Scatter Plots and the Line of Best Fit

What a scatter plot shows

A scatter plot displays paired measurements: for each subject you record two numbers and plot them as one point.

Measuring ten students’ study hours and test scores gives ten points, each carrying both facts about one student. Unlike a line graph, the points are not joined — there is nothing between two students.

• line of best fit

Describing an association

Three questions, in order.

Direction. As $x$ increases, does $y$ tend to rise or fall?

Pattern	Called
points trend upward	positive association
points trend downward	negative association
no tilt at all	no association

Form. Do the points follow a straight path or a curve? A line of best fit is only appropriate for a straight one.

Strength. Are the points tightly gathered around the trend, or widely scattered? Tight means strong; loose means weak.

The plot above is a strong positive linear association.

The line of best fit

A line of best fit is a straight line drawn through the middle of the cloud to summarise the trend. Drawn by eye, the aim is:

• follow the general direction of the points;
• keep roughly as many points above the line as below;
• ignore the temptation to connect any two particular points.

It is a summary, not a fact about any individual. A student who studied 5 hours scored 70 here, while the line predicts about 67 — the line describes the group, not the person.

Why the line does not go through the points

Students often expect a line of best fit to connect the dots, and are unsettled when it misses almost every one. Missing them is the job.

Each point is one individual. The line describes the group. Those are different claims, and no straight line can make both at once unless the data happens to be perfect.

In the plot above, the student who studied 5 hours scored 70, while the line predicts about 67. The line is not wrong about that student, and the student is not wrong about the line — the gap says this person did a little better than the group trend. That gap has a name: the residual.

So a line of best fit answers “what does this group tend to do?” and never “what will this person do?”. A prediction from it is a reasonable expectation, not a promise, and the more scattered the cloud, the looser the expectation.

This is also why a line only belongs on a plot whose points follow a straight path. Fitting a line to a curve summarises nothing — it draws a wrong answer confidently.

Using the line

Once you have the line, its equation works like any other, and the parts mean something:

Part	Means
slope	the rate of change — how much $y$ rises per unit of $x$
$y$-intercept	the predicted value when $x = 0$

For the plot above the line is roughly $y = 7x + 32$: about 7 extra marks per hour studied, starting from about 32 marks with no study.

Worked examples

Common mistakes

Practice problems