AP Statistics / Regression Analysis
Lesson
Residuals and what the model missed
A residual is observed y minus predicted y. The residual plot is the honesty check.
Learning goals
- Compute and interpret a residual in context.
- Use a residual plot to judge whether a linear model is reasonable.
Explanation
A residual is the leftover: actual y minus the line’s prediction. A positive residual means the model guessed too low. A negative residual means it guessed too high. One leftover is a story about one point; a pattern in many leftovers is a story about the model.
Plot residuals against x or against predicted y. Random scatter around zero supports a linear fit. A curve means a straight line is the wrong shape. A megaphone means the spread is changing, so the usual equal-variance story is shaky.
Key terms
- Residual. Observed y minus predicted y for a given x.
- Residual plot. A scatterplot of leftovers used to check leftover curve, outliers, or changing spread.
Common mistakes
- Calling a residual plot “fine” when a clear curve remains.
- Using the line to predict far outside the x-range without saying it is extrapolation.
Practice
Original Marlow Works items. Check the answer explanation after you try.
A reporter stands at one grocery-store exit and asks shoppers whether they support a new park tax. Why is this a poor way to estimate citywide support?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Residuals make a U-shape against x. What does that say about the linear model?
Answer. The linear model is the wrong form; a curve in the original scatterplot is being missed.
Systematic leftover pattern means the straight line is not capturing the relationship.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Statistics course page
Official revised 2026–27 five-unit framework.
College Board · All rights reserved · accessed 2026-10-01
Open sourceCC BY — attribution required
Collaborative Statistics
Open statistics text for distributions, probability, and inference.
Open Textbook Library listing · CC BY · accessed 2026-10-01
Open source