Correlation tracks linear co-movement—not proof of cause
When two quantities rise and fall together, statistics speaks of correlation—usually a linear link measured by Pearson's coefficient. Utilities may dial down generation on mild days because demand tracks weather, yet the slogan still holds: correlation does not imply causation.
Correlation describes a statistical relationship between random variables or bivariate data, typically how tightly they align on a straight line. Broader association covers any shared variability. Independence implies zero correlation, but zero correlation need not imply independence—nonlinear dependence can hide from a linear gauge. Mutual information and distance covariance generalize the idea beyond Pearson's line.
Pearson's product-moment coefficient, refined by Karl Pearson from Francis Galton's insight, divides covariance by the product of standard deviations, so it lives between −1 and +1 by Cauchy–Schwarz. Plus one marks a perfect increasing line, minus one a perfect decreasing line, and values near zero mean little linear link. Independence forces the coefficient to zero; jointly normal variables are the special case where uncorrelatedness and independence coincide. Spearman's and Kendall's rank coefficients watch monotonic rather than strictly linear co-movement and should be read as different association measures, not mere robust substitutes for Pearson.
From data, the sample correlation r_xy estimates the population ρ using paired deviations from the sample means. Measurement error shrinks the attainable range below ±1. In simple linear regression, R² equals the square of the Pearson coefficient. A standard textbook joint distribution on X ∈ {0,1} and Y ∈ {−1,0,1} yields variances 2/9 and 2/3 yet ρ = 0 even though the variables are dependent—an explicit warning against equating "uncorrelated" with "unrelated."
Predictive use remains fair game: if electricity load correlates with temperature, planners can act on the pattern whether or not every causal arrow is mapped. The intellectual hygiene is simply not to upgrade that pattern into a proven mechanism without more evidence.
Source: Correlation
