Introduction to regression

Residual analysis
y
Chapter 3 Introduction to regression 129
There are situations where the mere fitting of a regression line to some data is not
enough to convince us that the data set is truly linear. Even if the correlation is close to
+1 or −1 it still may not be convincing enough.
The next stage is to analyse the residuals, or deviations, of each data point from the
straight line. These are nothing more than the ‘error’ bars of the figure in worked
example 2.
A residual is the vertical difference between each data point and the regression line.
Calculating residuals
A sociologist gathers data on the heights of brothers and sisters in families from
different ethnic backgrounds. He enters his records in the table below.
x 2 3 5 8 9 9
y 3 7 12 10 12 16
He then plots each point, and fits a regression line as shown in figure 1, below. He then
decides to calculate the residuals.
The residuals are simply the vertical distances from the line to each point. These
lines are shown as blue and red bars in figure 2.
18
16
14
12
10
8
6
4
2
0
0 2 4 6 8 10
Figure 1
y
18
16
14
12
10
8
6
4
2
0
0 2 4 6 8 10 x
Figure 2
x