Simple Regression 231 An appealing benchmark that takes these considerations into account is the coefficient of determination, R 2 , which compares the model's sum of the squared prediction errors to the sum of the squared deviations of Y about its mean: ðY i - Y i Þ 2 R 2 = 1 - i=1 n n _ (8.7) 2 ðY i - Y Þ i=1 The numerator is the model's sum of squared prediction errors; the denominator is the sum of squared prediction errors if we ignore X and simply use the average value of Y to predict Y. The value of R 2 cannot be less than 0 nor greater than 1. An R 2 close to 1 indicates that the model's prediction errors are very small in relation to the variation in Y about its mean. An R 2 close to 0 indicates that the regression model is not an improvement over ignoring X and simply using the average value of Y to predict Y. Mathematically, the sum of the squared deviations of Y about its mean can be separated into the sum of the squared deviations of the model's predictions about the mean and the sum of the squared prediction errors: Y 2 Y 2 Y 2 n n _ n _