correlated

Correlated is an adjective used in statistics, probability, and related fields to describe a relationship between two quantities or events in which they tend to vary together. In statistical terms, correlation measures the strength and direction of a linear relationship between two variables. The most common summary statistic is Pearson's correlation coefficient r, which ranges from -1 to 1. Values near 1 indicate a strong positive linear relationship; values near -1 indicate a strong negative linear relationship; values near 0 indicate little to no linear association. Other measures include Spearman's rho and Kendall's tau, which assess monotonic relationships and are more robust to nonnormal data or outliers.

Correlation is not the same as causation. Two variables may be correlated for many reasons, including coincidence or a shared causal factor; conversely, a causal relationship may exist without strong observed correlation in some samples. Nonlinear relationships may exist with substantial dependence yet small Pearson r, and outliers can distort correlation estimates. The concept also appears in probability as correlation between events or random variables, where correlation indicates that the joint probability differs from the product of marginal probabilities in the simple independent case.

Correlation can be quantified using covariance, which measures joint variability but depends on the scales of the variables; correlation normalizes covariance to produce a scale-free measure. In practice, correlation is widely used in data analysis, finance (asset co-movements), signal processing, psychology, and meteorology, and a correlation matrix summarizes pairwise correlations among a set of variables.