In probability theory and statistics, correlation, also called correlation coefficient, indicates the strength and direction of a linear relationship between two random variables. In general statistical usage, correlation or co-relation refers to the departure of two variables from independence. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of data. Probability theory is the mathematical study of probability. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

A number of different coefficients are used for different situations. The best known is the Pearson product-moment correlation coefficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations. Despite its name, it was first introduced by Francis Galton. In mathematics, and in particular statistics, the Pearson product-moment correlation coefficient (r) is a measure of how well a linear equation describes the relation between two variables X and Y measured on the same object or organism. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... In probability and statistics, the standard deviation is the most common measure of statistical dispersion. ... Francis Galton Sir Francis Galton F.R.S. (February 16, 1822 – January 17, 1911), half-cousin of Charles Darwin, was a Victorian polymath, British anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician, and statistician. ...

Pearson's product-moment coefficient

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 ρ_{X, Y} between two random variables X and Y with expected values μ_X and μ_Y and standard deviations σ_X and σ_Y is defined as: Download high resolution version (672x672, 69 KB)Simple example of linear correlation. ... Download high resolution version (672x672, 69 KB)Simple example of linear correlation. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability and statistics, the standard deviation is the most common measure of statistical dispersion. ...

Since μ_X = E(X), σ_X² = E(X²) − E²(X) and likewise for Y, we may also write

The correlation is defined only if both standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value. In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

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. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

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 dependent, 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. In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution) is a specific probability density function. ...

A correlation between two variables is diluted in the presence of measurement error around estimates of one or both variables, in which case disattenuation provides a more accurate coefficient. In measurement and statistics, disattenuation of a correlation between two sets of parameters or measures is the estimation of the correlation in a manner that accounts for measurement error contained within the estimates of those parameters. ...

The sample correlation

If we have a series of n  measurements of X  and Y  written as x_i  and y_i  where i = 1, 2, ..., n, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X  and Y . The Pearson coefficient is also known as the "sample correlation coefficient". It is especially important if X  and Y  are both normally distributed. The Pearson correlation coefficient is then the best estimate of the correlation of X  and Y . The Pearson correlation coefficient is written: In mathematics, and in particular statistics, the Pearson product-moment correlation coefficient (r) is a measure of how well a linear equation describes the relation between two variables X and Y measured on the same object or organism. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...

where and are the sample means of x_i  and y_i , s_x  and s_y  are the sample standard deviations of x_i  and y_i  and the sum is from i = 1 to n. As with the population correlation, we may rewrite this as In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). ... In probability and statistics, the standard deviation is the most common measure of statistical dispersion. ...

Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1. Though the above formula conveniently suggests a single-pass algorithm for calculating sample correlations, it is notorious for its numerical instability (see below for something more accurate).

The sample correlation coefficient is the fraction of the variance in y_i  that is accounted for by a linear fit of x_i  to y_i . This is written

where σ_y|x²  is the square of the error of a linear fit of y_i  to x_i  by the equation y = a + bx. In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

and σ_y²  is just the variance of y

Note that since the sample correlation coefficient is symmetric in x_i  and y_i , we will get the same value for a fit of x_i  to y_i :

This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. Just as the above described sample correlation coefficient is the fraction of variance accounted for by the fit of a 1-dimensional linear submanifold to a set of 2-dimensional vectors (x_i , y_i ), so we can define a correlation coefficient for a fit of an m-dimensional linear submanifold to a set of n-dimensional vectors. For example, if we fit a plane z = a + bx + cy  to a set of data (x_i , y_i , z_i ) then the correlation coefficient of z  to x  and y  is 2-dimensional renderings (ie. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Non-parametric correlation coefficients

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. Parametric inferential statistical methods are mathematical procedures for statistical hypothesis testing which assume that the distributions of the variables being assessed belong to known parametrized families of probability distributions. ... The branch of statistics known as non-parametric statistics is concerned with non-parametric statistical models and non-parametric statistical tests. ... In statistics, Spearmans rank correlation coefficient, named for Charles Spearman and often denoted by the Greek letter ρ (rho), is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency... In statistics, rank correlation is the study of relationships between different rankings on the same set of items. ...

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. In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. ... In probability theory and, in particular, information theory, the mutual information, or transinformation, of two random variables is a quantity that measures the mutual dependence of the two variables. ...

Copulas and correlation

Most people erroneously believe that the information given by a correlation coefficient is enough to define the dependence structure between random variables. But to fully capture the dependence between random variables we must consider the copula between them. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the cumulative distribution functions are elliptic (as with, for example, the multivariate normal distribution). In statistics, a copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 1]n such that every marginal distribution is uniform on the interval [0, 1]. Sklars theorem is as follows. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ...

Correlation matrices

The correlation matrix of n random variables X₁, ..., X_n is the n  ×  n matrix whose i,j entry is corr(X_i, X_j). If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X_i /SD(X_i) for i = 1, ..., n. Consequently it is necessarily a non-negative definite matrix. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. ...

The correlation matrix is symmetrical (the correlation between X_i and X_j is the same as the correlation between X_j and X_i).

"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). However, correlations are not presumed to be acausal, though the causes may not be known. ... Correlation implies causation, also known as cum hoc ergo propter hoc (Latin for with this, therefore because of this) and false cause, is a logical fallacy by which two events that occur together are claimed to be cause and effect. ... An acausal system is a system that depends on both the past and the future. ...

Computing correlation accurately in a single pass

The following algorithm (in pseudocode) will estimate correlation with good numerical stability

 sum_sq_x = 0 sum_sq_y = 0 sum_coproduct = 0 mean_x = x[1] mean_y = y[1] last_x = x[1] last_y = y[1] for i in 2 to N: sweep = (i - 1.0) / i delta_x = x[i] - mean_x delta_y = y[i] - mean_y sum_sq_x += delta_x * delta_x * sweep sum_sq_y += delta_y * delta_y * sweep sum_coproduct += delta_x * delta_y * sweep mean_x += delta_x / i mean_y += delta_y / i pop_sd_x = sqrt( sum_sq_x / N ) pop_sd_y = sqrt( sum_sq_y / N ) cov_x_y = sum_coproduct / N correlation = cov_x_y / (pop_sd_x * pop_sd_y) 

For an enlightening experiment, check the correlation of {900,000,000 + i for i=1...100} with {900,000,000 - i for i=1...100}, perhaps with a few values modified. Poor algorithms will fail.

External links

• Understanding Correlation - Introductory material by a U. of Hawaii Prof
• Statsoft Electronic Textbook
• Pearson's Correlation Coefficient - How to calculate it fast
• Learning by Simulations - The distribution of the correlation coefficient