Calculating Pearson's Correlation Coefficient with Height and Weight Data

### Example: Height and Weight

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.