In

MathWorld page on the (cross-)correlation coefficient/s of a sample

Compute significance between two correlations

for the comparison of two correlation values. *

Proof that the Sample Bivariate Correlation has limits plus or minus 1

by Juha Puranen. * ttps://web.archive.org/web/20150407112430/http://www.biostat.katerynakon.in.ua/en/association/correlation.html Correlation analysis. Biomedical Statistics* R-Psychologis

Correlation

visualization of correlation between two numeric variables {{DEFAULTSORT:Correlation And Dependence Covariance and correlation Dimensionless numbers

statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...

, correlation or dependence is any statistical relationship, whether causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...

or not, between two random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

s or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are '' linearly'' related.
Familiar examples of dependent phenomena include the correlation between the height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abo ...

of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather
Extreme weather or extreme climate events includes unexpected, unusual, severe, or unseasonal weather; weather at the extremes of the historical distribution—the range that has been seen in the past. Often, extreme events are based on a locat ...

causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...

).
Formally, random variables are ''dependent'' if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, ''correlation'' is synonymous with ''dependence''. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficient
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two component ...

s, often denoted $\backslash rho$ or $r$, measuring the degree of correlation. The most common of these is the '' Pearson correlation coefficient'', which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as '' Spearman's rank correlation'' – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as ...

can also be applied to measure dependence between two variables.
Pearson's product-moment coefficient

The most familiar measure of dependence between two quantities is thePearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...

(PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides the covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

of the two variables by the product of their standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whi ...

s. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton
Sir Francis Galton, Fellow of the Royal Society, FRS Royal Anthropological Institute of Great Britain and Ireland, FRAI (; 16 February 1822 – 17 January 1911), was an English Victorian era polymath: a statistician, sociologist, psycholo ...

.
A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set.
The population correlation coefficient $\backslash rho\_$ between two random variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

$X$ and $Y$ with expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

s $\backslash mu\_X$ and $\backslash mu\_Y$ and standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whi ...

s $\backslash sigma\_X$ and $\backslash sigma\_Y$ is defined as:
$$\backslash rho\_\; =\; \backslash operatorname(X,Y)\; =\; =\; ,\; \backslash quad\; \backslash text\backslash \; \backslash sigma\_\backslash sigma\_>0.$$
where $\backslash operatorname$ is the expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

operator, $\backslash operatorname$ means covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

, and $\backslash operatorname$ is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms of moments is:
$$\backslash rho\_\; =$$
Correlation and independence

It is a corollary of theCauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...

that the absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...

of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), and some value in the open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

$(-1,1)$ in all other cases, indicating the degree of linear dependence
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
If the variables are independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...

, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables.
$$\backslash begin\; X,Y\; \backslash text\; \backslash quad\; \&\; \backslash Rightarrow\; \backslash quad\; \backslash rho\_\; =\; 0\; \backslash quad\; (X,Y\; \backslash text)\backslash \backslash \; \backslash rho\_\; =\; 0\; \backslash quad\; (X,Y\; \backslash text)\backslash quad\; \&\; \backslash nRightarrow\; \backslash quad\; X,Y\; \backslash text\; \backslash end$$
For example, suppose the random variable $X$ is symmetrically distributed about zero, and $Y=X^2$. Then $Y$ is completely determined by $X$, so that $X$ and $Y$ are perfectly dependent, but their correlation is zero; they are uncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...

. However, in the special case when $X$ and $Y$ are jointly normal, uncorrelatedness is equivalent to independence.
Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as ...

is 0.
Sample correlation coefficient

Given a series of $n$ measurements of the pair $(X\_i,Y\_i)$ indexed by $i=1,\backslash ldots,n$, the ''sample correlation coefficient'' can be used to estimate the population Pearson correlation $\backslash rho\_$ between $X$ and $Y$. The sample correlation coefficient is defined as :$r\_\; \backslash quad\; \backslash overset\; \backslash quad\; \backslash frac\; =\backslash frac\; ,$ where $\backslash overline$ and $\backslash overline$ are the sample means of $X$ and $Y$, and $s\_x$ and $s\_y$ are the corrected sample standard deviations of $X$ and $Y$. Equivalent expressions for $r\_$ are :$\backslash begin\; r\_\; \&=\backslash frac\; \backslash \backslash ;\; href="/html/ALL/s/pt.html"\; ;"title="pt">pt$ where $s\text{'}\_x$ and $s\text{'}\_y$ are the ''uncorrected'' sample standard deviations of $X$ and $Y$. If $x$ and $y$ are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of $r\_$, Pearson's product-moment coefficient.Example

Consider thejoint probability distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...

of and given in the table below.
:
For this joint distribution, the marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...

