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?
- Worked Example: First identify exactly what the question is asking: In a two-way table, 9 of 23 students in the first row chose yes. What is the row relative frequency for yes?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use the row total as the denominator.
- 9/23 simplifies to 9/23.
Answer: 9/23
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
- Warm-up: First identify exactly what the question is asking: In a two-way table, 9 of 23 students in the first row chose yes. What is the row relative frequency for yes?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use the row total as the denominator.
- 9/23 simplifies to 9/23.
- Check the result by substituting or estimating: the response should match 9/23 and make sense in the original problem.
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
- Warm-up: First identify exactly what the question is asking: A scatter plot with points rising left to right shows:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- When x increases and y tends to increase, the direction is positive.
- That is positive association.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: Correlation by itself can prove:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Correlation measures association.
- It does not prove causation without a stronger design.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: association, not causation
4. Practice case D: A regression line is y = 2x + 9. Predict y when x = 7.
Show solution
- Warm-up: First identify exactly what the question is asking: A regression line is y = 2x + 9. Predict y when x = 7.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Substitute x = 7.
- 2(7) + 9 = 23.
- Check the result by substituting or estimating: the response should match 23 and make sense in the original problem.
Answer: 23
5. Practice case E: A model predicts 29, but the actual value is 27. Find the residual using actual - predicted.
Show solution
- Warm-up: First identify exactly what the question is asking: A model predicts 29, but the actual value is 27. Find the residual using actual - predicted.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Residual = actual - predicted.
- 27 - 29 = -2.
- Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.
Answer: -2
Practice this interactively with instant feedback and an AI tutor.
Practice Residuals and Model Fit Take the free placement check