Scatterplot vs Correlation Coefficient

A scatterplot shows one point $(x, y)$ per individual, with the explanatory variable on the horizontal axis and the response variable on the vertical. Both numbers must come from the same individual: two lists of equal length measured on different units do not make a scatterplot. You read it for direction, form, strength, and any point that departs from the pattern.

Five students study 1, 2, 3, 4 and 5 hours and score 62, 68, 74, 76 and 85. The cloud rises from left to right and looks close to linear. The least-squares line is $\hat{y} = 56.8 + 5.4x$ (y-hat) and the correlation is $r = 0.986$.

The misreading is "the points look tight, so $r$ must be about 0.99." Strength cannot be read off reliably by eye, because how tight the cloud looks depends on the axis scaling and $r$ does not. Multiply every score by 3 and the plot stretches into a steeper, apparently tighter band, and $r$ is still 0.9859 to four places. Record hours in minutes instead and the plot flattens, and $r$ is unchanged again. Correlation is computed from standardized values, so multiplying either variable by a positive constant, or adding one to it, leaves the number alone while changing the picture completely.

A scatterplot also cannot report how many observations sit at one spot. Add four more students who each studied 3 hours and scored 74 and the plot looks identical, one dot at $(3, 74)$, while $r$ moves from 0.986 to 0.982. Overplotting like that can bury a whole subgroup in a dense region.

A correlation of zero is not the same as no relationship. The five points $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$ and $(2, 4)$ lie exactly on $y = x^2$ and give $r = 0$. The scatterplot shows that curve plainly, which is the reason to look at it before trusting any single summary number. Scatterplots are topic 5.1.

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.