Residuals and Model Fit
A free Statistics and Data Analysis lesson from the “Two-Variable Data” unit, with a worked example and practice problems including step-by-step solutions.
A residual is the difference between an actual value and the value predicted by a model. Residuals close to zero indicate better fit; patterns in residuals can warn that a linear model is missing structure. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Compute residuals as actual minus predicted
- Use residuals to judge fit
- Recognize patterns that suggest a poor linear model
Worked example
Problem. In a two-way table, 9 of 23 students in the first row chose yes. What is the row relative frequency for yes?
- Use the row total as the denominator.
- 9/23 simplifies to 9/23.
Answer: 9/23
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A: In a two-way table, 9 of 23 students in the first row chose yes. What is the row relative frequency for yes?
Show solution
- Use the row total as the denominator.
- 9/23 simplifies to 9/23.
Answer: 9/23
2. Practice case B: A scatter plot with points rising left to right shows:
Choices: a categorical variable only · positive association · negative association · no association
Show solution
- When x increases and y tends to increase, the direction is positive.
- That is positive association.
Answer: positive association
3. Practice case C: Correlation by itself can prove:
Choices: association, not causation · cause and effect every time · that the sample was random · that there are no outliers
Show solution
- Correlation measures association.
- It does not prove causation without a stronger design.
Answer: association, not causation
4. Practice case D: A regression line is y = 2x + 9. Predict y when x = 7.
Show solution
- Substitute x = 7.
- 2(7) + 9 = 23.
Answer: 23
5. Practice case E: A model predicts 29, but the actual value is 27. Find the residual using actual - predicted.
Show solution
- Residual = actual - predicted.
- 27 - 29 = -2.
Answer: -2
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