Scatter Plots & Correlation Revision Notes for Leaving Cert Mathematics

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.

Steps for Calculation

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:

$r = +1$ : Perfect positive correlation.
$r = -1$ : Perfect negative correlation.
$r = 0$ : No linear correlation.

Worked Examples

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Example 1: Positive Correlation

Problem: Calculate $r$ for the data below:

$x$	$y$
1	2
2	4
3	6
4	8

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).

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Example 2: Weak Negative Correlation

Problem: Calculate $r$ for the data below:

$x$	$y$
1	10
2	8
3	6
4	4

Solution:

Step 1: Create a table:

$x$	$y$	$x^2$	$y^2$	$xy$
1	10	1	100	10
2	8	4	64	16
3	6	9	36	18
4	4	16	16	16

Sums:

$\sum x = 10, \sum y = 28, \sum x^2 = 30, \sum y^2 = 216, \sum xy = 60$

Step 2: Substitute into the formula:

r = \frac{4(60) - (10)(28)}{\sqrt{[4(30) - 10^2][4(216) - 28^2]}}

r = \frac{240 - 280}{\sqrt{(120 - 100)(864 - 784)}}

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

Answer: $r = -1$ (Perfect negative correlation).

Summary

The correlation coefficient $r$ quantifies the strength and direction of a linear relationship between two variables.
Use the 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]}}

$r$ ranges from -1 to +1, with values near $0$ indicating weak or no correlation.
Steps to calculate:
1. Organise data into a table.
2. Calculate summations ( $\sum x$ , $\sum y$ , etc.).
3. Substitute into the formula.
Interpret $r$ : Positive values indicate direct correlation, negative values indicate inverse correlation.

Scatter Plots & Correlation (Leaving Cert Mathematics): Revision Notes