What does correlation measure?
You will be able to: Interpret the sign and magnitude of correlation for a linear association.
What does correlation measure?
Two class datasets can slope upward while differing in how tightly scores cluster around a line. Correlation gives a numerical summary of that linear association.
A useful starting point: How do you describe a pattern without overclaiming? →
Words and symbols before equations
- Correlation r
- A unitless measure of direction and strength of linear association between two quantitative variables.
- Magnitude of r
- The distance of r from zero, written as its absolute value.
- Standardized value
- An observation expressed in standard-deviation units from its mean.
- Linear association
- A pattern reasonably described by a straight trend.
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.
- Correlation r=0.861 (unitless). Inspect the plot; a numerical correlation alone cannot establish form or causation.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The correlation coefficient r lies between −1 and 1 when both variables have nonzero variation. Its sign gives linear direction; its magnitude gives linear strength. Values nearer either −1 or 1 indicate stronger linear association.
Standardizing both variables removes their units. Correlation does not measure points gained per hour; that is a slope. Exchanging the x and y variables leaves r unchanged, though their regression equations generally differ.
With r=−0.90, larger x tends to accompany smaller y with a strong linear association. With r=0.30, the positive linear association is weaker. Always inspect the scatterplot because r alone does not display curvature or unusual points.
A worked example, step by step
Compare r=−0.85 for temperature and jacket sales with r=0.40 for practice and scores.
- Both correlations are within the allowed range.
- The first is negative: higher temperatures tend to accompany lower sales.
- The first magnitude 0.85 exceeds 0.40, so its linear association is stronger.
- These numbers have no units and do not quantify sales per degree or points per hour.
A large negative r means strong negative linear association, not a weak relationship.
Can r=1.2 describe a very strong association?
Compare with an explanation
No. A correlation outside [−1,1] signals an error.
Predict. Change one thing. Explain.
Compare upward and downward versions of the same scatter. Read r and explain why reflection reverses its sign but preserves its magnitude.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Correlation r=0.861 (unitless). Inspect the plot; a numerical correlation alone cannot establish form or causation.
| Student | x (hours) | y (points) |
|---|---|---|
| P1 | 1 | 5 |
| P2 | 2 | 4 |
| P3 | 3 | 10 |
| P4 | 4 | 8 |
| P5 | 5 | 13 |
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 dataset has r=−0.72. Explain direction, strength relative to r=0.25, units, and the effect of reversing the variable roles.
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Compare with the answer and four-point rubric
- 1 point: Larger values of one variable tend to accompany smaller values of the other.
- 1 point: Its linear association is stronger in magnitude than 0.25.
- 1 point: Correlation has no measurement units.
- 1 point: Reversing variable roles leaves correlation at −0.72.
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 does the sign mean?
Direction of linear association.
RECALL 2What does magnitude mean?
Strength of linear association.
RECALL 3What happens if one variable is constant?
Correlation is undefined because that variable has zero spread.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does correlation measure?
- −1 ≤ r ≤ 1.
- Sign gives direction; magnitude gives linear strength.
- r is unitless and unchanged by swapping x and y.
Remember: A large negative r means strong negative linear association, not a weak relationship.
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 5.2.A · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 5.2, objectives 5.2.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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