Regression Effect

Whenever there is ANY spread about the linear regression line, for any x above {\bar x} , the y values will increase by less than the increase in x .

For example, for x one standard deviation above \bar{x} , the corresponding predicted y value (y-hat)  will be less than one standard deviation above \bar{y} .

Similarly, for x one standard deviation less than \bar{x} , the corresponding predicted y (y-hat) will decrease by less than one standard deviation below \bar{y} .

This is called the regression effect.

It happens because part of the unexplained variation in y is explained by the change of x (the regression).

The regression line is ALWAYS flatter than the SD-line. (Both lines pass through the point of averages, (\bar{x},\bar{y}) .