Calculation of Correlation Coefficient

Overview

The correlation coefficient, denoted by $r$ , measures the strength and direction of a linear relationship between two variables.

Remember: You can use your calculator to input the data and easily find the correlation coefficient.

However, you can also use the following formula:

r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}

Where:

$x$ and $y$ are the two variables.
$n$ is the number of data points.
$\sum x$ , $\sum y$ , $\sum xy$ , $\sum x^2$ and $\sum y^2$ are summations of the respective terms.

Organise the data: Create a table with columns for $x$ , $y$ , $x^2$ , $y^2$ and $xy$
Calculate the sums: Compute $\sum x$ , $\sum y$ , $\sum x^2$ , $\sum y^2$ and $\sum xy$
Substitute into the formula: Use the values to calculate $r$ . The value of $r$ will range from -1 to +1:

infoNote

Problem: Calculate $r$ for the data below:

Solution:

Step 1: Create a table:

$x$	$y$	$x^2$	$y^2$	$xy$
1	2	1	4	2
2	4	4	16	8
3	6	9	36	18
4	8	16	64	32

Sums:

$\sum x = 10, \sum y = 20, \sum x^2 = 30, \sum y^2 = 120, \sum xy = 60$

Step 2: Substitute into the formula:

r = \frac{4(60) - (10)(20)}{\sqrt{[4(30) - 10^2][4(120) - 20^2]}}

r = \frac{240 - 200}{\sqrt{(120 - 100)(480 - 400)}}

r = \frac{40}{\sqrt{20 \times 80}} = \frac{40}{40} = 1

Answer: $r = 1$ (Perfect positive correlation).