Suppose you have data for five people:
| Person | Height (in inches) | Weight (in pounds) |
|--------|---------------------|--------------------|
| A | 60 | 110 |
| B | 62 | 115 |
| C | 64 | 120 |
| D | 66 | 125 |
| E | 68 | 130 |
You want to calculate the correlation between height and weight.
1. **Calculate the mean of each variable:**
- Mean Height: `(60 + 62 + 64 + 66 + 68) / 5 = 64`
- Mean Weight: `(110 + 115 + 120 + 125 + 130) / 5 = 120`
2. **Find the deviations from the mean for each variable:**
For Height: `-4, -2, 0, 2, 4`
For Weight: `-10, -5, 0, 5, 10`
3. **Compute the product of these deviations for each pair and average them:**
- `(-4) * (-10) = 40`
- `(-2) * (-5) = 10`
- `0 * 0 = 0`
- `2 * 5 = 10`
- `4 * 10 = 40`
Average of these products: `(40 + 10 + 0 + 10 + 40) / 5 = 20`
4. **Compute the standard deviations:**
- For Height: `sqrt(((-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2) / 5) = sqrt((16 + 4 + 0 + 4 + 16) / 5) = sqrt(40 / 5) = sqrt(8) ≈ 2.83`
- For Weight: `sqrt(((-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2) / 5) = sqrt((100 + 25 + 0 + 25 + 100) / 5) = sqrt(250 / 5) = sqrt(50) ≈ 7.07`
5. **Calculate Pearson’s r:**
`r = 20 / (2.83 * 7.07) ≈ 20 / 20 = 1`
In this simplified example, Pearson's r = 1, indicating a perfect positive linear relationship between height and weight. As height increases, weight increases proportionally.
