Scatter Plots and Association

In this lesson you will: Plot paired numerical data and read a single dot as a single individual; Describe an association by its direction, form and strength, and point out outliers; Tell a weak association apart from no association at all.

One dot, one individual

Categories gave you a two-way table. Two numerical measurements give you a scatter plot instead, and the rule is simple: one dot for each thing measured, placed at its two values.

A phone came off the charger at 8 in the morning and its battery was read every hour. Nine readings, nine dots. The variable you think does the explaining — here, time — goes along the bottom. The variable you want to predict goes up the side.

Four things to say about any scatter plot

Describing a scatter plot is not a matter of taste. There are four things to report, and a full description touches all of them.

The battery plot, run through the list: negative direction, close to linear in form, strong, with nothing unusual. Four short phrases, and someone who has never seen the plot now knows what it looks like.

Weak is not the same as none

Twelve students reported their hours of sleep before a quiz and their score out of 20. The dots drift upward, but the band is wide: at seven hours the scores run from 15 to 18, and one student who slept eight hours scored below one who slept five.

That is a weak-to-moderate positive association. Sleep is not useless for predicting the score — the drift is real — but a prediction from this plot would come with a wide margin. Compare it with a plot where the dots are genuinely patternless, and the difference is visible.

And what no association looks like

The same twelve-ish idea, run on two things that have no business being related: shoe size and a history quiz percent. The dots sit in a horizontal smear. Move right and the heights do not trend either way.

Saying so is a real finding, not a failed one. A scatter plot that shows nothing has told you that one variable is no help in predicting the other, which is worth knowing before anybody builds a model on it.

Common mistake — Judging strength by how steep the trend is

Two plots. In the first, the dots creep upward very gently but every one of them sits almost exactly on the same line. In the second, the dots climb sharply but sprawl in a fat cloud. Kenji calls the first weak and the second strong.

He has swapped two different questions. Slope answers "how much does y change per unit of x?" — that is steepness. Strength answers "how close are the dots to the pattern?" — that is tightness.

A gentle, tight trend is a strong association with a small slope. A steep, sprawling trend is a weak association with a large slope. The two ideas are independent, and Lesson 11.4 gives strength its own number so the confusion has nowhere to hide.

How to catch it: Cover the numbers on both axes with your hand, so steepness stops being readable in units. Now ask one question only: how thick is the band of dots? Thin means strong. Then uncover the axes and talk about slope as a separate sentence.