Measuring correlation

In a nutshell

The strength and type of linear correlation between two variables can be calculated by the product moment correlation coefficient, $r$ , which takes values between $-1$ and $1$ .

Product moment correlation coefficient

There are some properties you must know regarding this measure:

It takes values between $-1$ and $1$ .
The greater the value of $\vert r\vert$ , the stronger the correlation.
If $r\gt 0$ , there is a positive linear correlation between the variables.
If $r\lt 0$ , there is a negative linear correlation between the variables.
If $r=0$ , there is no linear correlation between the variables.

Graphical interpretation of $r$

Using the scatter graph of bivariate data, you can find the sign of $r$ :


$r\lt0$	$r=0$	$r\gt 0$

Maths; Correlation and hypothesis testing; KS5 Year 13; Measuring correlation

Calculator tip

$\boxed{\begin{aligned}Statistics\\2: y=a+bx \\ \begin{array}{c|c|c|} & x & y \\ \hline 1 & ... & ... \\ 2 & ... & ...\\3 & ... & ...\\4 & ... & ...\\\end{array}\\ \end{aligned}}$

Example

A group of students registered the temperature of a substance over time. The results are shown below.

TEMPERATURE ( $ºC$ )	$11$	$15$	$18$	$20$	$24$
Time ( $min$ )	$1$	$2$	$3$	$4$	$5$

Find the product moment correlation coefficient for the data and comment on their correlation.

	$\boxed{\begin{aligned}Statistics\\2: y=a+bx \\ \begin{array}{c\|c\|c\|} & x & y \\ \hline 1 & 1 & 11 \\ 2 & 2 & 15\\3 &3 & 18\\4 &4& 20\\5&5&24\\\end{array}\\ \end{aligned}}$
	$\boxed{r=0.994}$

Given that $r$ is positive and very close to $1$ , you can conclude that there is a very strong positive linear correlation between temperature and time.