Values of the Pearson Correlation2.12002/12/022003/07/07 14:15:57.869 GMT-5ErinMaloneyemaloney@rice.eduErinMaloneyemaloney@rice.edu
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.
shows a scatter plot for which r1.
A perfect linear relationship, r1.
shows a perfect negative linear relationship.
Notice that as X increases,
Y decreases.
A perfect negative linear relationship, r-1.
shows a scatter plot for which r0.
Notice that there is no relationship between X and
Y.
There is no linear relationship between the variables, r0.
With real data, you would not expect to get values of r
of exactly -1, 0, or
1. The data for spousal ages shown in
and described in the introductory section has an
r of 0.97.
Scatter plot of spousal ages, r0.97.
The relationship between grip strength and arm strength depicted in
(also described in the introductory section) is
0.63.
Scatter plot of Grip Strength and Arm Strength, r0.63.
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 .
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. shows two variables (husband's age and wife's age) that have a strong but not a perfect linear relationship.