Thinking and Deciding

184 JUDGMENT OF CORRELATION AND CONTINGENCY
Table 8.1: Height and weight for five children
Child Height (in.) Weight (lb.)
A 48 45
B 43 35
C 51 55
D 46 44
E 45 40
correlated with weight among people, we mean that taller people tend to weigh
more than shorter people. As one figure goes up, so does the other, and vice versa.
We can measure the association between two different quantities on a scale of 1
to −1 by a correlation coefficient, abbreviated r. (Theformulaforr is found in most
statistics texts.) The coefficient r measures the extent to which one variable can be
predicted as a linear function of the other. A correlation coefficient of 1 indicates
that one measure is a perfect linear function of another, or has a perfect positive
correlation with it. An example is the relationship between degrees Fahrenheit (F)
and degrees Celsius (C): If we know one of these, we can calculate the other exactly
by a simple linear formula, F =1.8 · C +32. A correlation of 0 indicates no linear
relationship at all. In a long sequence of dice rolls, for example, there ought to be
no correlation between the number thrown on one roll of the dice and the number
thrown on the next roll. A correlation of −1 indicates a perfect negative correlation;
for example, there is a perfect negative correlation between a person’s height and
the distance of the person’s head from the ceiling of the room I am now in. The
correlation coefficient for the height column and the weight column in Table 14.1 is
0.98, very close to 1. (Again, consult a statistics book for the formula for r.)
The correlation coefficient is only one among many reasonable measures of correlation.
It has certain convenient mathematical properties, and it can be shown to be
the best measure for achieving certain goals. These issues are discussed in statistics
books. For our purposes, the most important property of the correlation coefficient
is that it treats all observations equally. For example, it does not weigh more heavily
those pairs of numbers that are consistent with the existence of a positive correlation,
as opposed to a negative or a 0 correlation.
We can investigate correlations, or associations, when each variable concerns
membership in a category. (Such variables are called dichotomous because membership
and nonmembership constitute a dichotomy. Another term is “binary.”) We
may ask, for example, whether “presence of blue eyes” (membership in the category
of blue-eyed people) is correlated with “presence of blond hair” in a population of
people. To calculate the correlation, we can assign a 1 to “presence” of each variable
(blue eyes or blond hair) and a 0 to “absence” of each variable for each person