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 $$1$$.

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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 $$r$$:

​$$r\lt0$$​​	​$$r=0$$​​	​$$r\gt 0$$​​

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.

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.