Scatterplot

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.