Russel-Research-Method-in-Anthropology

Sampling Theory 173
a lot of information about what is going on, but it doesn’t tell us why things
turned out the way they did. A sample with a bimodal or highly skewed distribution
is a hint that you might be dealing with more than one population or
culture.
The Central Limit Theorem
The fact that many variables are not normally distributed would make sampling
a hazardous business, were it not for the central limit theorem. According
to this theorem, if you take many samples of a population, and if the samples
are big enough, then:
1. The mean and the standard deviation of the sample means will usually approximate
the true mean and standard deviation of the population. (You’ll understand
why this is so a bit later in the chapter, when we discuss confidence intervals.)
2. The distribution of sample means will approximate a normal distribution.
We can demonstrate both parts of the central limit theorem with some examples.
Part 1 of the Central Limit Theorem
Table 7.1 shows the per capita gross domestic product (PCGDP) for the 50
poorest countries in the world in 1998.
Here is a random sample of five of those countries: Guinea-Bissau, Nepal,
Moldova, Zambia, and Haiti. Consider these five as a population of units of
analysis. In 2000, these countries had an annual per capita GDP, respectively
of $100, $197, $374, $413, and $443 (U.S. dollars). These five numbers sum
to $1,527 and their average, 1527/5, is $305.40.
There are 10 possible samples of two elements in any population of five
elements. All 10 samples for the five countries in our example are shown in
the left-hand column of table 7.2. The middle column shows the mean for
each sample. This list of means is the sampling distribution. And the righthand
column shows the cumulative mean.
Notice that the mean of the means for all 10 samples of two elements—that
is, the mean of the sampling distribution—is $305.40, which is exactly the
actual mean per capita GDP of the five countries in the population. In fact, it
must be: The mean of all possible samples is equal to the parameter that we’re
trying to estimate.
Figure 7.4a is a frequency polygon that shows the distribution of the five
actual GDP values. A frequency polygon is just a histogram with lines connecting
the tops of the bars so that the shape of the distribution is emphasized.