This article is about the correlation coefficient between two variables. The term correlation can also mean the cross-correlation of two functions or electron correlation in molecular systems. In Signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Electronic correlation refers to the interaction between Electrons in a quantum system whose Electronic structure is being considered
Several sets of (xy) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N. B. : the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

In probability theory and statistics, correlation, (often measured as a correlation coefficient), indicates the strength and direction of a linear relationship between two random variables. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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.

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. In Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values Despite its name, it was first introduced by Francis Galton. Sir Francis Galton FRS ( 16 February 1822 &ndash 17 January 1911) half-cousin of Charles Darwin, was an [1]

Pearson's product-moment coefficient

Mathematical properties

The correlation coefficient ρX, Y between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:

$\rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E((X-\mu_X)(Y-\mu_Y)) \over \sigma_X\sigma_Y},$

where E is the expected value operator and cov means covariance. In Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance Since μX = E(X), σX2 = E(X2) − E2(X) and likewise for Y, we may also write

$\rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^2)-E^2(X)}~\sqrt{E(Y^2)-E^2(Y)}}.$

The correlation is defined only if both of the 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, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky 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. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors 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. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other 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. In Probability theory and Statistics, two real-valued Random variables are said to be uncorrelated if their Covariance is zero However, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence. MVN redirects here For the airport with that IATA code in Mount Vernon Illinois, see Mount Vernon Airport.

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

The sample correlation

If we have a series of n  measurements of X  and Y  written as xi  and yi  where i = 1, 2, . . . , n, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X  and Y . In Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r The Pearson coefficient is also known as the "sample correlation coefficient". The Pearson correlation coefficient is then the best estimate of the correlation of X  and Y . The Pearson correlation coefficient is written:

$r_{xy}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.$
$r_{xy}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y},$

where $\bar{x}$ and $\bar{y}$ are the sample means of X  and Y , sx  and sy  are the sample standard deviations of X  and Y  and the sum is from i = 1 to n. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values As with the population correlation, we may rewrite this as

$r_{xy}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.$

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). In the mathematical subfield of Numerical analysis, numerical stability is a desirable property of numerical Algorithms The precise definition of stability

The square of the sample correlation coefficient, which is also known as the coefficient of determination, is the fraction of the variance in yi  that is accounted for by a linear fit of xi  to yi . In Statistics, the coefficient of determination, R 2, is the proportion of variability in a data set that is accounted for by a statistical model This is written

$r_{xy}^2=1-\frac{s_{y|x}^2}{s_y^2},$

where sy|x2  is the square of the error of a linear regression of xi  on yi  by the equation y = a + bx:

$s_{y|x}^2=\frac{1}{n-1}\sum_{i=1}^n (y_i-a-bx_i)^2,$

and sy2  is just the variance of y:

$s_y^2=\frac{1}{n-1}\sum_{i=1}^n (y_i-\bar{y})^2.$

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

$r_{xy}^2=1-\frac{s_{x|y}^2}{s_x^2}.$

This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. In statistics linear regression is a form of Regression analysis in which the relationship between one or more Independent variables and another variable called An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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 (xi , yi ), 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 (xi , yi , zi ) then the correlation coefficient of z  to x  and y  is

$r^2=1-\frac{s_{z|xy}^2}{s_z^2}.$

The distribution of the correlation coefficient has been examined by R. A. Fisher[2][3] and A. Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary K. Gayen. [4]

Geometric Interpretation of correlation

The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

Caution: This method only works with centered data, i. e. , data which have been shifted by the sample mean so as to have an average of zero. Some practitioners prefer an uncentered (non-Pearson-compliant) correlation coefficient. See the example below for a comparison.

As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0. 11, 0. 12, 0. 13, 0. 15, 0. 18).

