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Values of the Pearson Correlation

Module by: Erin Maloney

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

Pearson's rr can range from -1-1 to 11. An rr of -1-1 indicates a perfect negative linear relationship between variables, an rr of 00 indicates no linear relationship between variables, and an rr of 11 indicates a perfect positive relationship between variables. Figure 1 shows a scatter plot for which r=1 r1.

Figure 1: A perfect linear relationship, r=1 r1.
Figure 1 (r1.gif)

Figure 2 shows a perfect negative linear relationship. Notice that as XX increases, YY decreases.

Figure 2: A perfect negative linear relationship, r=-1 r-1.
Figure 2 (r2.gif)

Figure 3 shows a scatter plot for which r=0 r0. Notice that there is no relationship between XX and YY.

Figure 3: There is no linear relationship between the variables, r=0 r0.
Figure 3 (r3.gif)

With real data, you would not expect to get values of rr of exactly -1-1, 00, or 11. The data for spousal ages shown in Figure 4 and described in the introductory section has an rr of 0.970.97.

Figure 4: Scatter plot of spousal ages, r=0.97 r0.97.
Figure 4 (age_scatterplot.gif)

The relationship between grip strength and arm strength depicted in Figure 5 (also described in the introductory section) is 0.630.63.

Figure 5: Scatter plot of Grip Strength and Arm Strength, r=0.63 r0.63.
Figure 5 (strength.gif)

Glossary

Linear Relationship:
If the relationship between two variables is a perfect linear relationship, then a scatterplot of the points will fall on a straight line as shown in Figure 1. With real data, there is almost never a perfect linear relationship between two variables. The more the points tend to fall along a straight line the stronger the linear relationship. Figure 4 shows two variables (husband's age and wife's age) that have a strong but not a perfect linear relationship.

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