Path: janda.org/c10 > Syllabus > Topics and Readings > Relating Two Continuous Variables > Interpreting r

 Interpreting the Product-Moment Correlation:

Comparing measures of relationship: r and r2
r is the PEARSON PRODUCT-MOMENT 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
.r2 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 VariationY = Explained Variation + Unexplained Variation Total Sum of SquaresY = Explained Sum of Squares + Unexplained SS Total SSY = Regression SS + Error SS
Comparing features:
Like r, r2 = 0 when the variables are completely unrelated.
Unlike r2, intermediate values of r do not have a PRE interpretation unless they are squared and thus transformed into r2.
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, r2.
My suspicion: researchers have tended to report r rather than r2 simply because it produced "fatter" numbers, thus making their relationships seem stronger. The baseline represents the correlation coefficient, r, and the coordinate is r2

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. (2-tailed) . .000 Sum of Squares and Cross-products 3853.922 3554.437 Covariance 77.078 71.089 N 51 51 % vote for Reagan, 1980 Pearson Correlation 0.900* 1 Sig. (2-tailed) 0 . Sum of Squares and Cross-products 3554.437 4047.562 Covariance 71.089 80.951 N 51 51 ** Correlation is significant at the 0.01 level (2-tailed).

Computing Pearson product-moment 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 two-dimensional 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: 