AP Statistics · Exam guide
AP Statistics Regression Questions Explained: Interpreting Slope, r-Squared and Residuals
Regression questions give you computer output or a fitted line and ask you to interpret it. Every sentence has to name the variables and use units. Regression also shows up in the investigative task, so the habit pays off twice. The data below are made up for practice.
A made-up model
A teacher fits a line to predict a student's test score from the hours studied. The fitted line is: predicted score = 55 + 4.2 times hours. The coefficient of determination is r-squared = 0.64, and the standard deviation of the residuals is s = 6.0 points.
Interpreting each piece in context
| Piece | A full-credit sentence |
|---|---|
| Slope (4.2) | For each additional hour studied, the predicted test score increases by about 4.2 points, on average. |
| Intercept (55) | The predicted test score for a student who studies 0 hours is 55 points. (Mention it only if 0 hours makes sense.) |
| r-squared (0.64) | About 64 percent of the variation in test scores is accounted for by the linear relationship with hours studied. |
| s (6.0) | The actual test scores typically differ from the scores predicted by the line by about 6.0 points. |
Prediction and residuals
- Prediction for 5 hours: 55 + 4.2(5) = 76 points.
- A student who studied 5 hours and scored 82 has a residual of 82 - 76 = +6 points. The positive sign means the student scored higher than the line predicted.
- Prediction for 20 hours: 55 + 4.2(20) = 139 points. This is extrapolation far beyond the data, and a score above 100 is impossible, so the line should not be used there.
It does not give units, say that it is a prediction, or say per what.
It names both variables, uses units and says predicted.
In the investigative task
The last question can ask you to carry regression through several steps, such as checking the linear condition with a residual plot, doing a slope inference, or comparing two models. Treat each lettered part separately, and write every conclusion in context.
Common mistakes
- Saying that the score "will" go up. The model gives a predicted average.
- Interpreting r-squared as the percent of points that fall on the line.
- Confusing r (the correlation) with r-squared.
- Extrapolating far beyond the range of the data.
- Claiming that studying causes higher scores from an observational study.
Frequently asked questions
What does r-squared tell me?
The proportion of the variation in the response that is accounted for by the linear relationship with the explanatory variable.
How do I interpret a residual?
It is the actual value minus the predicted value. Positive means the point is above the line.
When is it safe to interpret the intercept?
When a value of 0 for the explanatory variable makes sense and is within or near the range of the data.
What does a pattern in a residual plot mean?
A curve or a fan shape suggests that a straight line is not an appropriate model.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every unit in the AP Statistics course ends with a ten-problem practice set, three multiple choice and seven free response, with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Statistics course Free lessons and lesson list
Other free-response guides:
- AP Statistics FRQ Guide: How Free-Response Answers Are Scored
- AP Statistics Inference for Proportions Explained: Intervals and Tests with a Worked Example
- AP Statistics Inference for Means Explained: The One-Sample t Test and Checking Conditions
- AP Statistics Chi-Square Test Explained: Homogeneity and Independence with a Worked Example
AP is a registered trademark of the College Board, which is not affiliated with and does not endorse this site. This guide is our own explanation of the question types and scoring. For the official scoring guidelines, see the College Board.