1. Introduction to Regression
3E Residuals

2. Definition
A residual is the vertical difference between each data point and the regression line.

3. To calculate the residuals for each data point:
Calculate the predicted value of y from the regression equation.
Calculate the difference between this predicted value and the original value.

4. Examples
Read Worked Example 8 on p.130.
Exercise 3E, Q.1

5. Note
Calculate the residuals by (y-ypred), i.e. the actual values minus the predicted values.
The sum of all the residuals always add to zero (or very close to zero after rounding), when least-squares regression is used.

6. Residual Analysis
Why do we calculate residuals? Residuals give us more information about the type of regression.
If we plot the residuals against the original x-values we get a pattern.

7. Types of residual plot
Points of residual randomly scattered above and below x-axis. Original data probably linear.

8. Points of residual show curved pattern with a series of negative, then positive and back to negative residuals. Original data probably non-linear.
Points of residual show curved pattern with a series of positive, then negative and back to positive residuals. Original data probably non-linear.

9. In other words, if you see any sort of pattern in the residual plot, there probably isn’t a linear relationship in the original data set.

10. Example. Ex 3E, Q.2, 4
You do Ex 3E, Q.5, 6, 7, 8

11. CAS calculator
Once you have done the least-squares regression analysis on the Lists & Spreadsheet page:
Move the cursor to the shaded cell in Column E and press:
Ctrl
MENU
4: Variables
3: Link To
And select the list stat?.resid. The stat number will vary depending on your calculator and previously stored data.
You can then plot the stat.resid against x-value by clicking on the y-axis label and selecting the residual list stat.resid