Pearson's correlation coefficient, when applied to a population, is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. The form of the definition involves a 'product moment', that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables hence the modifier product-moment in the name. Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The naming of the coefficient is thus an example of Stigler's Law. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844. N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero. The correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). Several sets of ( x, y) points, with the correlation coefficient of x and y for each set. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation). As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. It is the ratio between the covariance of two variables and the product of their standard deviations thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. In statistics, the Pearson correlation coefficient ( PCC, pronounced / ˈ p ɪər s ən/) ― also known as Pearson's r, the Pearson product-moment correlation coefficient ( PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ― is a measure of linear correlation between two sets of data. For broader coverage of this topic, see Correlation coefficient.

Negative correlation examples graph