By the usual procedure for finding the angle between two vectors (see dot product), the uncentered correlation coefficient is:

$\cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 2.93 } { \sqrt { 103 } \sqrt { 0.0983 } } = 0.920814711.$

Note that the above data were deliberately chosen to be perfectly correlated: y = 0. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 10 + 0. 01 x. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting x by E(x) = 3. 8 and y by E(y) = 0. 138) yields x = (−2. 8, −1. 8, −0. 8, 1. 2, 4. 2) and y = (−0. 028, −0. 018, −0. 008, 0. 012, 0. 042), from which

$\cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 0.308 } { \sqrt { 30.8 } \sqrt { 0.00308 } } = 1 = \rho_{xy},$

as expected.

Motivation for the form of the coefficient of correlation

Another motivation for correlation comes from inspecting the method of simple linear regression. In statistics linear regression is a form of Regression analysis in which the relationship between one or more Independent variables and another variable called As above, X is the vector of independent variables, xi, and Y of the dependent variables, yi, and a simple linear relationship between X and Y is sought, through a least-squares method on the estimate of Y:

$\ Y = X\beta + \varepsilon.\,$

Then, the equation of the least-squares line can be derived to be of the form:

$(Y - \bar{Y}) = \frac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2} (X - \bar{X})$

which can be rearranged in the form:

$(Y - \bar{Y})=\frac{r s_y}{s_x} (X-\bar{X})$

where r has the familiar form mentioned above :$\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.$

Interpretation of the size of a correlation

CorrelationNegativePositive
Small−0. 3 to −0. 10. 1 to 0. 3
Medium−0. 5 to −0. 30. 3 to 0. 5
Large−1. 0 to −0. 50. 5 to 1. 0

Several authors have offered guidelines for the interpretation of a correlation coefficient. Cohen (1988),[5] for example, has suggested the following interpretations for correlations in psychological research, in the table on the right.

As Cohen himself has observed, however, all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0. 9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater contribution from complicating factors.

Along this vein, it is important to remember that "large" and "small" should not be taken as synonyms for "good" and "bad" in terms of determining that a correlation is of a certain size. For example, a correlation of 1. 0 or −1. 0 indicates that the two variables analyzed are equivalent modulo scaling. Scientifically, this more frequently indicates a trivial result than an earth-shattering one. For example, consider discovering a correlation of 1. 0 between how many feet tall a group of people are and the number of inches from the bottom of their feet to the top of their heads.

Non-parametric correlation coefficients

Pearson's correlation coefficient is a parametric statistic and when distributions are not normal it may be less useful than non-parametric correlation methods, such as Chi-square, Point biserial correlation, Spearman's ρ and Kendall's τ. Parametric statistics are statistics where the population is assumed to fit any parametrized distributions (most typically the Normal distribution) Non-parametric statistics is a branch of Statistics concerned with non-parametric Statistical models and non-parametric inference, including non-parametric "Chi-square test" is often shorthand for Pearson's chi-square test. The point biserial correlation coefficient ( rpb) is a Correlation coefficient used when one variable (e In Statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendall's &tau or tau test(s) is a Non-parametric statistic 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 the entropy-based mutual information/total correlation which is capable of detecting 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 In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random In Probability theory and in particular in Information theory, total correlation (Watanabe 1960 is one of several generalizations of the Mutual information The latter are sometimes referred to as multi-moment correlation measures, in comparison to those that consider only 2nd moment (pairwise or quadratic) dependence.

The polychoric correlation is another correlation applied to ordinal data that aims to estimate the correlation between theorised latent variables. In Statistics, polychoric correlation is a technique for estimating the Correlation between two theorised normally distributed continuous Latent variables

Copulas and correlation

The information given by a correlation coefficient is not enough to define the dependence structure between random variables; to fully capture it we must consider a copula between them. In Statistics, a copula is used as a general way of formulating a multivariate distribution in such a way that various general types of dependence can be represented The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the cumulative distribution functions are the multivariate normal distributions. In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function MVN redirects here For the airport with that IATA code in Mount Vernon Illinois, see Mount Vernon Airport. In the case of elliptic distributions it characterizes the (hyper-)ellipses of equal density, however, it does not completely characterize the dependence structure (for example, the a multivariate t-distribution's degrees of freedom determine the level of tail dependence).