s are:
:$\backslash mathrm(X=x)=\; \backslash begin\; \backslash frac\; 1\; 3\; \&\; \backslash quad\; \backslash text\; x=0\; \backslash \backslash \; \backslash frac\; 2\; 3\; \&\; \backslash quad\; \backslash text\; x=1\; \backslash end$
:$\backslash mathrm(Y=y)=\; \backslash begin\; \backslash frac\; 1\; 3\; \&\; \backslash quad\; \backslash text\; y=-1\; \backslash \backslash \; \backslash frac\; 1\; 3\; \&\; \backslash quad\; \backslash text\; y=0\; \backslash \backslash \; \backslash frac\; 1\; 3\; \&\; \backslash quad\; \backslash text\; y=1\; \backslash end$
This yields the following expectations and variances:
:$\backslash mu\_X\; =\; \backslash frac\; 2\; 3$
:$\backslash mu\_Y\; =\; 0$
:$\backslash sigma\_X^2\; =\; \backslash frac\; 2\; 9$
:$\backslash sigma\_Y^2\; =\; \backslash frac\; 2\; 3$
Therefore:
: $\backslash begin\; \backslash rho\_\; \&\; =\; \backslash frac\; \backslash mathrm;\; href="/html/ALL/s/X-\backslash mu\_X)(Y-\backslash mu\_Y).html"\; ;"title="X-\backslash mu\_X)(Y-\backslash mu\_Y)">X-\backslash mu\_X)(Y-\backslash mu\_Y)$
Rank correlation coefficients

Rank correlation coefficients, such asSpearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation ( statistical dependence betwe ...

and Kendall's rank correlation coefficient (τ) measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other ''decreases'', the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...

, and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.Yule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co.
To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers $(x,y)$:
:(0, 1), (10, 100), (101, 500), (102, 2000).
As we go from each pair to the next pair $x$ increases, and so does $y$. This relationship is perfect, in the sense that an increase in $x$ is ''always'' accompanied by an increase in $y$. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if $y$ always ''decreases'' when $x$ ''increases'', the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared. For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.
Other measures of dependence among random variables

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is amultivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One ...

. (See diagram above.) In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence).
Distance correlation In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zer ...

was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence.
The Randomized Dependence Coefficient is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non-linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence.
For two binary variables, the odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (du ...

measures their dependence, and takes range non-negative numbers, possibly infinity: . Related statistics such as Yule's ''Y'' and Yule's ''Q'' normalize this to the correlation-like range . The odds ratio is generalized by the logistic model to model cases where the dependent variables are discrete and there may be one or more independent variables.
The correlation ratio In statistics, the correlation ratio is a measure of the curvilinear relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The measure is defined as the ''ratio'' of t ...

, entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

-based mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as ...

, total correlation, dual total correlation and polychoric correlation
In statistics, polychoric correlation{{Cite web, url=https://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/viewer.htm#procstat_corr_details14.htm, title=Base SAS(R) 9.3 Procedures Guide: Statistical Procedures, Second Edition, we ...

are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the coefficient of determination
In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
It is a statistic used ...

generalizes the correlation coefficient to multiple regression.
Sensitivity to the data distribution

The degree of dependence between variables and does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between and , most correlation measures are unaffected by transforming to and to , where ''a'', ''b'', ''c'', and ''d'' are constants (''b'' and ''d'' being positive). This is true of some correlationstatistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...

s as well as their population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using ...

analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to monotone transformations of the marginal distributions of and/or .
Most correlation measures are sensitive to the manner in which and are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.
Various correlation measures in use may be undefined for certain joint distributions of and . For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Measures of dependence based on quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile ...

s are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data were sampled.
Sensitivity to the data distribution can be used to an advantage. For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.
Correlation matrices

The correlation matrix of $n$ random variables $X\_1,\backslash ldots,X\_n$ is the $n\; \backslash times\; n$ matrix $C$ whose $(i,j)$ entry is :$c\_:=\backslash operatorname(X\_i,X\_j)=\backslash frac,\backslash quad\; \backslash text\backslash \; \backslash sigma\_\backslash sigma\_>0.$ Thus the diagonal entries are all identically one. 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 $X\_i\; /\; \backslash sigma(X\_i)$ for $i\; =\; 1,\; \backslash dots,\; n$. This applies both to the matrix of population correlations (in which case $\backslash sigma$ is the population standard deviation), and to the matrix of sample correlations (in which case $\backslash sigma$ denotes the sample standard deviation). Consequently, each is necessarily a positive-semidefinite matrix. Moreover, the correlation matrix is strictly positive definite if no variable can have all its values exactly generated as a linear function of the values of the others. The correlation matrix is symmetric because the correlation between $X\_i$ and $X\_j$ is the same as the correlation between $X\_j$ and $X\_i$. A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression. In statistical modelling, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, anautoregressive
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...

matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and Toeplitz.
In exploratory data analysis
In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but primar ...

, the iconography of correlations consists in replacing a correlation matrix by a diagram where the “remarkable” correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).
Nearest valid correlation matrix

In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed). In 2002, Higham formalized the notion of nearness using theFrobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m r ...

and provided a method for computing the nearest correlation matrix using the Dykstra's projection algorithm, of which an implementation is available as an online Web API.
This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure) and numerical (e.g. usage the Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...

for computing the nearest correlation matrix) results obtained in the subsequent years.
Uncorrelatedness and independence of stochastic processes

Similarly for two stochastic processes $\backslash left\backslash \_$ and $\backslash left\backslash \_$: If they are independent, then they are uncorrelated. The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.Common misconceptions

Correlation and causality

The conventional dictum that "correlation does not imply causation
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...

