The correlation coefficient only describes the strength of linear relationships
The correlation coefficient, r, is a good description of the strength of a relationship provided the crosses in a scatterplot of the data are not scattered round a curve.
If the data are scattered round a curve, the relationship is called nonlinear and r may seriously underestimate its strength.
The data in the diagram below lie on a straight line with positive slope, so the correlation coefficient is exactly 1.0.
Use the slider under the diagram to make the relationship nonlinear. Even though there is still a perfect relationship (the data lie exactly on a curve and the value of Y can be predicted perfectly from the value of X), r is less than 1.0. Indeed, it may even be 0.0 if the curvature is extreme enough!!
We will revisit nonlinear relationships in a later section of this chapter.