Pearson Product-Moment Correlation Coefficient

Definition

The population coefficient, which is typically abbreviated as ρ, or the Greek letter rho, is an index of the degree and direction of linear association between two continuous variables. These variables are usually denoted as X (commonly labeled as a predictor variable) and Y (commonly labeled as an outcome variable). Note, however, that the letters are arbitrary; the roles of the variables implied by these labels are irrelevant because the coefficient is a symmetric measure and takes on the same value no matter how two variables are declared.

The population value is estimated by calculating a sample coefficient, which is denoted by

Statistical Properties of the Coefficient

The coefficient can take on values between −1 and 1 (i.e., −1 ≤ ρ ≤ 1). The absolute value of the coefficient denotes the strength of the linear association [Page 1023]with |ρ| = 1 indicating a perfect linear association and ρ = 0 indicating no linear association. The sign of the coefficient indicates the direction of the linear association. When high scores on X correspond to high scores on Y and low scores on X correspond to low scores on Y the association is positive. When high scores on X correspond to low scores on Y and vice versa, the association is negative.

Bivariate scatterplots are often used to visually inspect the degree of linear association between two variables. Figure 1 demonstrates how an (X,Y) scatterplot might look when the population correlation between X and Y is negligible (e.g., ρ = .02). Note that there is no observable pattern in the dots; they appear to be randomly spaced on the plot.

Figure 2 demonstrates how an (X, Y) scatterplot might look when ρ = .90. There is an obvious pattern in the paired score points. That is, increases in X are associated with linear increases in Y at least on average.

Figure 3 demonstrates how an (X, Y) scatterplot might look when ρ = −.30. In this plot, increases in X are associated with linear decreases in Y at least on average. Due to the lower absolute value of the coefficient in this case, the pattern is not as easily identifiable.

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