Question Video: Understanding the Relationship between a Correlation Coefficient and a Scatter Plot of the Variables Mathematics

Video Transcript

If all points on a scatter diagram lie directly on a straight line of positive slope, what is the value of the product-moment correlation coefficient for this data set?

First, we will recall the definition of a product-moment correlation coefficient. The product-moment correlation coefficient is also known as Pearson’s correlation coefficient. The coefficient known as 𝑟 can take values in the closed interval from negative one to positive one and can tell us how strongly two continuous variables are linearly correlated. It will be helpful to picture 𝑟 on a number line from negative one to positive one. If 𝑟 is close to one, there is no correlation between the variables. If two variables have perfect positive or direct correlation, then 𝑟 equals one. If two variables have perfect negative or inverse correlation, then 𝑟 equals negative one.

All positive direct correlations are found to the right of zero, and all negative inverse correlations are found to the left of zero. Regardless of positive or negative, the weaker the correlation, the closer 𝑟 is to zero. And the stronger the correlation, the closer 𝑟 is to one or negative one.

In this example, we are told that all the points lie directly on a straight line of positive slope. This means that our product-moment correlation coefficient will be found somewhere above zero. We recall that the points on a scatter plot with a positive weak correlation coefficient generally increase from left or right, but the points are loosely spread apart. The points on a scatter plot with a positive strong correlation coefficient clump together closer to a straight line.

Finally, if the points on a scatter plot line up in a perfectly straight line with a positive slope, we have a correlation coefficient of 𝑟 equals one. This means that the scatter diagram being described in this example has a product-moment or Pearson’s correlation coefficient of positive one, whereas a negative correlation coefficient would represent points that follow more of a negative slope. And a positive 𝑟-value closer to zero means that the points of our scatter diagram are quite spread apart and only loosely follow the direction of a positive slope.