# Pearson product-moment correlation coefficient

While the modern theory of correlation and regression has its roots in the work of F. Galton, the version of the product-moment correlation coefficient in current use (2000) is due to K. Pearson [a2]. Pearson's product-moment correlation coefficient is a measure of the strength of a linear relationship between two random variables and (cf. also Random variable) with means , and finite variances , :

where is the covariance of and ,

It readily follows that , and that is equal to or if and only if each of and is almost surely a linear function of the other, i.e., () with probability (furthermore, and have the same sign). If , and are said to be uncorrelated. Independent random variables are always uncorrelated, however uncorrelated random variables need not be independent (cf. also Independence).

The term "product-moment" refers to the observation that , where denotes the th product moment of and about their means.

The coefficient also plays a role in linear regression (cf. also Regression analysis). If the regression of on is linear, then , and if the regression of on is linear, then . Note that the product of the two slopes is .

When and have a bivariate normal distribution (cf. also Normal distribution), is a parameter of the joint density function

with

Unlike the general situation, uncorrelated random variables with a bivariate normal distribution are independent.

For a random sample from a bivariate population, is estimated by the sample correlation coefficient (cf. also Correlation coefficient) , given by

If and denote, respectively, the vectors and , and denotes the angle between and , then

Further interpretations of can be found in [a3]. For details on the use of in hypothesis testing, and for large-sample theory, see [a1].

#### References

[a1] | O.J. Dunn, V.A. Clark, "Applied statistics: analysis of variance and regression" , Wiley (1974) |

[a2] | K. Pearson, "Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia" Philos. Trans. Royal Soc. London Ser. A , 187 (1896) pp. 253–318 |

[a3] | J.L. Rodgers, W.A. Nicewander, "Thirteen ways to look at the correlation coefficient" The Amer. Statistician , 42 (1988) pp. 59–65 |

**How to Cite This Entry:**

Pearson product-moment correlation coefficient. R.B. Nelsen (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pearson_product-moment_correlation_coefficient&oldid=18562