Correlation

Correlation quantifies how closely two variables move together in a straight-line relationship. A correlation of 1 means they move in perfect lockstep upward, -1 means one rises exactly as the other falls, and 0 means no linear relationship is detectable between them.

The most common version, Pearson's r, is calculated from the covariance of the two variables divided by the product of their standard deviations, which standardizes the result to the -1 to 1 range regardless of the variables' original units. Because it only captures linear relationships, two variables can be strongly related in a curved or non-monotonic way and still show a correlation near zero, so a scatter plot alongside the number is essential.

Correlation is widely used to screen for relationships worth investigating further, but the single most important caveat in statistics applies directly to it: correlation does not imply causation. A high correlation can arise from one variable causing the other, the reverse, a third variable driving both, or coincidence, the extreme version of the last is spurious correlation. Aggregating data before checking correlation can also reverse or mask the true relationship, a pattern known as simpson's paradox. Establishing that a relationship is causal requires causal inference methods, not correlation alone.