Correlation coefficient

The correlation coefficient $r$ reports two things and nothing else: which way a cloud of points tilts, given by its sign, and how tightly the cloud hugs a straight line, given by its size. It is confined to the interval from $-1$ to $1$, and it reaches either end only when every point lies exactly on one line. It has no units and it does not change if you swap which variable is explanatory.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. For these points $r = 0.8107$, a fairly strong positive linear association, and squaring it gives $r^2 = 0.657$, so the least-squares line accounts for about 66 percent of the variation in the scores.

Two readings of 0.8107 are wrong. It is not a percentage of points on the line, and $r$ is not on a ratio scale, so "$r = 0.8$ is twice as strong as $r = 0.4$" fails. Halve this correlation to $r = 0.4054$ and the variation explained falls from 0.657 to 0.164, roughly a quarter as much rather than half. Strength comparisons belong to $r^2$, not to $r$.

The boundary is the word linear. An $r$ near 0 rules out a net straight-line trend and rules out nothing else. When the $x$-values are spaced symmetrically about $\bar{x}$ (x-bar) and $y$ depends only on the distance from that center, as on a parabola with its vertex at $\bar{x}$, the positive and negative contributions cancel exactly and $r = 0$ while $y$ is a perfect function of $x$. The symmetry is part of the condition, not decoration: the $x$-values 0, 1, 2, 3, 6 have $\bar{x} = 2.4$, and the parabola $y = -(x - 2.4)^2$ centered there gives $r = -0.6118$ instead. That case is worked through in does a correlation of zero mean no relationship.

Correlation is topic 5.2 in Unit 5, Regression Analysis.