Regression Analysis

What Unit 5 covers

Unit 5, Regression Analysis, is about relationships in two-variable quantitative data. You describe an association between two quantitative variables, measure it with correlation, and, when a straight line fits, build a linear model that predicts one variable from the other.

The explanatory variable (the predictor) goes on the x-axis. The response variable (what you predict) goes on the y-axis. Once you fit a line, you judge how well it fits by looking at residuals, the gaps between what actually happened and what the line predicted.

This is the only unit built entirely around bivariate quantitative data, and all of it is descriptive: you fit models and check them rather than run formal inference about the slope.

How Unit 5 fits the exam

Unit 5 is worth 10-20% of the multiple-choice section and takes about 9 class periods, the fewest of the five units. It reuses ideas from earlier one-variable work, such as describing a pattern and finding a mean.

The exam is 3 hours long. Section I has 42 multiple-choice questions (50% of the score, 90 minutes). Section II has 4 free-response questions (50% of the score, 90 minutes), each worth 10 points and 12.5% of the exam score. A graphing calculator with statistical capabilities is expected, and formulas and tables are provided for both sections.

Regression content shows up in individual multiple-choice questions, a multiple-choice question set, and as part of a free-response question. Expect to read calculator output and translate it into a sentence about the real context.

The five topics in Unit 5

Unit 5 has five topics, numbered 5.1 to 5.5:

• 5.1 Graphical Representations Between Two Quantitative Variables. Build a scatterplot from bivariate quantitative data, then describe the association by its form (linear or non-linear), direction (positive or negative), strength (strong, moderate, or weak), and unusual features such as clusters or points that do not fit the pattern.
• 5.2 Correlation. The correlation coefficient $r$ (a unit-free number from -1 to 1) summarizes the strength and direction of a linear association. Values near -1 or 1 mean a strong linear association, and $r = 0$ means no linear association. A value close to -1 or 1 does not by itself prove a line is the right model, and correlation does not necessarily imply causation.
• 5.3 Linear Regression Models. When the form looks linear, predict the response with $\hat{y} = a + bx$, where $\hat{y}$ is the predicted response, $a$ is the y-intercept, and $b$ is the slope. Predicting inside the observed range of x-values is interpolation; predicting beyond it is extrapolation, which grows less reliable the further out you go.
• 5.4 Residuals. A residual is the observed response minus the predicted response, $\text{residual} = y - \hat{y}$. A positive residual means the model underpredicts, and a negative residual means it overpredicts. A residual plot graphs residuals against predicted values (or the explanatory variable): apparent randomness supports a linear model, while clear curvature suggests a line is not the best fit.
• 5.5 Least-Squares Regression. The least-squares regression line (LSRL) is the line that minimizes the sum of the squared residuals, and it passes through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ and $\bar{y}$ are the means of x and y. Technology reports the slope $b$, the intercept $a$, the correlation $r$, and the coefficient of determination $r^2$.

Reading slope, intercept, and r-squared in context

Every number the calculator reports needs a sentence tied to the scenario.

• The slope $b$ is the predicted change in the response variable for a one-unit increase in the explanatory variable.
• The y-intercept $a$ is the predicted response when the explanatory variable equals 0. Sometimes that has no sensible meaning, because $x = 0$ falls outside the data (extrapolation) or would give an impossible value such as a negative height.
• The coefficient of determination $r^2$, the square of the correlation, is the proportion of the variation in the response variable that is explained by the linear relationship with the explanatory variable.

For more on these ideas, see least-squares regression line, r vs r-squared, how to interpret residual plots, and correlation vs causation.

What was removed from this unit

The Fall 2026 redesign trimmed regression down to description. Inference for regression slopes, meaning confidence intervals and hypothesis tests for the slope of a regression line, is no longer on the AP exam. Analyzing departures from linearity beyond reading a residual plot was also removed.

These topics are still standard in most college introductory statistics courses, so you may meet them later even though they are off the AP exam. On the exam, everything in Unit 5 stays at the level of fitting a line and checking it, not testing claims about the slope.

Tools for practicing regression

Work through the calculations and the visual intuition with these:

• Regression calculator to find the slope, intercept, $r$, and $r^2$ for your own data.
• Influential point interactive to see how a single unusual point can pull the least-squares line.

You can also return to the AP Statistics hub or review the earlier Unit 4 on inference for means.