations where you have one population and your sample size
is n, the degrees of freedom for the corresponding t-distribution is n – 1. For
example, a sample of size 10 uses a t-distribution with 10 – 1, or 9, degrees of
freedom, denoted t (pronounced tee sub-nine). Situations involving two popu-
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lations use different degrees of freedom and are discussed in Chapter 15.

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