A residual is actual minus predicted. A positive residual means the model underestimated that point.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why least squares
The regression line minimises the sum of the squared residuals, which is what makes it the least-squares line.
The residual plot
Plot residuals against x. Random scatter means a line was appropriate; a curve means it was not.
A good r can still be wrong
A high correlation with a curved residual plot means the linear model is misfitting despite looking good.
r squared
r squared is the proportion of variation in y explained by the linear relationship with x.
Interpreting it
An r squared of 0.8 means 80% of the variation is explained. It does not mean 80% of points lie on the line.
Residual is observed minus predicted
A positive residual means the actual value exceeded the prediction. Getting the subtraction the right way round matters, because the sign is what the interpretation rests on.
The residual plot is the diagnostic
A random scatter of residuals supports a linear model. A curved pattern means the relationship is not linear and the line is the wrong choice, however high r² happens to be.
Changing spread is also a problem
Residuals fanning out mean the variability is not constant, which undermines the inference procedures even when the line itself fits the shape. Both patterns are worth naming when they appear.
Influential points
A point with an extreme x-value can pull the line substantially. An outlier in y inflates the residuals without moving the line much. The two have different effects and should be distinguished.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Marcus Read a Residual Plot
Marcus fits a line, gets r = 0.95, and sees a clear U shape in the residual plot.
- Step 1
An r of 0.95 suggests a very strong linear relationship.
Step 4: Your Turn
Practice makes it stick.
The Residual
Problem 1 of 2
Predicted 9, actual 11. What is the residual?
The Explained Variation
Problem 2 of 2
r = 0.8. What is r squared?
Check the Fit
1 of 8
Predicted 20, actual 17. Residual?
2 of 8
A negative residual means the model did what? 1 overestimated, 2 underestimated.
3 of 8
r = 0.6. What is r squared?
4 of 8
r squared = 0.49. What percentage of variation is explained?
5 of 8
A curved residual plot. Is a line appropriate? 1 yes, 0 no.
6 of 8
What does least squares minimise? 1 the sum of squared residuals, 2 the largest residual.
7 of 8
Which statements about r squared are true?
8 of 8
r = 0.5. What is r squared?
Step 5: Quick Check
Show what you know.
Question 1 of 2
r = 0.9. What is r squared?
Question 2 of 2
What should a residual plot look like for a good linear model?
What You Learned
- A residual is actual minus predicted.
- A patterned residual plot rejects the linear model, however high r is.
- r squared is the proportion of variation in y explained by x.