Russel-Research-Method-in-Anthropology

Sampling Theory 171
In normal distributions—that is, distributions that have a mean of 0 and a
standard deviation of 1—exactly 34.135% of the area under the curve (the
white space between the curve and the baseline) is contained between the perpendicular
line that represents the mean in the middle of the curve and the line
that rises from the baseline at 1 standard deviation above and 1 standard deviation
below the mean.
Appendix B is a table of z-scores, orstandard scores. These scores are
the number of standard deviations from the mean in a normal distribution, in
increments of 1/100th of a standard deviation. For each z-score, beginning
with 0.00 standard deviations (the mean) and on up to 3.09 standard deviations
(on either side of the mean), appendix B shows the percentage of the physical
area under the curve of a normal distribution. That percentage represents the
percentage of cases that fall within any number of standard deviations above
and below the mean in a normally distributed set of cases.
We see from appendix B that 34.13% of the area under the curve is one
standard deviation above the mean and another 34.13% is one standard deviation
below the mean. (The reason that this is so is beyond what we can go into
here. For more on this, consult a standard statistics text.) Thus, 68.26% of all
scores in a normal distribution fall within one standard deviation of the mean.
We also see from appendix B that 95.44% of all scores in a normal distribution
fall within two standard deviations and that 99.7% fall within three standard
deviations.
Look again at figure 7.1. You can see why so many cases are contained
within 1 sd above and below the mean: The normal curve is tallest and fattest
around the mean and much more of the area under the curve is encompassed
in the first sd from the mean than is encompassed between the first and second
sd from the mean.
If 95.44% of the area under a normal curve falls within two standard deviations
from the mean, then exactly 95% should fall within slightly less than
two standard deviations. Appendix B tells us that 1.96 standard deviations
above and below the mean account for 95% of all scores in a normal distribution.
And, similarly, 2.58 sd account for 99% of all scores. This, too, is shown
in figure 7.1.
The normal distribution is an idealized form. In practice, many variables
are not distributed in the perfectly symmetric shape we see in figure 7.1. Figure
7.2 shows some other shapes for distributions. Figure 7.2a shows a
bimodal distribution. Suppose the x-axis in figure 7.2a is age, and the y-axis
is strength of agreement, on a scale of 1 to 5, to the question ‘‘Did you like
the beer commercial shown during the Superbowl yesterday?’’ The bimodal
distribution shows that people in their 20s and people in their 60s liked the
commercial, but others didn’t.