AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/5/5-2)
Unit 5 · Topic 5.2
5.2 Correlation
The correlation coefficient r measures the direction and strength of a linear association with one unit-free number between −1 and 1. It's useful, but it's easy to misuse: a strong r doesn't prove a line is the right model, and correlation alone never shows causation.
Key terms
- correlation coefficient (r)
- linear association
- strength and direction
- correlation vs. causation
What r measures
r summarizes how tightly the points cluster around a straight line and which way the line slopes. Its sign gives the direction: positive r for a positive association, negative r for a negative one. Its size gives the strength: the closer to −1 or 1, the stronger.
r = 1 or r = −1 means every point lies exactly on a line. r = 0 means no linear association, though there could still be a strong curved one. You'll calculate r with technology.
The table gives rough guides, not official cutoffs. Always look at the scatterplot too.
| r (roughly) | Linear association |
|---|---|
| 0.8 to 1 or −0.8 to −1 | Strong |
| 0.5 to 0.8 or −0.5 to −0.8 | Moderate |
| 0 to 0.5 or 0 to −0.5 | Weak |
| 0 | None |
Properties of r
Four properties to know:
- r has no units, so changing units (inches to centimeters, dollars to euros) doesn't change it.
- r is the same whichever variable is on which axis.
- r only measures linear association. It's meaningless as a summary of a curved pattern.
- r is not resistant: a single unusual point can raise or lower it a lot.
A big r doesn't mean a line fits
For the points (1, 1), (2, 4), (3, 9), …, (8, 64), which follow y = x² exactly, r ≈ 0.976. That looks like a strong linear association, but the data are curved. A high r only says the points rise together fairly steadily; you need a scatterplot or residual plot (5.4) to judge whether a line is the right model.
How one point can change r
Because r isn't resistant, a single point can move it a lot. A point far from the pattern (say, a student who studied 8 hours and scored 60) weakens the association and pulls r toward 0. A point far out in x that falls right in line with the pattern can make r closer to 1 or −1 than the rest of the data would.
So report r together with a scatterplot, and mention any unusual points that might be affecting it.
Correlation and causation
A strong correlation doesn't show that changes in x cause changes in y. Possible explanations include a confounding variable affecting both, coincidence, or causation running the other way. Only a well-designed experiment with random assignment can establish cause and effect.
Interpret r in context: "r = 0.97 indicates a strong, positive, linear association between hours studied and exam score for these students."
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Interpret r
For the hours-and-scores data (5.1), technology gives r ≈ 0.974. Interpret this value.
Show the solutionHide the solution
- Step 1: Sign: positive, so scores tend to rise as hours studied rise.
- Step 2: Size: 0.974 is close to 1, so the linear association is strong.
- Step 3: Context: name both variables and the group.
Answer: r ≈ 0.974 means there is a strong, positive, linear association between hours studied and exam score for these 10 students.
- Example 2Calculator allowed
Effect of changes on r
Scores are converted from points to percentages of a 120-point maximum (divide by 1.2), and hours are converted to minutes (multiply by 60). What is the new correlation?
Show the solutionHide the solution
- Step 1: Changing units multiplies x and y by positive constants.
- Step 2: r is unit-free, so it doesn't change.
Answer: Still r ≈ 0.974.
- Example 3Calculator allowed
Trap: correlation as causation
Across cities, the number of churches and the number of crimes have r = 0.85. A blogger claims churches cause crime. Evaluate the claim.
Show the solutionHide the solution
- Step 1: The data are observational, and r only measures association.
- Step 2: City population is a likely confounding variable: bigger cities have more churches and more crimes.
- Step 3: No cause-and-effect conclusion is justified.
Answer: Not justified. Population size likely drives both counts. Correlation doesn't show causation.
Common mistakes
- Saying r = 0 means no relationship at all. It means no linear relationship.
- Using a high r as proof that a linear model is appropriate.
- Thinking that switching x and y or changing units changes r.
- Treating correlation as evidence of causation.
On the exam
- "Interpret r" requires direction, strength, the word linear and context.
- Multiple-choice questions often test what changes r (adding an unusual point) and what doesn't (units, switching axes).
Connected topics
Videos
Check yourself
4 questions on 5.2 Correlation. Pick an answer to see if you got it, and why.
The correlation between the heights (in inches) and weights (in pounds) of a group of adults is r = 0.68. If heights are converted to centimeters and weights to kilograms, what is the new correlation?
The correlation between the number of hours worked per week and weekly earnings for a group of employees is r = 0.74. What is the correlation if weekly earnings is used as the explanatory variable and hours worked as the response?
A scatterplot of a car's speed and its fuel efficiency shows a clear upside-down U shape: efficiency rises at low speeds, peaks around 50 mph, and then falls. The correlation is r = 0.05. Which statement is correct?
Which correlation indicates the strongest linear relationship?
0 of 4 answered