# 12.4 - Coefficient of Determination

The amount of variation in the response variable that can be explained by (i.e. accounted for) the explanatory variable is denoted by $$R^2$$. This is known as the coefficient of determination or R-squared

## Example: $$R^2$$ From Output Section

Model Summary
9.71152 37.04% 35.73% 29.82%

In our Exam Data example this value is 37.04% meaning that 37.04% of the variation in the final exam scores can be explained by quiz averages.

The coefficient of determination is equal to the correlation coefficient squared. In other words $$R^2=(Pearson's\;r)^2$$

## Example: $$R^2$$ From Pearson's r Section

The correlation between quiz averages and final exam scores was $$r=.608630$$

Coefficient of determination: $$R^2=.608630^2=.3704$$

 Pearson correlation of Quiz_Average and Final = 0.608630 P-Value = <0.0001

When going from $$r$$ to $$R^2$$ you can simply square the correlation coefficient. $$R^2$$ will always be a positive value between 0 and 1.0.  When going from $$R^2$$ to $$r$$, in addition to computing $$\sqrt{R^2}$$, the direction of the relationship must also be taken into account. If the relationship is positive then the correlation will be positive. If the relationship is negative then the correlation will be negative.

## Examples: From $$R^2$$ to $$r$$ Section

Quiz Averages and Final Exam Scores

There is a direct (i.e., positive) relationship between quiz averages and final exam scores. The coefficient of determination ($$R^2$$) is 37.04%. What is the correlation between quiz averages and final exam scores?

$$r=\sqrt{R^2}=\sqrt{.3704}= \pm .6086$$

The correlation is $$r=+.6086$$ because we are told that there is a positive relationship between the two variables.

Daily High Temperatures and Hot Chocolate Sales

As the daily high temperature decreases, hot chocolate sales increase at a restaurant. 49% of the variance in hot chocolate sales can be attributed to variance in daily high temperatures. What is the correlation between daily high temperatures and hot chocolate sales?

$$R^2=.49$$              $$r=\sqrt{.49}= \pm .7$$

The correlation between daily high temperatures and hot chocolate sales is $$r=-.7$$. Because there is an indirect relationship between the two variables, the correlation is negative.