Interpret scatterplots

A scatterplot shows two-variable data: each dot is one observation, measured two ways. How far right the dot sits (its $x$-coordinate) gives one variable. How high it sits (its $y$-coordinate) gives the other. Both numbers belong to the same observation.

In the opening plot, $(8,41)$ is one delivery that went $8$ miles and took $41$ minutes. It doesn’t mean there were $8$ deliveries or $41$ distances.

So when a question names one delivery, you read one dot. When it asks what happens as $x$ increases, step back and read the whole cloud of points.

An association is the overall pattern in how two variables go together. When one is large, what does the other tend to be? You can describe any association by asking three questions: which way, what shape, how tight?

1. Direction

Which way does the cloud go? Read it from left to right.

• Positive association: as $x$ gets larger, $y$ tends to get larger too.
• Negative association: as $x$ gets larger, $y$ tends to get smaller.
• No clear association: the points don’t lean consistently up or down.

The word tend matters. In a positive association, not every dot has to be higher than the one before it.

2. Form

What shape do the points follow?

• Linear: the cloud runs along a roughly straight path.
• Nonlinear: the cloud bends into a curve or some other shape that isn’t straight.

Direction and form are separate questions. A cloud can rise and curve at the same time, so it’s positive and nonlinear.

3. Strength

How tightly do the points hug that shape?

• Strong: the points stay close to one clear path.
• Weak: the points spread out widely but still lean one way.

Strength is about how close the points stay to the pattern, not how steep the pattern is. A gentle rise can be strong, and a steep one can be weak.

Once you know which way, what shape and how tight, look for two things those labels can miss.

A cluster is a group of points bunched together in one part of the plot. Two clusters might mean two different groups or conditions, but the scatterplot alone can’t tell you what made them.

An outlier is a point that sits far from the overall pattern. Compare it with the pattern, not only with the dot next to it. A point can have a very large $x$-value or $y$-value and still fit the trend.

To check a point you’re unsure about:

1. Cover it up in your mind.
2. Describe the pattern the other points make.
3. Uncover it and ask whether it’s unusually far from that pattern.

Covering a point is only a way to read the plot. Don’t drop it from a calculation unless the question tells you to.

You can usually answer a scatterplot question by reading the plot. Here’s the routine:

1. Read the axis labels and the scale, so you know what each dot’s two numbers mean.
2. If the question names one value, like a distance of $8$ miles, find that dot and read its other coordinate.
3. If the question asks about the relationship, scan the whole cloud for which way, what shape and how tight.
4. Check for clusters and outliers.
5. Put the pattern in the variables’ own words, like “longer deliveries tend to take more time.”

Typing the points into Desmos won’t make this reading any better, and every point you copy is a chance for a typo. Read the plot you’re given.

Some questions go further and ask you to fit an equation, make a prediction from a model, interpret a residual, compare how well models fit, or decide whether linear or exponential growth fits better. Those are modeling questions, and they’re covered in Fit and interpret data models.