Visual illustration of Pearson correlation coefficient r

[dhiraj]

From what I have understood about Pearson correlation coefficient I have created a visual illustration, I would like to know if this understanding looks correct.

Say I have a sample with 5 data points:-

x y
8 6
16 8
20 16
28 12
32 20

My goal is to calculate Pearson correlation coefficient between x and y.

So this is how the diagram I created looks like:-

View attachment 6472

I have done appropriate color coding.

So in this case the covariance between x and y is:-

\(\displaystyle cov(x,y) = \frac {\sum d_x d_y}{n-1} \)

\(\displaystyle d_x\) and \(\displaystyle d_y\) are the deviations (not standard deviation) from \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) respectively, these mean lines are shown in the diagram (red line for \(\displaystyle \bar{x}\) and the green line for \(\displaystyle \bar{y}\)).

Pearson correlation coefficient \(\displaystyle r = \frac{cov(x,y)}{S_x S_y} \)

Based on the diagram, standard deviations of x and y are:-

\(\displaystyle S_x = \sqrt{ \frac{\sum d_x^2}{n-1} }\)

\(\displaystyle S_y = \sqrt{ \frac{\sum d_y^2}{n-1} }\)

So replacing these in the formula for the correlation coefficient we get:-
\(\displaystyle r = \frac {\sum d_x d_y} { (n-1) \sqrt{ \frac{\sum d_x^2}{n-1} } \sqrt{ \frac{\sum d_y^2}{n-1} } } \)Is this interpretation correct with respect to the diagram I have shown? I know the signs of \(\displaystyle d_x\) and \(\displaystyle d_y\) will depend on which side of \(\displaystyle \bar{x}\) and \(\displaystyle \bar{y}\) , \(\displaystyle x\) and \(\displaystyle y\) appear.

 

[I like Serena]

Hi dhiraj!

It's all correct.
And note that the formula for $r$ can be simplified to:
$$ r = \frac {\sum d_x d_y} {\sqrt{ \sum d_x^2 } \sqrt{ \sum d_y^2 }}$$