Question

a. Describe what the Pearson correlation coefficient is and what the coefficient of determination is.

compare it to a real world example or a example problem

Answer 1

Pearson correlation coefficient : - It is the test statistics that measures the statistical relationship, or association, between two continuous variables. In which one variable is dependent variable and other variable is independent variable.

We can consider

Y = dependent variable

X = independent variable

We predict the value of variable Y using the corresponding value of variable X.

There are 3 types correlation occurs between these two variables (Y and X)

We use the notation for Pearson correlation coefficient is r

1) If r = 0, there is no correlation between Y and X.

2) If r < 0, there is negative correlation between Y and X.

3) If r > 0, there is positive correlation between Y and X.

We can find  coefficient of determination using the value of Pearson correlation coefficient(r) is as below

coefficient of determination = r2

For example : - If value of Pearson correlation coefficient is r = 0. 8423

then  coefficient of determination = r2 = (0. 8423)2 = 0.7094

Here, r = 0. 8423 and r2 =0.7094 =70.94%

Using value of coefficient of determination we can interpret : - the 70.94% of the variation of Y explained by the linear relationship with X.

The formula for finding  Pearson correlation coefficient (r) is as below

where

For example

Find Pearson correlation coefficient(r) and coefficient of determination(r2) for the following data

X	1004	975	992	935	985	932
Y	40	100	65	145	80	150

Solution : -

We construct the table is as below

X	Y	X2	Y2	XY
1004	40	1008016	1600	40160
975	100	950625	10000	97500
992	65	984064	4225	64480
935	145	874225	21025	135575
985	80	970225	6400	78800
932	150	868624	22500	139800
5823	580	5655779	65750	556315

  

  

We have to find Pearson correlation coefficient (r) is as below

1) We can say that Pearson correlation coefficient (r) = -0.9897 that is there is negative correlation between variable Y and variable X.

Pearson correlation coefficient (r) = -0.9897

2) coefficient of determination : -

coefficient of determination = r2 = (-0.9897)2 = 0.9795

Here, r = -0.9897 and r2 = 0.9795 =97.95%

Using value of coefficient of determination we interpret : - the 97.95% of the variation of Y explained by the linear relationship with X.