Pearson's Correlation Coefficient

Origin of Pearson's Correlation Coefficient

In the 19th century, Darwin's theory of evolution stimulated interest in making various measurements of living organisms. One of the best-known contributors of that time, Sir Francis Galton, offered the statistical concept of correlation in addition to the law of heredity. In his 1888 article (cited in Stigler, 1986), “Co-Relations and Their Measurement, Chiefly from Anthropometric Data,” Galton demonstrated that the regression lines for the lengths of the forearm and head had the same slope (denoted as r) when both variables used the same scale for the measurement. He labeled the r as an index of co-relation and characterized its identity as a regression coefficient. The spelling of co-relation was deliberately chosen to distinguish it from the word correlation, which was already being used in other disciplines, although the spelling was changed to correlation within a year. However, it was not until Karl Pearson (1857–1936) that Pearson's correlation coefficient was introduced.

Computational Formulas

The Pearson's correlation coefficient is also called the Pearson product-moment correlation, defined by

because it is obtained by multiplying the z-scores of two variables (i.e., the products), followed by taking the average (i.e., the moment) of these products. Various computational formulas for Pearson's correlation coefficient can be derived from the above formula (Chen & Popovich, 2002), as demonstrated as follows:

Interpretation of Pearson's Coefficient

Pearson's correlation coefficient does not suggest a cause-and-effect relationship between two variables. It cannot be interpreted as a proportion, nor does it represent the proportionate strength of a relationship. For example, a correlation of .40 is not twice as strong a relationship compared to a correlation of .20. In contrast to Pearson's correlation coefficient, r2xy (also labeled as the coefficient of determination) should be interpreted as a proportion. Say the Pearson's correlation coefficient for class grade and intelligence is .7. The r2xy value of .49 suggests that 49% of the variance is shared by intelligence and class grade. It can also be interpreted that 49% of the variance in intelligence is explained or predicted by class grade and vice versa.

Special Cases of Pearson's Correlation Coefficient

There are several well-known simple correlations such as the point-biserial correlation coefficient, thephi coefficient, the Spearman correlation coefficient, and the eta coefficient. All of them are special cases of Pearson's correlation coefficient. Suppose we use Pearson's correlation coefficient to assess the relationship between a dichotomous variable (e.g., gender) and a continuous variable (e.g., income). This correlation is often referred to as the point-biserial correlation coefficient. We can also use it to gauge the relationship between two dichotomous variables (e.g., gender and infection), and we refer to this coefficient as the Phi coefficient. If both variables are measured at the ordinal level and ranks of their characteristics are assigned, Pearson's correlation coefficient can also be applied to gauge the relationship. This coefficient is often labeled as the Spearman correlation coefficient. Finally, Pearson's correlation coefficient, when applied to measure the relationship between a multichotomous variable (e.g., races, schools, colors) and a continuous variable, is referred to as the Eta coefficient (Wherry, 1984).

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