Scatterplots and Lines of Best Fit — SAT Math

Introduction

Scatterplots show the relationship between two numerical variables, and the SAT frequently asks you to read a line of best fit drawn through that data. You'll be tested on what the slope and y-intercept mean in context, how to use the equation to make a prediction, and how to compare an actual data value to a predicted one. This falls under Problem-Solving and Data Analysis (≈15% of the SAT Math section). Questions are almost always set in a real-world context, so translating algebra into plain English is the central skill here.

Prerequisite assumed: You know how to identify slope and y-intercept from an equation. If those ideas in context feel shaky, review Slope and Intercept in Context first. Sibling note: Deciding whether a linear or exponential model fits a scatterplot better is covered in Linear vs. Exponential Models — this note focuses purely on interpreting and applying a linear model once it's given.

Core Concept

Association

A scatterplot shows a positive association when larger x-values tend to pair with larger y-values (the cloud of points rises left to right). It shows a negative association when larger x-values tend to pair with smaller y-values (the cloud falls left to right). If there's no clear trend, we say there is no association.

The Line of Best Fit

A line of best fit (also called a regression line or trend line) is placed through a scatterplot to model the overall trend. Its equation is written in slope-intercept form:

y = mx + b

Slope ( $m$ ): The predicted change in $y$ for each one-unit increase in $x$ . A positive slope → positive association; a negative slope → negative association.

y-intercept ( $b$ ): The predicted value of $y$ when $x = 0$ . This has a real-world meaning only when $x = 0$ is a reasonable value in context.

Making a Prediction

Substitute the given $x$ -value into the equation to get the predicted $y$ -value.

Actual vs. Predicted (Residual)

Every real data point may not sit exactly on the line. The difference between the actual value and the predicted value is the residual:

\text{residual} = \text{actual} - \text{predicted}

Positive residual → the point sits above the line.
Negative residual → the point sits below the line.

SAT questions usually ask for the positive difference, or phrase it as "how much greater" or "how much less" — always read carefully.

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