21.2.2 Product Moment Correlation Coefficient

Product Moment Correlation Coefficient

The PMCC is a mathematical measure of how linearly correlated two variables are. It is calculated as follows:

For a sample of n pairs of observations ($x_i, y_i$):

• $S_{xx} = \sum(x_i - \overline{x})^2 = \sum x_i^2 - \frac{\left(\sum x_i\right)^2}{n}$
• $S_{yy} = \sum(y_i - \overline{y})^2 = \sum y_i^2 - \frac{\left(\sum y_i\right)^2}{n}$
• $S_{xy} = \sum(x_i - \overline{x})(y_i - \overline{y}) = \sum x_iy_i - \frac{\sum x_i \sum y_i}{n}$

Formula

The Product Moment Correlation Coefficient is:

$$r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}= \frac{\sum{x_iy_i} - \frac{\sum{x_i} \sum{y_i}}{n}}{\sqrt{\Bigg[\bigg(\sum{x_i^2} - \frac{(\sum{x_i})^2}{n}\bigg)\bigg(\sum{y_i^2} - \frac{(\sum{y_i})^2}{n}\bigg)\Bigg]}}$$

As we can see from inside the $S_{xx}$ bracket, $x - \overline{x}$ measures the distance of each point from the mean.

The Meaning of $r$

• $r$ takes values between -1 and 1. A value of 1 corresponds to perfect positive correlation between $x$ and $y$.
• A value of -1 corresponds to perfect negative correlation between $x$ and $y$.

$r = 1 \Rightarrow$ Perfect positive linear correlation

$r = 1 \Rightarrow$ Perfect positive linear correlation

("Gradient has no bearing on correlation coefficient.")

$r = 0 \Rightarrow$ No linear correlation between $x$ and $y$

Here $r$ is close to $-1$

$\therefore$ Strong negative correlation

These vertical distances from the line are called residuals.

It is not always appropriate to calculate r, e.g., when the data is not linear. All of the above have the same value of $r$.