What does a prediction error tell you?
You will be able to: Calculate a residual and interpret its sign in context.
What does a prediction error tell you?
A student practices 3 hours and scores 10 points. The line predicts 8. The observed score is 2 points above the prediction.
A useful starting point: When is a prediction beyond the evidence? →
Words and symbols before equations
- Residual e
- Observed response minus predicted response, y−ŷ.
- Underprediction
- The prediction is below the observation; the residual is positive.
- Overprediction
- The prediction is above the observation; the residual is negative.
- Vertical distance
- The difference in response values at the same x, measured in response units.
What this picture assumes
Synthetic five-student data illustrate reasoning, not population inference or causal effects. x is practice time in hours; y is quiz score in points. All plots keep fixed scales across controls. The curve and reflected data are deliberate comparison scenarios. With only five points, judgments of pattern are tentative.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- P3: observed 10, predicted 8, residual 2 points. Positive residual: the line underpredicts. Look for structure in the residuals; five observations offer limited diagnostic evidence.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Calculate the prediction before subtracting. At x=3, ŷ=2+2(3)=8. Then e=10−8=2 points. The line underpredicts this observed score by 2 points.
At x=4, the observed score is 8 while the prediction is 10. Its residual is 8−10=−2 points: the line overpredicts by 2 points.
A residual is a signed vertical gap. It is not the shortest perpendicular distance to the line, and it is not an error in x. Its units are the response units because both subtracted quantities are response values.
| Quantity | Meaning | Units |
|---|---|---|
| y | Observed response | Points |
| ŷ | Response predicted by the line | Points |
| e=y−ŷ | Signed vertical prediction error | Points |
A worked example, step by step
A delivery model predicts 18 minutes, while a delivery takes 15 minutes. Calculate and interpret the residual.
- Identify y=15 minutes and ŷ=18 minutes.
- Use observed minus predicted: e=y−ŷ.
- Calculate 15−18=−3 minutes.
- The actual delivery was 3 minutes shorter than predicted; the model overpredicted this time.
Reversing the subtraction reverses the interpretation. Residual always means observed minus predicted here.
A residual is zero. What does that mean?
Compare with an explanation
For that observation, the predicted and observed responses match exactly.
Predict. Change one thing. Explain.
Keep the upward dataset and change the selected student. Compare its vertical gap with its signed residual. Find one underprediction and one overprediction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
P3: observed 10, predicted 8, residual 2 points. Positive residual: the line underpredicts. Look for structure in the residuals; five observations offer limited diagnostic evidence.
| Student | x (hours) | y (points) | ŷ (points) | e=y−ŷ (points) |
|---|---|---|---|---|
| P1 | 1 | 5 | 4 | 1 |
| P2 | 2 | 4 | 6 | -2 |
| P3 | 3 | 10 | 8 | 2 |
| P4 | 4 | 8 | 10 | -2 |
| P5 | 5 | 13 | 12 | 1 |
Synthetic five-student data illustrate reasoning, not population inference or causal effects. x is practice time in hours; y is quiz score in points. All plots keep fixed scales across controls. The curve and reflected data are deliberate comparison scenarios. With only five points, judgments of pattern are tentative.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the paired observations, predictions, residuals, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionAt x=2 the line ŷ=3+4x predicts a score and the residual is −2 points. Find the prediction, observed score and contextual interpretation.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Substitute x=2 to get ŷ=3+8=11 points.
- 1 point: Use e=y−ŷ, so y=ŷ+e.
- 1 point: Calculate y=11+(−2)=9 points.
- 1 point: The model overpredicted this observed score by 2 points.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is the residual formula?
Observed minus predicted: y−ŷ.
RECALL 2What are residual units?
Response units.
RECALL 3How can you recover y?
Add the residual to the prediction.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does a prediction error tell you?
- e=y−ŷ.
- Positive: underprediction. Negative: overprediction.
- y=ŷ+e; residual units are response units.
Remember: Reversing the subtraction reverses the interpretation. Residual always means observed minus predicted here.
Conditions: Synthetic five-student data illustrate reasoning, not population inference or causal effects. x is practice time in hours; y is quiz score in points. All plots keep fixed scales across controls. The curve and reflected data are deliberate comparison scenarios. With only five points, judgments of pattern are tentative.
Refresh Kid · AP Statistics Unit 5 · Objectives 5.4.A, 5.4.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 5.4, objectives 5.4.A, 5.4.B. Framework effective Fall 2026, checked September 17, 2026. Unit 5 is Regression Analysis in the revised five-unit course: scatterplots, correlation, linear prediction, residuals and least squares. Inference tests for regression slopes are outside this unit.
Examples and datasets are synthetic, independently authored teaching material. Correlation describes linear association, not causation. Least squares uses squared vertical residuals and a fitted intercept. Check scatterplots, residual patterns, unusual observations, explanatory range and population applicability before interpreting a model. Five-point demonstrations do not support population or causal claims.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was located, but full lesson content was not available to the research tool. OpenStax is optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original least-squares surface uses self-hosted Three.js with its MIT license. The two base axes are candidate intercept and slope; vertical height is the resulting sum of squared errors. Rotation helps inspect the valley and its minimum as two parameters change. This is a parameter surface, not three measured variables. Fixed scales, an exact 2D parameter table and a labeled scatterplot provide alternatives. No autoplay or external 3D dependency is required.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
Want to work through this with a tutor?
Bring your question about What does a prediction error tell you? Your explanation and answers remain free to access.