Between which two values does the product-moment correlation coefficient lie?
The product-moment correlation coefficient, or PMCC as it’s often abbreviated, is a measure of the strength of linear correlation between two variables. It tells us how closely the points on a scatter plot of the data lie to their statistical line of best fit. Let’s consider some different types of scatter plots.
In the first scatter plot, we may conclude that there is relatively strong negative correlation between the two variables because the points on the scatter plot lie relatively close to a straight line with a negative slope. In the second scatter plot, we may conclude that there is no linear correlation between the two variables because the points do not lie close to any straight line. In the third scatter plot, we may say there is moderate positive correlation between 𝑥 and 𝑦 because, this time, the points lie fairly close to a straight line with a positive slope.
If, however, the points lie exactly on a straight line, then we say there is perfect negative linear correlation between the two variables if the line slopes downwards and perfect positive correlation if the line slopes upwards. The value of the product-moment correlation coefficient, which we denote using the letter 𝑟, is negative one for perfect negative correlation and positive one for perfect positive correlation. All other bivariate data sets will have a value of 𝑟 somewhere between these two values.
So, for example, the value of 𝑟 for the first data set may be approximately negative 0.8. It may be close to zero for the second data set and may be close to positive 0.6 for the third. The closer the value of the product-moment correlation coefficient is to negative or positive one, the stronger the correlation. And the closer it is to zero, the weaker the correlation. To answer the question then, the value of the product-moment correlation coefficient lies between negative one and positive one inclusive.