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Chapter 4: Getting in Line with Simple Linear Regression 75

residual at or beyond +3 or –3. If this occurs, you can consider that point an
outlier, which warrants further investigation. (For more on outliers, see the
section “Scoping for outliers” later in this chapter.)

The residuals should also occur at random — some above the line, and some
below the line. If a pattern occurs in the residuals, the line may not be fitting right.

The plots in Figure 4-6 seem to have an issue with the very last observation,
the one for 12th graders. In this observation, the average student weight
(142) seemed to follow the pattern of increasing with each grade level, but
the textbook weight (16.06) was less than for 11th graders (20.79) and is the
first point to break the pattern.

You can also see in the plot in the upper-right corner of Figure 4-6 that the very
last data value has a standardized residual that sticks out from the others and
has a value of –3 (something that should be a very rare occurrence). So the value
you expected for y based on your line was off by a factor of 3 standard devia-
tions. And because this residual is negative, what you observed for y was much
lower than you may have expected it to be using the regression line.

The other residuals seem to fall in line with a normal distribution, as you can
see in the upper-right plot of Figure 4-6. The residuals concentrate around
zero, with fewer appearing as you move farther away from zero. You can also
see this pattern in the upper-left plot of Figure 4-6, which shows how close to
normal the residuals are. The line in this graph represents the equal-to-normal
line. If the residuals follow close to the line, then normality is okay. If not, you
have problems (in a statistical sense, of course). You can see the residual with
the highest magnitude is –3, and that number falls outside the line quite a bit.

The lower-left plot in Figure 4-6 makes a histogram of the standardized resid-
uals, and you can see it doesn’t look much like a bell-shaped distribution. It
doesn’t even look symmetric (the same on each side when you cut it down the
middle). The problem again seems to be the residual of –3, which skews the
histogram to the left.

The lower-right plot of Figure 4-6 plots the residuals in the order presented
in the data set in Table 4-1. Because the data was ordered already, the lower-
right residual plot looks like the upper-right residual plot in Figure 4-6, except
the dots are connected. This lower-right residual plot makes the residual of
–3 stand out even more.

Checking the spread of the y’s for each x
The graph in the upper-right corner of Figure 4-6 also addresses the homosce-
dasticity condition. If the condition is met, then the residuals for every x-value
have about the same spread. If you cut a vertical line down through each x-value,
the residuals have about the same spread (standard deviation) each time, except
for the last x-value, which again represents grade 12. That means the condition
of equal spread in the y-values is met for the textbook-weight example.

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