Give the symbols for Pearson's correlation in the sample and in the
population

State the possible range for Pearson's correlation

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 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 linear relationship between
variables. Figure 1 shows a scatter plot for which r = 1.

Figure 1. A perfect positive linear relationship,
r = 1.

Figure 2. A perfect negative linear relationship, r = -1.

Figure 3. A scatter plot for which r = 0. Notice that there
is no relationship between X and Y.

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 Figure 4 and
described in the introductory section
has an r of 0.97.

Figure 4. Scatter plot of spousal ages,
r = 0.97.

Figure 5. Scatter plot of Grip Strength
and Arm Strength, r = 0.63.

The relationship between grip strength and arm
strength depicted in Figure 5 (also described in the introductory
section) is 0.63.