When is a prediction beyond the evidence?
You will be able to: Distinguish interpolation from extrapolation and qualify predictions.
When is a prediction beyond the evidence?
Our students practiced between 1 and 5 hours. A prediction for 3.5 hours uses the observed range; a prediction for 9 hours assumes the same trend continues beyond it.
A useful starting point: How does a line turn hours into a predicted score? →
Words and symbols before equations
- Observed range
- The smallest through largest explanatory values represented in the data.
- Interpolation
- Prediction at an x value within the observed range.
- Extrapolation
- Prediction outside the observed range.
- Model applicability
- Whether the data and assumptions support using the model in a new situation.
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.
- Fitted line: ŷ=2 + (2)x. At x=3 hours, predicted score=8 points. Interpolation: within the observed 1–5 hours. This is not a guaranteed score.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Interpolation uses x within the range actually observed. For a suitable linear pattern, it has support from nearby measurements. It is still uncertain, especially in gaps or for a different population.
Extrapolation extends the pattern beyond measured x values. Practice benefits may level off, a quiz has a maximum score, or a different process may take over. The algebra can produce an answer even when the context makes it unreliable.
Check the range and population before interpreting a prediction. A model for one type of assessment is not automatically valid for another. A high correlation does not authorize unlimited extrapolation.
A worked example, step by step
For data from 1–5 hours and ŷ=2+2x, compare predictions at 3.5 and 9 hours.
- At 3.5 hours, ŷ=2+2(3.5)=9 points.
- 3.5 lies inside [1,5], so this is interpolation.
- At 9 hours, ŷ=2+2(9)=20 points.
- 9 lies outside [1,5], so the numerical result is extrapolation with uncertain model validity.
A calculator returning a number does not make the prediction supported by data.
Is interpolation always accurate?
Compare with an explanation
No. It depends on the model’s suitability, variability, coverage and applicability to the target situation.
Predict. Change one thing. Explain.
Move predicted hours to 6 while holding everything else fixed. Read the range warning and explain why the model can calculate a value without validating it.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Fitted line: ŷ=2 + (2)x. At x=3 hours, predicted score=8 points. Interpolation: within the observed 1–5 hours. This is not a guaranteed score.
| Student | x (hours) | y (points) | ŷ (points) |
|---|---|---|---|
| P1 | 1 | 5 | 4 |
| P2 | 2 | 4 | 6 |
| P3 | 3 | 10 | 8 |
| P4 | 4 | 8 | 10 |
| P5 | 5 | 13 | 12 |
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 questionA model based on children aged 8–12 predicts height in centimeters by ŷ=80+5x. Compare predictions at ages 10 and 20.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Age 10 gives 130 cm and lies inside the observed age range.
- 1 point: Age 20 gives 180 cm by substitution but is outside the range.
- 1 point: The first is interpolation and the second is extrapolation.
- 1 point: Growth patterns and population applicability can change, so the second prediction lacks support from these data.
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 defines extrapolation?
An x value outside the observed range.
RECALL 2Does high correlation remove its risk?
No.
RECALL 3What else matters besides range?
Model form, data gaps, response variability and population applicability.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When is a prediction beyond the evidence?
- Compare x with the observed range.
- Inside = interpolation; outside = extrapolation.
- Check population, gaps and physical limits too.
Remember: A calculator returning a number does not make the prediction supported by data.
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 5.3.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 5.3, objectives 5.3.A. 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.
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