Calculate Karl Pearson's coefficient of correlation between x and y for the following data and interpret the result:

• (1, 6)
• (2, 5)
• (3, 7)
• (4, 9)
• (5, 8)
• (6, 10)
• (7, 11)
• (8, 13)
• (9, 12)

To calculate Karl Pearson's coefficient of correlation (r) between the variables x and y, we first need to gather the data points and perform some calculations. The data provided is:

• (1, 6)
• (2, 5)
• (3, 7)
• (4, 9)
• (5, 8)
• (6, 10)
• (7, 11)
• (8, 13)
• (9, 12)

Step-by-Step Calculation

1. **Calculate the means** of x and y:

The mean of x (x̄) = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 9 = 5

The mean of y (ȳ) = (6 + 5 + 7 + 9 + 8 + 10 + 11 + 13 + 12) / 9 = 9

2. **Calculate the deviations** from the mean:

For each x and y, subtract the mean:

• x - x̄: -4, -3, -2, -1, 0, 1, 2, 3, 4
• y - ȳ: -3, -4, -2, 0, -1, 1, 2, 4, 3

3. **Calculate the products of the deviations**:

• (-4)(-3) = 12
• (-3)(-4) = 12
• (-2)(-2) = 4
• (-1)(0) = 0
• (0)(-1) = 0
• (1)(1) = 1
• (2)(2) = 4
• (3)(4) = 12
• (4)(3) = 12

4. **Sum of products**: 12 + 12 + 4 + 0 + 0 + 1 + 4 + 12 + 12 = 55

5. **Calculate the squares of the deviations**:

• For x: 16, 9, 4, 1, 0, 1, 4, 9, 16 (sum = 60)
• For y: 9, 16, 4, 0, 1, 1, 4, 16, 9 (sum = 60)

6. **Calculate r** using the formula:

r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² * Σ(y - ȳ)²]

Substituting the values:

r = 55 / √(60 * 60) = 55 / 60 = 0.9167

Interpretation of the Result

The coefficient of correlation, r = 0.9167, indicates a strong positive linear relationship between x and y. This means that as the values of x increase, the values of y also tend to increase significantly. In practical terms, this suggests that there is a consistent pattern where higher x values correspond to higher y values.