Russel-Research-Method-in-Anthropology

568 Chapter 19
TABLE 19.6
Calculating the Mean for the Frequency Table of the Grouped Variable AGE
AGE range midpoint (x) f fx
20–29 25 6 150
30–39 35 6 210
40–49 45 5 225
50–59 55 8 440
60 65 5 325
n 30 fx 1,350
x 1,350/30
45.00
these grouped data, I’ve assigned the midpoint to be 65, as if the range were
60–69, even though the actual range is 60–78, and the real midpoint of the
60 category is
(60676778)/5 68.2
If you have grouped data, however, it will usually be because the data were
collected in grouped form to begin with. In that case, there is no way to know
what the real range or the real midpoint is for the 60 category, so we have
to assign a midpoint that conforms to the midpoints of the other ranges.
Applying the midpoint for all the other classes, the midpoint for the 60 category
would be 65, which is what I’ve done in table 19.6.
You can see the result of all this distortion: The grouped data have a mean
of 45.00, while the ungrouped data have a calculated mean of 45.033. In this
case, the difference is teeny, but it won’t always be that way. If you collect
data in groups about interval variables like age, you can never go back and see
how much you’ve distorted things. It’s always better to collect interval data at
the interval level, if you can, rather than in grouped form. You can group the
data later, during the analysis, but you can’t ‘‘ungroup’’ them if you collect
data in grouped form to begin with.
A Mathematical Feature of the Mean
The arithmetic mean has an important feature: The sum of the deviations
from the mean of all the scores in a distribution is zero. While the median and
the mean are both midpoints in sets of scores, the mean is the point in a distribution
at which the two halves balance each other out. You can see this in
table 19.7, which shows data from 10 countries on the percentage of women