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In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. It is found by dividing their covariance by the product of their standard deviations.
Mathematical properties
Linear correlations between 1000 pairs of numbers. The data are graphed on the lower left and their correlation coefficients listed on the upper right. Each set of points correlates maximally with itself, as shown on the diagonal (all correlations = +1). The correlation <math>\rho_{xy}<math> between two random variables X and Y with expected values <math>\mu_X<math> and <math>\mu_Y<math> and standard deviations <math>\sigma_X<math> and <math>\sigma_Y<math> is defined as:
- <math>
\rho_{xy}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E((X-\mu_X)(Y-\mu_Y)) \over \sigma_X\sigma_Y}.<math>
It is defined only if both standard deviations are finite and at least one of them is nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.
The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from −1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are as far from being independent as two random variables can be, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are jointly normal, independence is equivalent to uncorrelatedness.
"Correlation does not imply causation"
The conventional dictum that "correlation does not imply causation" is treated in the article titled spurious relationship. See also Correlation implies causation (logical fallacy).
Statistical estimation of population correlations by sample correlations
If several values of X and Y have been measured, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X and Y.
The coefficient is especially important if X and Y are both normally distributed and follow the linear regression model.
Non-parametric statistics
Pearson's correlation coefficient is a parametric statistic, and it may be less useful if the underlying assumption of normality is violated. Non-parametric correlation methods, such as Spearman's ρ and Kendall's τ may be useful when distributions are not normal; they are a little less powerful than parametric methods if the assumptions underlying the latter are met, but are less likely to give distorted results when the assumptions fail.
Other measures of dependence among random variables
To get a measure for more general dependencies in the data (also nonlinear) it is better to use the correlation ratio which is able to detect almost any functional dependency, or mutual information which detects even more general dependencies. To fully capture the dependence between random variables we must consider the copula between them.
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