Values of the Pearson Correlation

David M. Lane

Prerequisites

Introduction to Bivariate Data

Learning Objectives
1. Describe what Pearson's correlation measures
2. Give the symbols for Pearson's correlation in the sample and in the population
3. State the possible range for Pearson's correlation
4. Identify a perfect linear relationship

The Pearson product-moment correlation coefficient is a measure of the strength of the linear relationship between two variables. It is referred to as Pearson's correlation or simply as the correlation coefficient. If the relationship between the variables is not linear, then the correlation coefficient does not adequately represent the strength of the relationship between the variables.

The symbol for Pearson's correlation is "ρ" when it is measured in the population and "r" when it is measured in a sample. Because we will be dealing almost exclusively with samples, we will use r to to represent Pearson's correlation unless otherwise noted.

Pearson's r can range from -1 to 1. An r of -1 indicates a perfect negative linear relationship between variables, an r of 0 indicates no linear relationship between variables, and an r of 1 indicates a perfect positive relationship between variables. Figure 1 shows a scatter plot for which r = 1.

 Figure 1. A perfect linear relationship, r = 1. Figure 2 shows a perfect negative linear relationship. Notice that as X increases, Y decreases.

 Figure 2. A perfect negative linear relationship, r = -1. Figure 3 shows a scatter plot for which r = 0. Notice that there is no relationship between X and Y.

 Figure 3. There is no linear relationship between the variables, r = 0. With real data, you would not expect to get values of r of exactly -1, 0, or -1.