Correlation describes a statistical relationship between variables in which changes in one variable are associated with changes in another according to a specified pattern. In its common statistical use, correlation measures the degree and direction of association between variables, with the Pearson correlation coefficient commonly used for linear relationships between quantitative variables. A positive correlation indicates that higher values of one variable tend to occur with higher values of another, while a negative correlation indicates that higher values of one tend to occur with lower values of the other. A correlation near zero indicates little or no linear association but does not necessarily mean that no relationship exists.
Correlation coefficients are commonly scaled from negative one to positive one for measures such as Pearson correlation. A value near positive one indicates a strong positive linear association, a value near negative one indicates a strong negative linear association and a value near zero indicates weak linear association. The magnitude describes association under the assumptions of the selected coefficient, while the sign describes direction. Different correlation measures have different requirements. Spearman correlation is based on ranks and can be useful for monotonic relationships or ordinal data, while other measures are appropriate for different variable types or analytical settings.
Correlation is distinct from Causation. If two variables are correlated, the observation alone does not establish that changes in one variable produce changes in the other. A correlation may arise because one variable influences another, because the second influences the first, because both are influenced by another factor or because of sampling variation. For example, two measurements may increase together because both respond to a separate environmental condition. Establishing causation requires stronger evidence and reasoning than demonstrating statistical association, often involving experimental design, temporal evidence, control of confounding and a plausible causal mechanism.
Correlation can be affected by outliers and by the shape of the relationship between variables. Pearson correlation measures linear association and can therefore be low when variables have a strong nonlinear relationship. A single extreme observation can substantially alter a correlation coefficient. Researchers should inspect data visually and consider whether the assumptions of the selected measure are reasonable. Scatterplots are particularly useful because they can reveal clusters, curvature, heteroscedasticity and influential observations that a single numerical coefficient can conceal.
Correlation also depends on the population and sampling range. Restricting the range of one variable can reduce the observed correlation even when a relationship exists in a broader population. Aggregated data can produce relationships that differ from relationships observed at the individual level. This relates to ecological inference and aggregation effects, where relationships among groups do not necessarily represent relationships among individuals. Researchers should therefore identify the unit of analysis and population represented by a correlation rather than treating a coefficient as a universal property of two variables.
Statistical significance of a correlation concerns whether the observed association is compatible with a specified null model under particular assumptions. It does not indicate the practical importance of the relationship and does not establish causation. With a very large sample, a very small correlation can be statistically detectable, while a meaningful association can fail to reach conventional significance in a small sample. Confidence intervals around correlation estimates provide information about sampling uncertainty and can be more informative than reporting a coefficient alone.
Correlation is widely used in exploratory analysis, scientific research, finance, social science, psychology, epidemiology, engineering and many other fields. Researchers use it to identify potentially related variables, develop predictive models and investigate patterns that may justify further study. However, correlations found through extensive searching across many variables can arise by chance, creating an increased risk of false discoveries. Predefined hypotheses, correction for multiple comparisons, replication and appropriate validation can help distinguish robust relationships from patterns produced by extensive exploratory searching.
Correlation is most informative when its mathematical definition, variable types, sampling design and research context are clear. A correlation coefficient is not the same as a regression coefficient, although the concepts are related. Correlation is symmetric in the sense that the ordinary Pearson coefficient between two variables does not designate one as the explanatory cause and the other as the outcome. Regression can instead specify a dependent variable and predictor variables, although regression alone still does not automatically establish causation. Correct interpretation therefore requires attention to design, assumptions, context and the distinction between association and causal effect.
