Properties Of Correlation Coefficient

The correlation coefficient $r$ has several important properties.

• It has no unit — $r$ is a pure number, independent of the units of measurement. For instance, $r$ between height in feet and weight in kilograms could still be a plain 0.7.
• A negative $r$ means an inverse relation — the two variables move in opposite directions (when a commodity's price rises its demand falls; when the interest rate rises the demand for funds falls, funds being costlier).
• A positive $r$ means same-direction movement — e.g. a rise in the price of coffee (a substitute) raising the demand for tea, or better irrigation going with higher yield.

• $r$ always lies between $-1$ and $+1$, i.e. $-1 \le r \le 1$. A value outside this range signals a calculation error.
• $r$ is unaffected by change of origin and change of scale. Defining $U = \dfrac{X-A}{B}$ and $V = \dfrac{Y-C}{D}$ (with A, C assumed means and B, D common factors of the same sign) gives $r_{XY} = r_{UV}$. This property underlies the step-deviation short-cut.
• If $r = 0$, the variables are uncorrelated — there is no linear relation, though some other (non-linear) relation may still exist.
• If $r = 1$ or $r = -1$, the correlation is perfect and the relation is exactly linear.
• A high $|r|$ (close to $\pm 1$) means a strong linear relationship; a low $|r|$ (close to 0) means a weak one — though a weak linear value can still hide a non-linear relation.

A cautionary example: doctors sent to villages hit by an epidemic may show a positive correlation between the number of doctors and the number of deaths, the opposite of what one expects. The explanation lies outside the numbers — many reported deaths were terminal cases, the doctors' benefit shows only after a lag, and some deaths (e.g. from a sudden tsunami) may be unrelated to the epidemic altogether. Understanding the data must come before interpreting $r$.

Interpreting the size of $r$. Suppose the correlation coefficient between marks secured in English and Statistics works out to $0.1$ — the two subjects are positively correlated, but only weakly: a student's English marks and Statistics marks do not reliably move together, so someone strong in English may well be weak in Statistics. Had the coefficient instead been $0.9$, marks in the two subjects would move together almost in lock-step — a student doing well in English would almost invariably do well in Statistics too. …