Scatterplots, Association, and Lines of Best Fit

One point only tells you about one case. Thirty points let you see a pattern: does y tend to rise as x rises, fall as x rises, or do the two have no real relationship at all? That pattern is association, and it's only visible once you have enough points to see a trend instead of noise.

Three things describe an association: direction (positive means both variables rise together, negative means one rises as the other falls), form (does the cloud look like a straight line, or does it curve), and strength (how tightly the points hug that line or curve, versus scattering widely around it). A best-fit line is the straight line drawn through the cloud that stays as close as possible to every point at once. Its slope tells you, in context, how much y changes for each one-unit change in x. For points (2, 14) and (10, 46), slope is (46-14)/(10-2) = 32/8 = 4: each one-unit increase in x goes with a 4-unit increase in y on average.

Strength of association is not proof of cause. Two variables can rise together because one causes the other, because both are driven by a third factor, or by coincidence. A scatterplot alone never settles which. This sits right next to outlier questions: because a best-fit line answers to every point, including distant ones, one far-off outlier in a corner drags the whole line toward it, changing the slope and weakening how well the line fits the rest of the cloud.