Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- The correlation coefficient r summarises the direction and strength of a linear association between two quantitative variables.
- You find r with technology; you interpret it in context.
- Model sentence: “r = 0.93 shows a strong, positive, linear association between trunk diameter and apple yield: trees with wider trunks tend to produce more apples.”
Key relationships
| Fact | Meaning |
|---|---|
| −1 ≤ r ≤ 1 | Any value outside this interval is an error |
| r > 0 / r < 0 | positive / negative association |
| |r| close to 1 | strong linear association; |r| close to 0 means weak or no linear association |
| r = ±1 | perfect linear association: all points on one line |
| r = 0 | no linear association (a curve is still possible) |
| Units | none; changing units does not change r |
| Swap x and y | r is unchanged |
| Rough working guide (not official) | |r| > 0.8 strong; 0.5 to 0.8 moderate; < 0.5 weak |
Assumptions and conventions
- Both variables are quantitative.
- r describes linear association only, so look at the scatterplot first.
- r is not resistant: one unusual point can change it a lot.
- An association in an observational study cannot show cause and effect; a well-designed experiment can.
Mistakes to avoid
- Judging strength by sign: −0.85 is stronger than 0.60.
- “r = 0 means no relationship”: only no linear relationship.
- “r near 1 means linear”: a curve can give r = 0.98. Check the plot.
- Giving r units, or saying it changes when units change.
- Confusing r with the slope: r is about closeness to a line, not steepness.
- Claiming cause from correlation; name a possible lurking variable instead.
- No context: name strength, direction, “linear” and both variables.
Quick self-check
- Heights are converted from cm to inches. The correlation with weight was 0.66. What is it now? (Still 0.66: r has no units)
- Which is the stronger linear association: r = −0.72 or r = 0.45? (r = −0.72, because 0.72 > 0.45)
- A scatterplot shows a perfect U-shape and r = 0. Is there an association? (Yes, a strong non-linear one; there is no linear association)
- Towns with more hospitals have more deaths (r = 0.88). Do hospitals cause deaths? (No. Population is a likely lurking variable; this is observational data)
Next: practice questions.