In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables 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 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 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 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).

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 relationship between the conditional expectation of one variable given the other is not constant as the conditioning variable changes; broadly correlation in this specific sense is used when ${\displaystyle E(Y|X=x)}$ is related to ${\displaystyle x}$ in some manner (such as linearly, monotonically, or perhaps according to some particular functional form such as logarithmic). Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted ${\displaystyle \rho }$ or ${\displaystyle r}$, 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 coefficient – have been developed to be more robust than Pearson's and to detect less structured relationships between variables. Mutual information can also be applied to measure dependence between two variables.

The most familiar measure of dependence between two quantities is the Pearson product-moment 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 of the two variables by the product of their standard deviations. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton.

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.^{[citation needed]}

The population correlation coefficient ${\displaystyle \rho _{X,Y}}$ between two random variables ${\displaystyle X}$ and ${\displaystyle Y}$ with expected values ${\displaystyle \mu _{X}}$ and ${\displaystyle \mu _{Y}}$ and standard deviations ${\displaystyle \sigma _{X}}$ and ${\displaystyle \sigma _{Y}}$ is defined as:

$${\displaystyle \rho _{X,Y}=\operatorname {corr} (X,Y)={\operatorname {cov} (X,Y) \over \sigma _{X}\sigma _{Y}}={\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})] \over \sigma _{X}\sigma _{Y}},\quad {\text{if}}\ \sigma _{X}\sigma _{Y}>0.}$$

where ${\displaystyle \operatorname {E} }$ is the expected value operator, ${\displaystyle \operatorname {cov} }$ means covariance, and ${\displaystyle \operatorname {corr} }$ 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: