What does r-squared explain?
You will be able to: Interpret r² as the proportion of response variation explained by a fitted line.
What does r-squared explain?
Predicting the same score for every student ignores practice time. A regression line may reduce the squared prediction errors compared with this constant-mean prediction.
A useful starting point: What do the slope and intercept mean in context? →
Words and symbols before equations
- Response mean ȳ
- The average observed y value.
- Total squared variation SST
- Sum of squared deviations from the response mean, Σ(y−ȳ)².
- Unexplained squared variation SSE
- Sum of squared residuals from the fitted line.
- Coefficient of determination r²
- The proportion of SST explained by simple least-squares regression with an intercept.
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. r=0.861; r²=0.741. x̄=3 hours; ȳ=8 points. SSE=14 points². SST=54 points²; reduction=40 points². 74.1% of observed score variation is explained by this linear model, not necessarily caused by practice.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For our five quiz scores, the mean is 8 points. Predicting 8 for everyone gives squared deviations 9,16,4,0,25, totaling SST=54 points squared. The least-squares line leaves SSE=14.
The reduction is 54−14=40 points squared. Dividing by the original 54 gives r²=1−14/54≈0.741. About 74.1% of the observed variation in quiz scores is explained by the linear model using practice hours.
“Explained” is statistical, not causal. This does not mean 74.1% of students lie on the line or that predictions are 74.1% accurate. Squaring removes direction: positive and negative correlations with equal magnitudes have the same r².
| Feature | r | r² |
|---|---|---|
| Range | −1 through 1 | 0 through 1 |
| Direction | Sign gives direction | Sign is lost |
| Meaning | Linear association | Proportion of response variation explained by simple linear regression |
A worked example, step by step
A fitted simple regression with an intercept has SST=200 and SSE=50. Interpret r².
- SST measures squared variation around the response mean.
- The line reduces this variation by 200−50=150.
- Compute r²=150/200=0.75.
- 75% of the observed response variation is explained by the linear model using its explanatory variable; 25% remains unexplained by that model.
r² is not a percent of accurate predictions or the percent of y caused by x. The formula here concerns a least-squares line with an intercept.
If r=−0.8, what is r²?
Compare with an explanation
0.64. About 64% of observed response variation is explained by the fitted simple linear model with an intercept.
Predict. Change one thing. Explain.
Compare upward and downward datasets. Observe equal r² but opposite r signs. Then inspect the curved dataset and explain why r²=0 does not rule out a nonlinear model.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Fitted line: ŷ=2 + (2)x. r=0.861; r²=0.741. x̄=3 hours; ȳ=8 points. SSE=14 points². SST=54 points²; reduction=40 points². 74.1% of observed score variation is explained by this linear model, not necessarily caused by practice.
| Student | x (hours) | y (points) | ŷ (points) | e=y−ŷ (points) | e² (points²) |
|---|---|---|---|---|---|
| P1 | 1 | 5 | 4 | 1 | 1 |
| P2 | 2 | 4 | 6 | -2 | 4 |
| P3 | 3 | 10 | 8 | 2 | 4 |
| P4 | 4 | 8 | 10 | -2 | 4 |
| P5 | 5 | 13 | 12 | 1 | 1 |
Technology convention used here: ŷ=a+bx. The fitted line passes through (x̄,ȳ). Correlation and r² are undefined when response variation is zero.
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 questionFor a simple least-squares line predicting plant height from light exposure, SST=90 and SSE=30. Calculate and interpret r², and state what it does not establish.
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Compare with the answer and four-point rubric
- 1 point: Compute the explained squared variation: 90−30=60.
- 1 point: Divide: r²=60/90≈0.667.
- 1 point: About 66.7% of the observed height variation is explained by the linear model using light exposure.
- 1 point: This alone does not establish that light caused that percentage of height variation.
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 variation is explained?
Variation in the observed response around its mean.
RECALL 2What baseline does SST use?
Predicting the response mean for every observation.
RECALL 3Does r² retain direction?
No. Consult r or the slope.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does r-squared explain?
- r²=1−SSE/SST for simple least squares with an intercept.
- Percent explained =100r².
- r² loses the sign of r; check slope or r for direction.
Remember: r² is not a percent of accurate predictions or the percent of y caused by x. The formula here concerns a least-squares line with an intercept.
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 5.5.A, 5.5.B · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 5.5, objectives 5.5.A, 5.5.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.
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