Pearson Product-Moment Correlation Coefficient

Among his many accomplishments, Sir Francis Galton first introduced the concept of correlation in a book titled Natural Inheritance, which was published in 1889. However, Karl Pearson is credited for extending the concept of correlation and for developing the product-moment correlation coefficient. Pearson's product-moment correlation coefficient is by far the most common index of the relationship between two variables, or bivariate relationship.

Pearson's product-moment correlation coefficient measures the degree to which the points in the scatterplot tend to cluster about a straight line. In other words, the product-moment correlation coefficient measures the degree of linear relationship between two variables. If we label the two variables of interest as x and y, then Pearson's product-moment correlation coefficient, denoted by r, is given by the following formula:

where

The first element of Equation 1 indicates that Pearson r is the ratio of the covariance between variables X and Y to the square root of the product of the x and y variances. Interestingly, only the numerators[Page 751] of the variance and covariance equation appear in the latter part of Equation 1 because the denominators cancel each other out. Furthermore, Equation 2 could be used to rewrite the Pearson r formula as

However, an even more interpretable way of expressing Pearson r is as follows:

where

That is, Sx and Sy are the standard deviations of the two variables. Thus, Equation 4 indicates that Pearson r is the average cross-product of the standardized x and y variable scores. The fact that Pearson r can be expressed as a product of two variables measured in standard units means that we can express the relationship between two variables even if they are measured on different scales. For example, x can be measured in inches and y can be measured in acres, or in pounds, or in dollars. Provided that both variables are converted to standard units, r can measure their association.

If only the numerator in Equation 4 was used to compute Pearson r, then r would increase as the number of observations in the data set increased, regardless of the true relationship between the two variables. Therefore, n − 1 is needed in the denominator of Equation 4 to provide a statistic that is independent of the sample size. This denominator guarantees that r always lies between −1 and +1, inclusive.

Pearson's correlation coefficient helps to determine both the magnitude and direction of pairwise variable relationships. The sign of the coefficient tells us whether the relationship is positive or negative. A positive correlation means that as the values of one variable increases, so do the values of the second variable. Conversely, a negative relationship indicates that an increase in one variable is associated with a decrease in the other variable. The numerical part of the correlation coefficient indicates the magnitude of the relationship. A value of zero indicates no linear relationship between the two variables, whereas the closer the correlation coefficient is to 1 or –1, the greater the relationship between the variables.

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