Correlation
Correlation measures the strength and direction of a linear relationship between two numerical variables, expressed as a single number — the Pearson correlation coefficient, most commonly — ranging from -1 (a perfect negative linear relationship, where one variable increases exactly as the other decreases) through 0 (no linear relationship) to +1 (a perfect positive linear relationship, where both variables increase together). Developed by Karl Pearson in the 1890s building on earlier work by Francis Galton, it remains the single most widely used measure for summarizing how two numerical variables move together, condensing a relationship that might involve thousands of individual data points into one easily communicated number.
The word "linear" in that definition is doing more load-bearing work than it might first appear, and overlooking it is the single most common way correlation gets misused: Pearson correlation specifically measures only the strength of a straight-line relationship, and it can report a correlation near zero even when two variables have a strong, obvious, but curved (non-linear) relationship — a U-shaped or perfectly sinusoidal pattern can produce a near-zero Pearson correlation despite the two variables clearly and predictably relating to each other, just not in a way a straight line captures. This is precisely the trap Anscombe's quartet was constructed to illustrate: datasets with nearly identical correlation coefficients that look completely different when actually plotted, which is why checking a scatter plot before trusting a reported correlation value is standard, required practice rather than an optional nicety.
Pearson correlation is also highly sensitive to outliers, since its underlying calculation involves each point's deviation from the mean of both variables, meaning a single extreme point can meaningfully inflate or deflate the reported correlation away from what the bulk of the data actually shows — which is why Spearman's rank correlation (developed by Charles Spearman in 1904), a related but distinct measure built on the ranks of values rather than their raw magnitudes, is often used as a more outlier-resistant alternative, and is also the technically correct choice for ordinal data, where Pearson's assumption of genuinely numerical, equally-spaced values doesn't actually hold.
Correlation strength is conventionally described in rough bands — weak, moderate, strong — but there's no single, universally agreed numeric cutoff for each category, and appropriate thresholds vary meaningfully by field: a correlation of 0.3 might be considered a notably strong and practically important finding in social science or psychology research, where human behavior is influenced by countless interacting factors, while the same 0.3 would be considered weak and unremarkable in a physics context, where tightly controlled experiments often produce correlations close to 0.9 or higher — which is why a reported correlation's practical significance always depends on domain context rather than any fixed numeric threshold applying universally.
Perhaps the single most important caveat attached to correlation, repeated so often across statistics education it's become something of a cliché — "correlation does not imply causation" — exists precisely because a correlation coefficient makes no distinction at all between three structurally different explanations for an observed relationship: one variable genuinely causing the other, the causal direction running the opposite way from what's assumed, or a third, unmeasured variable (a confounder) independently driving both variables in a way that produces an association between them with no direct causal link between the two variables actually being compared.
Read correlation as a single-number summary of linear association, powerful for its simplicity but limited exactly where that simplicity becomes a liability: its blindness to non-linear relationships is why a scatter plot should always accompany any reported correlation coefficient, its sensitivity to outliers is why Spearman's rank correlation exists as a more robust and, for ordinal data, more technically appropriate alternative, and its complete inability to distinguish causation from coincidence or confounding is the single most important limitation to keep in view whenever a correlation is used to justify a real-world decision.