Pearson correlation coefficient

The Pearson correlation coefficient (r) is a measure of linear association between two continuous variables, introduced by Karl Pearson in 1896. It ranges from −1 to +1, where +1 indicates a perfect positive linear relationship, −1 indicates a perfect negative linear relationship, and 0 indicates the absence of a linear relationship. The coefficient is computed as the covariance of the two variables divided by the product of their standard deviations, which normalizes the measure so that it is dimensionless and comparable across domains.

The Pearson coefficient is simultaneously one of the most widely used and most widely misused statistics in science. Its popularity rests on its computational simplicity and intuitive interpretation. Its misuse rests on the assumptions that are rarely checked: that the relationship is linear, that the variables are approximately normally distributed, that outliers are absent, and that the correlation reflects a direct relationship rather than a confounded one. In practice, these assumptions are frequently violated, and the resulting correlation coefficients are interpreted as evidence of causal relationships that do not exist. The coefficient measures only linear dependence; two variables can be perfectly deterministically related — for example, as a circle or a parabola — and still have a Pearson correlation of zero.

The Pearson correlation coefficient is the statistical equivalent of a hammer: indispensable when the problem is a nail, dangerous when the problem is anything else. The fact that it is taught as the default measure of association in most introductory courses is not a testament to its universality but to the pedagogical convenience of linear algebra. The world is full of non-linear relationships, and the Pearson coefficient systematically blinds us to them by design.