2 - The Regression Line

Summary

Introduction

This chapter is about the regression line. The regression line is important on its own (to statisticians), and it will help us with multiple regression in chapter 4. The first example is a scatter diagram showing the heights of 1078 fathers and their sons (figure 1). Each pair of fathers and sons becomes a dot on the diagram. The height of the father is plotted on the x-axis; the height of his son, on the y-axis. The left hand vertical strip (inside the chimney) shows the families where the father is 64 inches tall to the nearest inch; the right hand vertical strip, families where the father is 72 inches tall. Many other strips could be drawn too. The regression line (solid) approximates the average height of the sons, given the heights of their fathers. This line goes through the centers of all the vertical strips. The regression line is flatter than the SD line, which is dashed. “SD” is shorthand for “standard deviation”; definitions come next.

The regression line

We have n subjects indexed by i = 1, …, n, and two data variables x and y. A data variable stores a value for each subject in a study. Thus, x_i is the value of x for subject i, and y_i is the value of y.