
 Comparing measures of relationship: r and
r^{2}
 r is the PEARSON PRODUCTMOMENT CORRELATION
COEFFICIENT
 It ranges from 1.0 to +1.0  indicating perfect
negative and positive relationships.
 Thus, the SIGN of r reveals the direction of the
relationship.
 The magnitude of r "indicates" the strength of the
relationship, measured by r2
 .r^{2} is the COEFFICIENT OF DETERMINATION
 It ranges from 0 to +1.0  indicating,
respectively, the ABSENCE of a systematic relationship
to a PERFECT relationship.
 Intermediate values have a PRE interpretation as
the proportion of variance in Y that is "explained" by
X  by the regression line.

Total Variation_{Y }
=

Explained Variation +

Unexplained Variation

Total Sum of Squares_{Y }
=

Explained Sum of Squares +

Unexplained SS

Total SS_{Y } =

Regression SS +

Error SS

 Comparing features:
 Like r, r^{2} = 0 when the
variables are completely unrelated.
 Unlike r^{2}, intermediate values of r do
not have a PRE interpretation unless they are squared
and thus transformed into r^{2}.
 Thus the correlation coefficient, r, simply
suggests the strength of a relationship between
variables; the exact strength can be expressed only by
the coefficient of determination, r^{2}.
 My suspicion: researchers have tended to report r
rather than r^{2} simply because it produced
"fatter" numbers, thus making their relationships seem
stronger.
 The baseline represents the correlation
coefficient, r, and the coordinate is
r^{2}


How about your analyses of the STATES data?
 Formulate a hypothesis relating political outcomes to
socioeconomic characteristics.
 Examples and findings?


Illustration of CORRELATION output from SPSS for
REAGAN84 and REAGAN80
Go to plot of the %
vote for Reagan in 1984 against % vote for Reagan in
1980
Correlations



% vote for Reagan, 1984

% vote for Reagan, 1980

% vote for Reagan, 1984

Pearson Correlation

1.000

0.900^{*}


Sig. (2tailed)

.

.000


Sum of Squares and Crossproducts

3853.922

3554.437


Covariance

77.078

71.089


N

51

51

% vote for Reagan, 1980

Pearson Correlation

0.900^{*}

1


Sig. (2tailed)

0

.


Sum of Squares and Crossproducts

3554.437

4047.562


Covariance

71.089

80.951


N

51

51

** Correlation is significant at the
0.01 level (2tailed).

Computing Pearson productmoment correlation from above
computer printout


 The values in the formula above are taken
from the boldface entries in the CORRELATION output
above
How to produce a "scatterplot" showing the
twodimensional plot of a bivariate correlation
The Graph Menu in SPSS lists the option,
Scatter. . ., which produces this box, showing
"Simple" as the default:
Click on the "Define" button and you get this dialog box,
asking you to enter variables for the Y and X axes:
Click on the "OK" button, and you get this
scatterplot:

