Understanding and Calculating Karl Pearson's Correlation Coefficient

Karl Pearson's correlation coefficient, often denoted as r, is a widely used statistical measure to determine the strength and direction of the linear relationship between two variables. This article will guide you through the formula, steps, and interpretation of r, as well as provide an example calculation for clarity.

What is Karl Pearson's Correlation Coefficient?

r measures the linear dependence between two variables. A value of r close to 1 indicates a strong positive linear relationship, while a value close to -1 indicates a strong negative linear relationship. If r is close to 0, it suggests no linear relationship between the variables.

Formula for Calculating r

The formula for calculating r is given by:

r

Where:

n number of pairs of scores x values of the first variable y values of the second variable sum xy sum of the product of paired scores sum x sum of the x values sum y sum of the y values sum x^2 sum of the squares of the x values sum y^2 sum of the squares of the y values

Steps to Calculate r

Collect Data

The first step is to gather pairs of data points for the two variables you want to analyze. These pairs will be used to form the sums in the r formula.

Calculate Sums

The next step is to compute the following sums:

sum x sum y sum xy sum x^2 sum y^2

Substitute Values

Plug the sums into the formula for calculating r and simplify.

Calculate r

Finally, perform the arithmetic to find the value of r.

Interpretation of r

The value of r can range from -1 to 1:

r 1: Indicates a perfect positive correlation r -1: Indicates a perfect negative correlation r 0: Indicates no correlation

Example Calculation

Let's go through an example calculation.

Step 1: Collect Data

We have the following data points:

x y 1 2 2 3 3 5 4 4 5 5

Step 2: Calculate Sums

Using the data, we calculate the following sums:

n 5

sum x 1 2 3 4 5 15

sum y 2 3 5 4 5 19

sum xy (1 * 2) (2 * 3) (3 * 5) (4 * 4) (5 * 5) 2 6 15 16 25 64

sum x^2 1^2 2^2 3^2 4^2 5^2 1 4 9 16 25 55

sum y^2 2^2 3^2 5^2 4^2 5^2 4 9 25 16 25 79

Step 3: Substitute Values into the Formula

Substitute these values into the formula:

r

Step 4: Calculate r

Calculate the final value:

r

This value of 0.848 indicates a strong positive correlation between the two variables.

By following these steps, you can calculate r for any given dataset, providing valuable insights into the relationship between variables.