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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 most distributions encountered in practice.
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.
In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with , element
For example, a number of subjects might each be given three trials at the same task, and it is predicted that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page [16] and is usually referred to as Page's trend test for ordered alternatives.
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
For example, in a paired data set where each "pair" is a single measurement made for each of two units (e.g., weighing each twin in a pair of identical twins) rather than two different measurements for a single unit (e.g., measuring height and weight for each individual), the ICC is a more natural measure of association than Pearson's correlation.
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities.