Calculating Karl Pearson's Product Moment of Correlation

To calculate the Karl Pearson's Product Moment of correlation using the step-deviation method, we need to follow these steps:

Step 1: Calculate the mean of X and Y
- Calculate the mean of X (X̄) by summing up all the values of X and dividing by the number of values.
X = 4 + 6 + 8 + 10 + 12 = 40
X̄ = 40 / 5 = 8

- Calculate the mean of Y (Ȳ) by summing up all the values of Y and dividing by the number of values.
Y = 6 + 8 + 10 + 12 + 14 = 50
Ȳ = 50 / 5 = 10

Step 2: Calculate the deviation of X and Y from their respective means
- Calculate the deviation of X (dX) by subtracting the mean of X (X̄) from each value of X.
dX = 4-8, 6-8, 8-8, 10-8, 12-8 = -4, -2, 0, 2, 4

- Calculate the deviation of Y (dY) by subtracting the mean of Y (Ȳ) from each value of Y.
dY = 6-10, 8-10, 10-10, 12-10, 14-10 = -4, -2, 0, 2, 4

Step 3: Calculate the step-deviation for X and Y
- Determine the range of X by subtracting the lowest value from the highest value.
Range of X = 12 - 4 = 8

- Determine the range of Y by subtracting the lowest value from the highest value.
Range of Y = 14 - 6 = 8

- Determine the step size for X by dividing the range of X by the number of values.
Step size of X = 8 / 5 = 1.6

- Determine the step size for Y by dividing the range of Y by the number of values.
Step size of Y = 8 / 5 = 1.6

- Calculate the step-deviation for X (dx) by dividing the deviation of X (dX) by the step size of X.
dx = -4 / 1.6, -2 / 1.6, 0 / 1.6, 2 / 1.6, 4 / 1.6 = -2.5, -1.25, 0, 1.25, 2.5

- Calculate the step-deviation for Y (dy) by dividing the deviation of Y (dY) by the step size of Y.
dy = -4 / 1.6, -2 / 1.6, 0 / 1.6, 2 / 1.6, 4 / 1.6 = -2.5, -1.25, 0, 1.25, 2.5

Step 4: Calculate the product of step-deviations