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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.
The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is only partially correct. [4] The Pearson correlation can be accurately calculated for any distribution that has a finite covariance matrix, which includes
Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If Y always takes on the same values as X , we have the covariance of a variable with itself (i.e. σ X X {\displaystyle \sigma _{XX}} ), which is called the variance and is more commonly denoted as σ X 2 , {\displaystyle ...
The coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories." [ 6 ] The tetrachoric correlation coefficient should not be confused with the Pearson correlation coefficient computed by assigning, say, values 0.0 and 1.0 to represent the two levels of ...
An entity closely related to the covariance matrix is the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector , which can be written as = ( ()) ( ()), where is the matrix of the diagonal elements of (i.e., a diagonal matrix of the variances of for =, …,).
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables. [a] The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. [citation needed]
The coefficient of multiple correlation is known as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been ...
The classical measure of dependence, the Pearson correlation coefficient, [1] is mainly sensitive to a linear relationship between two variables. Distance correlation was introduced in 2005 by Gábor J. Székely in several lectures to address this deficiency of Pearson's correlation, namely that it can easily be zero for dependent variables.