ABSTRACT
When X and Y are continuous variables the familiar Pearson correlation coefficient, r, provides an obvious estimator of effect size in terms of the size (magnitude of r) and direction (sign of r) of a linear relationship between X and Y. However, thus far in this book, although the Y variable has been continuous the independent variable (X) has been a dichotomous variable such as membership in Group a or Group b. Although computational formulas and software for r obviously require both X and Y to be quantitative variables, calculating an r between a truly dichotomous categorical X variable and a quantitative Y variable does not present a problem. By a truly dichotomous variable we mean a naturally dichotomous (or nearly so) variable, such as gender, or an independent variable that is created by assigning participants into two different treatment groups to conduct an experiment. We are not referring to the problematic procedure of creating a dichotomous variable by arbitrarily dichotomizing originally continuous scores into two groups, say, those above the median versus those below the median. When an originally continuous variable is dichotomized it will nearly always correlate lower with another variable than if it had not been dichotomized (Hunter & Schmidt, 2004). Similarly, as Hunter and Schmidt (2004) discussed, when a continuous variable has been dichotomized it cannot attain the usual maximum absolute value of correlation with a continuous variable, |1|.
