Measuring relationships
Correlation describes how strongly two numerical variables are linearly related, and in which direction.
The correlation coefficient r
The correlation coefficient r ranges from −1 to 1.
| r | Meaning |
|---|---|
| 1 | perfect positive correlation |
| 0.7 to 0.99 | strong positive |
| 0.3 to 0.7 | moderate positive |
| 0 to 0.3 | weak or none |
| 0 | no linear correlation |
| −0.3 to −0.7 | moderate negative correlation |
| −0.7 to −0.99 | strong negative |
| −1 | perfect negative |
(The cut-offs are rough guides.)
Seeing it in scatter plots
- r ≈ 0.95: points hug an upward line.
- r ≈ 0.5: an upward trend with lots of scatter.
- r ≈ 0: a cloud with no direction.
- r ≈ −0.85: points cluster along a downward line.
Examples
| Variables | Likely r |
|---|---|
| height and arm span | strong positive (≈ 0.9) |
| hours of practice and free-throw % | moderate positive |
| outdoor temperature and heating bill | strong negative |
| shoe size and IQ | ≈ 0 |
Calculating r
r is usually found with technology (CORREL in spreadsheets, LinReg on calculators). The formula compares how x and y vary together:
r = Σ[(x − x̄)(y − ȳ)] / √[Σ(x − x̄)² × Σ(y − ȳ)²]
r² (coefficient of determination)
r² tells the proportion of the variation in y explained by the linear relationship with x. If r = 0.8, r² = 0.64: about 64% of the variation is explained.
Correlation is not causation
A strong correlation doesn’t prove that x causes y.
| Correlation | Real explanation |
|---|---|
| ice cream sales and drownings | lurking variable: hot weather |
| number of firefighters and fire damage | bigger fires need more firefighters |
| shoe size and reading level (children) | age affects both |
To show causation, you need a well-designed experiment (random assignment, control group).
Limits of r
- r only measures linear relationships. A perfect U-shaped curve can have r ≈ 0.
- Outliers can strongly change r.
- r has no units and doesn’t change if you rescale the variables.
Worked examples
Example 1: r = −0.92. Describe it. Strong negative linear correlation.
Example 2: r = 0.6. What percent of variation is explained? r² = 36%.
Example 3: Cities with more libraries have more crime. Does reading cause crime? No; population size is a lurking variable.
Common mistakes
- Claiming causation from correlation.
- Thinking r = 0 means no relationship at all (there might be a non-linear one).
- Ignoring outliers.
Try it yourself
- Which shows a stronger relationship: r = −0.8 or r = 0.6?
- Describe r = 0.15.
- If r = 0.9, what is r²?
- Name a possible lurking variable: hours of TV watched and grades are negatively correlated.
Answers: 1) r = −0.8 2) weak positive 3) 0.81 4) e.g. time spent on homework, or sleep
Quick recap
- Correlation measures linear relationships; the correlation coefficient r ranges from −1 to 1.
- Sign gives direction (positive correlation or negative correlation); size gives strength.
- Correlation is not causation; watch for lurking variables.