The fundamental point to remember is that statistical numbers depend on
the process by which they were derived, and are informative only
to the extent that the process is logical.
A statistic gives us numerical information about some class of
things. In statistics, that class is called a population
(regardless of whether or not it's a class of people). The
different types of statistics--totals, frequencies, averages--
tell us about certain quantitative properties of the class. We
know from our study of the basic nature of induction that there
are two ways to support a statement about a class. We can
examine each and every member--the method of complete
enumeration--or we can study a sample of the class and then
generalize our findings to the class as a whole. Statisticians
use both methods.
Samples can be used to estimate a ratio, a frequency, a frequency
distribution, or an average value in a population.
Statistical generalizations have much in common with the
universal generalizations that we studied earlier (e.g., all S are
P). In both cases, we infer that what is true of a sample is
true of the entire population. In both cases, the inference is
valid only to the extent that the sample is representative of the
For statistical generalization, we are interested in a
quantitative property of the population: the proportion of Ss
that are P. What we need is a method of choosing our samples so
that the proportion of Ps in the sample will reflect the
proportion of Ps in the population.
The method statisticians have devised is the use of random
samples. The reason is that if we choose our sample randomly,
then every member of the population has an equal chance of being
included in the sample.
The use of random samples has certain implications that we should
be aware of.
First, there is always a specific margin of error attached to the
conclusion we draw about the population.
Second, the margin of error depends on the size of the sample.
The larger the sample, the smaller the margin of error, and vice
Finally, we should be aware that even when we take the margin of
error into account, we are still only dealing with probabilities.