Karl Pearson's coefficient of correlation
Karl Pearson's r measures how closely two variables move together in a straight line. It runs from −1 to +1; 0 means no linear link.
Formulas
Reading the value of r (one common scheme)
| Value of r | Degree | What it means |
|---|---|---|
| +1 | Perfect positive | All points lie on a rising straight line |
| +0.75 to below +1 | High positive | Both rise together, closely |
| +0.25 to below +0.75 | Moderate positive | Both rise together, loosely |
| Above 0 to below +0.25 | Low positive | A weak link |
| 0 | No correlation | No straight-line link (a curved link may still exist) |
| Below 0 to −1 | Same bands, negative | One rises as the other falls, e.g. exercise and BMI |
| −1 | Perfect negative | All points lie on a falling straight line |
Example: exercise and BMI
Hours of exercise per week (x) and BMI (y) of 5 adults: (1, 29), (2, 27), (3, 26), (4, 24), (5, 24). Find Karl Pearson's r and interpret it.
- Means: x̄ = 15 ÷ 5 = 3; ȳ = 130 ÷ 5 = 26.
- dx = x − 3: −2, −1, 0, 1, 2
- dy = y − 26: 3, 1, 0, −2, −2
- dx²: 4, 1, 0, 1, 4, so Σdx² = 10
- dy²: 9, 1, 0, 4, 4, so Σdy² = 18
- dxdy: −6, −1, 0, −2, −4, so Σdxdy = −13
- r = −13 ÷ √(10 × 18) = −13 ÷ √180 = −13 ÷ 13.42 = −0.97 (rounded to 2 decimals).
- Check by the direct method: Σx = 15, Σy = 130, Σxy = 377, Σx² = 55, Σy² = 3398. r = (5 × 377 − 15 × 130) ÷ √[(5 × 55 − 15²) × (5 × 3398 − 130²)] = −65 ÷ √(50 × 90) = −65 ÷ 67.08 = −0.97.
r = −0.97, a high negative correlation: adults who exercised more had a lower BMI. This shows association only; it does not prove that exercise caused the lower BMI.
Merits and demerits
| Merits | Demerits |
|---|---|
| Gives both direction (+ or −) and strength in one number | Measures only straight-line links; can miss a curved one |
| Uses every pair of values | Strongly affected by extreme values (outliers) |
| Has no units, so any two variables can be compared | Takes longer to calculate than rank correlation |
| Not changed by a change of origin or scale, e.g. weight in kg or g | A high r does not prove cause and effect |
Nutrition researchers use r to study links such as sodium intake and blood pressure, or BMI and waist circumference. A strong r shows that two things move together, but only a controlled trial can show that one causes the other.
2 marks: “What is negative correlation? Give an example.” When one variable rises as the other falls, r lies between 0 and −1. Example: hours of exercise and BMI.
5 marks: “Calculate r for x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5.” x̄ = 3, ȳ = 4. Σdxdy = 6, Σdx² = 10, Σdy² = 6. r = 6 ÷ √60 = 6 ÷ 7.75 = 0.77 (2 decimals): high positive correlation.
10 marks: “Explain Karl Pearson's coefficient of correlation with its merits, demerits and an example.” Define correlation and its types, give the formula, explain the −1 to +1 scale, solve a 5-pair sum, list merits and demerits, and state that correlation is not causation.
The sign of r gives the direction; the size gives the strength. Close to ±1 is strong, close to 0 is weak.