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Concept
Pearson correlation coefficient
Summary
In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. 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. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. 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).
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. The naming of the coefficient is thus an example of Stigler's Law.
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. Given a pair of random variables (for example, Height and Weight), the formula for ρ is
where
is the covariance
is the standard deviation of
is the standard deviation of .
The formula for can be expressed in terms of mean and expectation. Since
the formula for can also be written as
where
and are defined as above
is the mean of
is the mean of
is the expectation.
The formula for can be expressed in terms of uncentered moments.
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Related concepts (38)
Correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Karl Pearson
Karl Pearson (ˈpɪərsən; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university statistics department at University College London in 1911, and contributed significantly to the field of biometrics and meteorology. Pearson was also a proponent of social Darwinism and eugenics, and his thought is an example of what is today described as scientific racism.
Uncorrelatedness (probability theory)
In probability theory and statistics, two real-valued random variables, , , are said to be uncorrelated if their covariance, , is zero. If two variables are uncorrelated, there is no linear relationship between them. Uncorrelated random variables have a Pearson correlation coefficient, when it exists, of zero, except in the trivial case when either variable has zero variance (is a constant). In this case the correlation is undefined.
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