Make a guess, then check it.
The correlation r is a single number that measures the direction and strength of a linear association. Train your eye with the guessing game, compute r from z-scores with every step shown, see which changes leave r alone, and practice deciding when a correlation can support cause and effect.
The correlation r measures the direction and strength of a linear association between two quantitative variables. It is always between −1 and 1.
Study the scatterplot, set your guess, and check it. Play several rounds to train your eye.
Make a guess, then check it.
Each point's z-score product is positive in the upper-right and lower-left quadrants (around the point of means) and negative in the other two. r adds up those products and divides by n − 1.
For each study, decide what the evidence supports.
The correlation r measures the direction and strength of the linear association between two quantitative variables. The sign gives the direction; the size gives the strength.
r = 1 or r = −1 only when every point lies exactly on a line. r near 0 means a weak linear association. Values near ±1 mean the points hug a line closely.
r is computed from z-scores, so it has no units. Converting inches to centimeters, or adding a constant to every value, leaves r unchanged. Swapping x and y also leaves r unchanged.
r only describes linear relationships. A strong curved pattern can have r near 0, and a curve can also have r near 1. Always look at the scatterplot before interpreting r.
Because r uses means and standard deviations, a single unusual point can make r much stronger or much weaker than the pattern of the rest of the data.
A high r does not mean x causes y. A lurking variable may drive both. Only a well-designed randomized experiment supports a cause-and-effect conclusion.
How common changes to the data affect the correlation.
| Change | Effect on r | Why |
|---|---|---|
| Convert units (multiply x or y by a positive number) | No change | z-scores are unchanged |
| Add a constant to every x or y | No change | Every point shifts the same amount |
| Swap the explanatory and response variables | No change | The formula is symmetric in x and y |
| Multiply x (or y) by a negative number | Sign flips, same size | The direction reverses |
| Add an outlier that fits the line but is far away | |r| usually increases | It strengthens the linear pattern |
| Add an outlier far from the pattern | |r| usually decreases | It weakens the linear pattern |
| Data follow a strong curve | r may be near 0 (or misleading) | r only measures linear association |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 5.2. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 5 terminology guide.