Correlation matrices

The correlation matrix of n random variables X1, . . . , Xn is the n  ×  n matrix whose i,j entry is corr(XiXj). 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 Xi /SD(Xi) for i = 1, . In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector . . , n. Consequently it is necessarily a positive-semidefinite matrix. In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number.

The correlation matrix is symmetric because the correlation between Xi and Xj is the same as the correlation between Xj and Xi.

Removing correlation

It is always possible to remove the correlation between zero-mean random variables with a linear transform, even if the relationship between the variables is nonlinear. Suppose a vector of n random variables is sampled m times. Let X be a matrix where Xi,j is the jth variable of sample i. Let Zr,c be an r by c matrix with every element 1. Then D is the data transformed so every random variable has zero mean, and T is the data transformed so all variables have zero mean, unit variance, and zero correlation with all other variables. The transformed variables will be uncorrelated, even though they may not be independent. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other

$D = X -\frac{1}{m} Z_{m,m} X$

$T = D (D^T D)^{-\frac{1}{2}}$

where an exponent of -1/2 represents the matrix square root of the inverse of a matrix. In Mathematics, the square root of a matrix extends the notion of Square root from numbers to matrices. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- The covariance matrix of T will be the identity matrix. If a new data sample x is a row vector of n elements, then the same transform can be applied to x to get the transformed vectors d and t:

$d = x - \frac{1}{m} Z_{1,m} X$

$t = d (D^T D)^{-\frac{1}{2}}.$

Correlation and causality

The conventional dictum that "correlation does not imply causation" means that correlation cannot be validly used to infer a causal relationship between the variables. Correlation does not imply causation is a phrase used in the Sciences and Statistics to emphasize that Correlation between two variables does not imply Correlation does not imply causation is a phrase used in the Sciences and Statistics to emphasize that Correlation between two variables does not imply This dictum should not be taken to mean that correlations cannot indicate causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).

Here is a simple example: hot weather may cause both a reduction in purchases of warm clothing and an increase in ice-cream purchases. Therefore warm clothing purchases are correlated with ice-cream purchases. But a reduction in warm clothing purchases does not cause ice-cream purchases and ice-cream purchases do not cause a reduction in warm clothing purchases.

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health? Or does good health lead to good mood? Or does some other factor underlie both? Or is it pure coincidence? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

Correlation and linearity

Four sets of data with the same correlation of 0. 81

While Pearson correlation indicates the strength of a linear relationship between two variables, its value alone may not be sufficient to evaluate this relationship, especially in the case where the assumption of normality is incorrect.

The image on the right shows scatterplots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. A scatter graph or scatter plot is a type of Display using Cartesian coordinates to display values for two Variables for a set of data Anscombe's quartet comprises four Datasets which have identical simple statistical properties yet which are revealed to be very different when inspected graphically Francis John (Frank Anscombe ( 1918-05-13 – 2001-10-17) was an English Statistician. [6] The four y variables have the same mean (7. 5), standard deviation (4. 12), correlation (0. 81) and regression line (y = 3 + 0. 5x). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0. In Statistics, an outlier is an observation that is numerically distant from the rest of the data. 81. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as a summary statistic, cannot replace the individual examination of the data.

Computing correlation accurately in a single pass

The following algorithm (in pseudocode) will calculate Pearson correlation with good numerical stability. Pseudocode is a compact and informal high-level description of a Computer programming Algorithm that uses the structural conventions of some Programming language In Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r

sum_sq_x = 0sum_sq_y = 0sum_coproduct = 0mean_x = x[1]mean_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 / Ncorrelation = cov_x_y / (pop_sd_x * pop_sd_y)