If the coefficient of determination (R^2) is positive, what does this imply about the coefficient of correlation?

The coefficient of determination, ( R^2 ), quantifies the proportion of variance in the dependent variable that can be predicted from the independent variable(s). It varies from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect predictive power.

When ( R^2 ) is positive, it suggests that the model explains a portion of the variability in the dependent variable. Since ( R^2 ) is derived from the square of the correlation coefficient, there are implications for the coefficient of correlation. The coefficient of correlation can indeed be either positive or negative. A positive correlation indicates that as one variable increases, the other variable also tends to increase, while a negative correlation indicates that as one variable increases, the other variable tends to decrease.

Thus, a positive ( R^2 ) means there is some level of association (either positive or negative) between the variables, which confirms that the coefficient of correlation can take on either sign under those circumstances. This understanding highlights that the nature of the correlation coefficient itself depends on the direction of the relationship, which is why a positive ( R^2 ) does not constrain the correlation coefficient to be purely positive.