Russel-Research-Method-in-Anthropology

630 Chapter 20
The good news is that if you double a correlation coefficient, you quadruple
the variance accounted for. For example, if you get an r of .25, you’ve
accounted for 6.25% of the variance, or error, in predicting the score of a
dependent variable from a corresponding score on an independent variable.
An r of .50 is twice as big as an r of .25, but four times as good, because .50 2
means that you’ve accounted for 25% of the variance.
Accounting for Variance
What does ‘‘accounting for variance’’ mean? It simply means being able to
make a better prediction with an independent variable than we could without
one. Recall from chapter 19 that the total amount of variance in an observed
set of scores is the sum of the squared deviations from the mean, divided by
n1. This is shown in Formula 20.23.
2
xx
Variance
Formula 20.23
n1
Well, if the correlation between the two variables in table 20.14 is r
0.943, and its square is r 2 0.889, then we can say ‘‘we have accounted for
89% of the variance.’’ I’ll define what I mean by this.
The mean TFR for the 10 countries in table 20.15 is 3.114, and the total
variance (sum of squares) is 26.136. When we predict TFR, using the regression
formula, from the infant mortality rate, the variance in the predicted
scores will be
(1.889)26.1362.90
Except for rounding error, this is the total ‘‘new error’’ in table 20.17 and
this is what it means to say that ‘‘about 89% of the variance in TFR is
[accounted for] [determined by] [predicted by] the infant mortality rate across
10 countries.’’
By the way, in case you’re wondering, the correlation between infant mortality
and TFR across the 50 countries in table 19.8 is .89 and the correlation
between these two variables across all countries of the world is .86.
Testing the Significance of r
Just as with gamma, it is possible to test whether or not any value of Pearson’s
r is the result of sampling error, or reflects a real covariation in the larger