What is the correlation coefficient r?

In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of r is always between +1 and –1.

The closer the value of r is to +1, the stronger the linear relationship. For example, suppose the value of Diesel prices are directly related to the prices of Bus tickets, with a correlation coefficient of +0.8. The relationship between Diesel prices and Bus fares has a very strong positive correlation since the value is close to +1. So if the price of Diesel decreases, Bus fares follow in tandem. If the price of Diesel increases, so does the prices of Bus fares.

The correlation coefficient, denoted by r, is a measure of the strength of the straight-line or linear relationship between two variables. Values between 0.4 and 0.8 (-0.4 and -0.8) indicate a moderate positive (negative) linear relationship via a fuzzy-firm linear rule.

We may interpret r values as under:

Exactly –1 is interpreted as a perfect downhill (negative) linear relationship

–0.80 is interpreted as a strong downhill (negative) linear relationship

–0.50 is interpreted as a moderate downhill (negative) relationship

–0.20 is interpreted as a weak downhill (negative) linear relationship

0 No linear relationship

+0.20 is interpreted as a weak uphill (positive) linear relationship

+0.50 is interpreted as a moderate uphill (positive) relationship

+0.80 is interpreted as a strong uphill (positive) linear relationship

Exactly +1 is interpreted as a perfect uphill (positive) linear relationship

Conclusion: Anytime the correlation coefficient, denoted as r, is greater than zero, it’s a positive relationship. Conversely, anytime the value is less than zero, it’s a negative relationship. A value of zero indicates that there is no relationship between the two variables. However, this is only for a linear relationship; it is possible that the variables have a strong curvilinear relationship.

Illustration:

Consider two variables crop yield (Y) and rainfall (X). Here construction of regression line of Y on X would make sense and would be able to demonstrate the dependence of crop yield on rainfall. We would then be able to estimate crop yield given rainfall.

The coefficient of X in the line of regression of Y on X is called the regression coefficient of Y on X. It represents change in the value of dependent variable (Y) corresponding to unit change in the value of independent variable (X).  For instance if the regression coefficient of Y on X is 0.48 units, it would indicate that Y will increase by 0.48 if X increased by 1 unit. A similar interpretation can be given for the regression coefficient of X on Y.

Related article:

What is Regression Analysis?