Mentor: In order to see whether a line is a good fit or a bad fit for a set of data we can examine
the
residuals of that line.

Student: Why are the residuals related to determining if the line is a good fit?

Mentor: Well, the residuals express the difference between the data on the line and the actual data
so the values of the residuals will show how well the residuals represent the data.

Student: OK, well what do I look for when I'm examining the residuals?

Mentor: Well, if the line is a good fit for the data then the residual plot will be random. However,
if the line is a bad fit for the data then the plot of the residuals will have a pattern.

Student: How would data that forms a pattern look compared to random data?

Mentor: Well, let's take a look at a set of data with a good fit and a set of data with a bad fit to
see the difference. First, let's look at the residuals of a line that is a good fit for a data
set. Using the
Regression Activity, graph the data points: {(1, 3) (2, 4) (3, 3) (4, 7) (5, 6) (6, 6) (7, 7) (8, 9)}. Now, select
Display line of best fit and select
Show Residuals. Now you can see the Residual Plot of all of the residuals found when the predicted values of
the line of best fit are subtracted from the actual values.

Student: The residuals appear randomly placed along the graph. I can see how this would be a random
pattern of residuals. What would a residual plot look like for a line that was a bad fit for
the data?

Mentor: Well, let's look at another graph. Using the
Regression Activity, plot the following points: {(4, -11), (3, -6), (2, -3), (1, -2), (0, -3), (-1, -6), (-2,
-11)}. These points graph the quadratic equation -x^2 +2x-3. Now, select
Line of Best Fit to plot a line to fit the data. Now select
Show Residuals in order to view the residual plot that you want to examine.

Student: Hey, the residuals form a pattern! They are definitely not randomly scattered, but instead
they are making a curve. This line was not a good fit. Will there be times when I won't be
able to tell if the residuals form a pattern or not?

Mentor: Sometimes you will not have enough residuals to be able to see a definite pattern in the
plot, but in most cases you will be able to look at the residual plot and, using this
criteria, determine whether the line is a good fit or a bad fit for the data.

Student: I noticed that the residual values (the values under
Line of best fit) seem to have a sum of about 0. Does the sum of these residuals help determine whether a line
is a good fit for the data or not?

Mentor: The sum of the residuals does not necessarily determine anything. The
line of best fit will often have a sum of about 0 because it is including all data points and therefore it
will be a bit too far above some data points and a bit too far below some data points.
Therefore, in the case of the line of best fit often the positive error will balance out the
negative error so that the sum of the residuals will be approximately 0. However, this does
not mean that the line is a good fit for the data; it only means that the line is equally
above and below the actual data.

Student: OK, now I know that in order to find out if a line is a good fit for a set of data I can look
at the residual plot and if the residuals are a pattern then the line is not a good fit.