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, and was introduced by Francis Galton. Probability theory is the mathematical study of probability. ... Statistics is the science and practice of developing knowledge through the use of empirical data expressed in quantitative form. ... 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 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 commonly used measure of statistical dispersion. ... Francis Galton Sir Francis Galton FRS (February 16, 1822 - January 17, 1911) was an English explorer, statistician, anthropologist, creator of modern eugenics (he coined the term), and investigator of the human mind. ...

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_xy 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 (and especially gambling), the expected value (or 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 odds are... In probability and statistics, the standard deviation is the most commonly used 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 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. 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... The graph of the absolute value function In mathematics, the absolute value (or modulus) 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.

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. 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. ...

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, especially in physics and engineering. ...

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. ... In probability and statistics, the standard deviation is the most commonly used 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.

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.

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 In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

"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 have causes. In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no logical connection, yet it may be implied that they do, due to a certain third, unseen factor (referred to as a confounding factor or lurking variable). The spurious relationship gives an... 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. ...

Correlation matrices

The correlation matrix of n random variables X₂, ..., X_n is the n  ×  n matrix whose i,j entry is corr(X_i, X_j). It 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. ...

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. 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. ... Non-parametric (or distribution-free) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics, make no assumptions about the frequency distributions of the variables being assessed. ... 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...

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. 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 information theory, the mutual information of two random variables is a quantity that measures the independence of the two variables. ... In statistics, a copula is a probability distribution on a unit cube [0, 1]n for which every marginal distribution is uniform on the interval [0, 1]. Sklars theorem is as follows. ...

External links

• Statsoft Electronic Textbook (http://www.statsoft.com/textbook/stathome.html)
• Pearson's Correlation Coefficient (http://www.vias.org/tmdatanaleng/cc_corr_coeff.html) - How to calculate it fast