360 Statistical Methods
The R Square value in cell B5 (.372) is the coeffi cient of determination
R^2 discussed in the previous chapter. This value indicates that 37% of the
variance in calculus scores can be attributed to the regression. In other
words, 37% of the variability in the fi nal calculus score is due to differ-
ences among students (as quantifi ed by the values of the predictor vari-
ables) and the rest is due to random fl uctuation. Although this value might
seem low, it is an unfortunate fact that decisions are often made on the
basis of weak predictor variables, including decisions about college ad-
missions and scholarships, freshman eligibility in sports, and placement
in college classes.
The Multiple R (0.610) in cell B4 is just the square root of the R^2 ; this
is also known as the multiple correlation. It is the correlation among
the response variable, the calculus score, and the linear combination of
the predictor variables as expressed by the regression. If there were only
one predictor, this would be the absolute value of the correlation between
the predictor and the dependent variable. The Adjusted R Square value in
cell B6 (0.320) attempts to adjust the R^2 for the number of predictors. You
look at the adjusted R^2 because the unadjusted R^2 value either increases or
stays the same when you add predictors to the model. If you add enough
predictors to the model, you can reach some very high R^2 values, but not
much is to be gained by analyzing a data set with 200 observations if the
regression model has 200 predictors, even if the R^2 value is 100%. Adjusting
the R^2 compensates for this effect and helps you determine whether adding
additional predictors is worthwhile.
The standard error value, 9.430 (cell B7), is the estimated value of s, the
standard deviation of the error term e, in other words, the standard devia-
tion of the calculus score once you compensate for differences in the predic-
tor variables. You can also think of the standard error as the typical error
for prediction of the 80 calculus scores. Because a span of 10 points cor-
responds to a difference of one letter grade (A vs. B, B vs. C, and so on), the
typical error of prediction is about one letter grade.Figure 9-4
Multiple
regression
statistics