Linear regression and correlation
The regression line describes the linear relation between two variables; the correlation coefficient shows its strength and direction.
Formulas
- b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)²
- a = ȳ − b · x̄
- r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²), −1 ≤ r ≤ 1
Your turn
Three problems on this topic. Comma or point, fractions like 1/6 work too.
Task 1 of 3Regression & correlation
Task
Work out Pearson's correlation coefficient for these pairs.
| x | y |
|---|---|
| 1 | 5 |
| 3 | 3 |
| 4 | 0 |
| 5 | 1 |
| 7 | -6 |
r (2 decimal places)
Comma or point; fractions like 1/6 work too.
Worked example
Study hours x and points y were measured. Find the regression line y = a + b·x.
Worked solution
- 1.x̄ = 3.40, ȳ = 25.20
- 2.b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² = 75.60 / 17.20 = 4.40
- 3.a = ȳ − b · x̄ = 25.20 − 4.40 · 3.40 = 10.26
Typical mistake: Dividing by Σ(yᵢ − ȳ)² instead of Σ(xᵢ − x̄)² for b.
Work out Pearson's correlation coefficient for these pairs.
Worked solution
- 1.r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²)
- 2.= 52.80 / √(22.80 · 134.80) = 0.95
- 3.|r| close to 1 means a strong linear relation; the sign shows the direction.
Typical mistake: Reading correlation as causation.
Frequently asked questions
Does high correlation mean causation?
No. A relation can come from a third variable or chance.