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Math foundations

SAT scatterplots: read the line of best fit like a pro

A scatterplot’s line of best fit turns a cloud of points into a prediction. Learn to interpret its slope and intercept in context, make predictions, and avoid the classic causation trap.

SPSAT Practice Editorial7 min read

The short version

What you'll take away

  • The slope of a line of best fit is the predicted change in y for each one-unit increase in x.
  • Points above the line have actual values greater than predicted; points below have smaller ones.
  • A strong association does not show cause, and predictions far outside the data are unreliable.
In this article

From a cloud of points to a line

A scatterplot shows pairs of values as points. Here, each point is a student: x is hours of practice in a week and y is that student’s score on a 20-point quiz. The points rise from left to right, so more practice is associated with higher scores.
A line of best fit summarizes the trend. For these points, the line \(\displaystyle y = 1.1x + 5\) runs through the middle of the cloud, with some points above it and some below.

The study lab

Practice hours vs. quiz score

Nine students (illustrative data) and the line of best fit y = 1.1x + 5.

What to notice: Illustrative data. The points sit close to the line, which indicates a strong positive association. The line predicts, but individual students vary.

Interpret slope and intercept in context

In \(\displaystyle y = 1.1x + 5\), the slope 1.1 means the model predicts about 1.1 more quiz points for each additional hour of practice. The y-intercept 5 is the predicted score for a student with 0 hours of practice.
SAT questions often ask for the “best interpretation” of a slope. The right answer uses the units of both variables and the word “predicted” or “estimated,” because the line describes a trend, not a guarantee for each person.

Predictions and residuals

To predict, substitute x into the equation. For 6 hours, \(\displaystyle y = 1.1(6) + 5 = 11.6\). The actual student who practiced 6 hours scored 12, so that point sits 0.4 above the line. The difference between the actual and predicted values (actual minus predicted) is called a residual.
Predictions are most trustworthy within the range of the data. The model predicts a score of 27 for 20 hours, which is impossible on a 20-point quiz. Extending a line far beyond the data is called extrapolation, and it can produce unrealistic results.

Your turn

Quick practice

Using the line of best fit y = 1.1x + 5, what is the predicted quiz score for a student who practices 6 hours?

Hint: The question asks for the value from the line, not the plotted point.

Choose one answer

Direction, strength, and the causation trap

Describe an association by its direction and its strength. Then remember what a scatterplot cannot show: that one variable causes the other.

See the skill

Describe what you see

Points rise left to right

Positive

As x increases, y tends to increase.

Points fall left to right

Negative

As x increases, y tends to decrease.

Tight around the line

Strong

Predictions from the line are fairly reliable.

Association only

Not causation

An observational study shows a link, not a cause. Only a randomized experiment can support a causal claim.

Use the visual as a reasoning aid: tie each choice to the relationship, grammar, or evidence the passage actually gives.

Practice: try these before checking the answers

A line of best fit for phone battery life is y = −0.5x + 40, where x is hours of screen time and y is the remaining battery percent.
  • 1. Interpret the slope −0.5 in context.
  • 2. What remaining battery percent does the model predict after 20 hours?
  • 3. A phone has 37% left after 10 hours. Is it above or below the line, and by how much?
  • 4. Should you use the model to predict battery after 100 hours? Why or why not?

Sources & further reading

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