" means that correlation cannot be used by itself to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations ( tautologies), where no causal process exists. Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).
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 both? Or does some other factor underlie both? 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.
Simple linear correlations

The Pearson correlation coefficient indicates the strength of a ''linear'' relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of $Y$ given $X$, denoted $\backslash operatorname(Y\; \backslash mid\; X)$, is not linear in $X$, the correlation coefficient will not fully determine the form of $\backslash operatorname(Y\; \backslash mid\; X)$. The adjacent image showsscatter plot
A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data ...

s of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four $y$ variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (''y'' = 3 + 0.5''x''). 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. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...

which exerts enough influence to lower the correlation coefficient from 1 to 0.816. 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 visual examination of the data. 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. The Pearson correlation can be accurately calculated for any distribution that has a finite covariance matrix, which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a sufficient statistic
In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the par ...

if the data is drawn from a multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One ...

. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.
Bivariate normal distribution

If a pair $(X,Y)$ of random variables follows a bivariate normal distribution, the conditional mean $\backslash operatorname(X\; \backslash mid\; Y)$ is a linear function of $Y$, and the conditional mean $\backslash operatorname(Y\; \backslash mid\; X)$ is a linear function of $X$. The correlation coefficient $\backslash rho\_$ between $X$ and $Y$, along with the marginal means and variances of $X$ and $Y$, determines this linear relationship: :$\backslash operatorname(Y\backslash mid\; X)\; =\; \backslash operatorname(Y)\; +\; \backslash rho\_\; \backslash cdot\; \backslash sigma\_Y\backslash frac,$ where $\backslash operatorname(X)$ and $\backslash operatorname(Y)$ are the expected values of $X$ and $Y$, respectively, and $\backslash sigma\_X$ and $\backslash sigma\_Y$ are the standard deviations of $X$ and $Y$, respectively. The empirical correlation $r$ is anestimate
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...

of the correlation coefficient $\backslash rho$. A distribution estimate for $\backslash rho$ is given by$$\backslash pi\; (\backslash rho\; ,\; r)\; =\; \backslash frac\; (1\; -\; r^2)^\; \backslash cdot\; (1\; -\; \backslash rho^2)^\; \backslash cdot\; (1\; -\; r\; \backslash rho\; )^\; F\backslash !\backslash left(\backslash frac,-\backslash frac;\; \backslash nu\; +\; \backslash frac;\; \backslash frac\backslash right)$$where $F$ is the Gaussian hypergeometric function and $\backslash nu\; =\; N-1\; >\; 1$ . This density is both a Bayesian posterior density and an exact optimal confidence distribution density.
See also

*Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable a ...

* Canonical correlation
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors ''X'' = (''X''1, ..., ''X'n'') and ''Y'' ...

* Coefficient of determination
In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
It is a statistic used ...

* Cointegration
Cointegration is a statistical property of a collection of time series variables. First, all of the series must be integrated of order ''d'' (see Order of integration). Next, if a linear combination of this collection is integrated of order less ...

* Concordance correlation coefficient In statistics, the concordance correlation coefficient measures the agreement between two variables, e.g., to evaluate reproducibility or for inter-rater reliability.
Definition
The form of the concordance correlation coefficient \rho_c as
:\rho_c ...

* Cophenetic correlation
* Correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables re ...

* Correlation gap
* Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...

* Covariance and correlation
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in sim ...

* Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...

* Ecological correlation
In statistics, an ecological correlation (also ''spatial correlation'') is a correlation between two variables that are group means, in contrast to a correlation between two variables that describe individuals. For example, one might study the corr ...

* Fraction of variance unexplained
* Genetic correlation In multivariate quantitative genetics, a genetic correlation (denoted r_g or r_a) is the proportion of variance that two traits share due to genetic causes, the correlation between the genetic influences on a trait and the genetic influences on a ...

* Goodman and Kruskal's lambda
* Iconography of correlations
* Illusory correlation
* Interclass correlation
* Intraclass correlation
* Lift (data mining)
* Mean dependence
* Modifiable areal unit problem
* Multiple correlation
In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions th ...

* Point-biserial correlation coefficient
* Quadrant count ratio
* Spurious correlation
* Statistical arbitrage
* Subindependence
References

Further reading

* * *External links

MathWorld page on the (cross-)correlation coefficient/s of a sample

Compute significance between two correlations

for the comparison of two correlation values. *

Proof that the Sample Bivariate Correlation has limits plus or minus 1

by Juha Puranen. * ttps://web.archive.org/web/20150407112430/http://www.biostat.katerynakon.in.ua/en/association/correlation.html Correlation analysis. Biomedical Statistics* R-Psychologis

Correlation

visualization of correlation between two numeric variables {{DEFAULTSORT:Correlation And Dependence Covariance and correlation Dimensionless numbers