correlations

Correlation is a statistical measure that describes how two variables relate to each other. It quantifies both the strength and the direction of an association, whether the relationship is approximately linear or monotonic, and it is symmetric: the correlation between X and Y is the same as between Y and X. Correlation does not by itself imply that one variable causes changes in the other.

Several coefficients are used. Pearson r assesses linear relationships for interval or ratio data and ranges from -1 to 1, with values near 1 or -1 indicating a strong straight-line relationship and values near 0 indicating little linear association. Spearman rho and Kendall tau are nonparametric measures based on ranks and are suitable for ordinal data or nonnormal relationships; both also range from -1 to 1. Interpretation follows that direction is positive or negative; magnitude is strength but context matters.

Correlation is often summarized in a correlation matrix for many variables and visualized with scatter plots. Partial or control correlation removes the effect of one or more confounding variables. In time series analysis, autocorrelation measures how a variable relates to itself at a later time.

Limitations include sensitivity to outliers, restricted data ranges, and nonlinearity that can mask relationships. Spurious correlations can arise from coincidences or confounding factors. Causality requires additional evidence from experiments or causal modeling rather than correlation alone.

Common uses span science, engineering, economics, and finance, including feature selection, risk assessment, and pattern discovery. Researchers report the chosen coefficient, sample size, and confidence intervals to convey precision and reliability.