Thirteen Ways to Look at the Correlation Coefficient
1988/02/01 by Joseph Lee Rodgers, W. Alan Nicewander · 2,974 citations
Mathematics · #Advanced Statistical Methods and Models #Correlation #Correlation coefficient #Correlation ratio #Fisher transformation #Geometry #Mathematics #Statistics
paper · doi:10.1080/00031305.1988.10475524
published in The American Statistician 42(1), 59-66 (Taylor & Francis)
openalex publication_date 1988/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract
In 1885, Sir Francis Galton first defined the term “regression” and completed the theory of bivariate correlation. A decade later, Karl Pearson developed the index that we still use to measure correlation, Pearson's r. Our article is written in recognition of the 100th anniversary of Galton's first discussion of regression and correlation. We begin with a brief history. Then we present 13 different formulas, each of which represents a different computational and conceptual definition of r. Each formula suggests a different way of thinking about this index, from algebraic, geometric, and trigonometric settings. We show that Pearson's r (or simple functions of r) may variously be thought of as a special type of mean, a special type of variance, the ratio of two means, the ratio of two variances, the slope of a line, the cosine of an angle, and the tangent to an ellipse, and may be looked at from several other interesting perspectives.
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