Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced /ˈpɪərsən/), also referred to as the Pearson's r or Pearson product-moment correlation coefficient (PPMCC), is a measure of the linear dependence (correlation) between two variables X and Y. It has a value between +1 and −1 inclusive, where 1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation. It is widely used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s. Early work on the distribution of the sample correlation coefficient was carried out by Anil Kumar Gain and R. A. Fisher.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. 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 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 formula for ρ is:
The formula for ρ can be expressed in terms of mean and expectation. Since
Then the formula for ρ can also be written as
The formula for ρ can be expressed in terms of uncentered moments. Since
the formula for ρ can also be written as
Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above. So if we have one dataset {x1,...,xn} containing n values and another dataset {y1,...,yn} containing n values then that formula for r is:
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