AP Statistics Unit 2: Exploring Two-Variable Data
Study scatterplots, correlation, regression, residuals, transformations with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
This unit explores relationships between two variables using scatterplots, correlation, and least-squares regression. You learn to describe, quantify, and model linear associations.
Why it matters
Regression analysis appears on every AP Statistics exam. Free-response questions regularly ask you to interpret slope, r-squared, and residual plots. Understanding the distinction between correlation and causation is essential for the investigative task.
Key concepts
- Scatterplots display the relationship between two quantitative variables; describe direction, form, and strength.
- The correlation coefficient r measures the strength and direction of a linear relationship (-1 to 1).
- The least-squares regression line minimizes the sum of squared residuals and passes through (x-bar, y-bar).
- r-squared gives the proportion of variation in y explained by the linear relationship with x.
Describing Bivariate Relationships
A scatterplot shows each (x, y) pair as a point. Describe the relationship by direction (positive or negative), form (linear, curved, or no pattern), and strength (how tightly points cluster around a pattern). Outliers and influential points deserve mention. Categorical variables can be incorporated by using different symbols or colors for subgroups. Always describe the relationship in context — for example, "as study hours increase, exam scores tend to increase in a moderately strong, linear pattern."
Correlation and Regression
The correlation r quantifies the strength of a linear relationship. Values near 1 or -1 indicate strong linear patterns; values near 0 suggest no linear association. The LSRL y-hat = a + bx minimizes the sum of squared vertical distances from points to the line. The slope b represents the predicted change in y for a one-unit increase in x. The y-intercept a is the predicted y when x = 0, which may or may not have practical meaning. Always interpret slope and intercept in the specific context of the problem.
Residuals and Assessing Fit
A residual is actual minus predicted: y - y-hat. A residual plot graphs residuals against x-values. A good linear model produces residuals that scatter randomly around zero with no pattern. Curved patterns in the residual plot suggest a nonlinear relationship. r-squared is the proportion of variability in y accounted for by the regression on x. High r-squared does not mean the model is appropriate — always check the residual plot. Correlation does not imply causation; only controlled experiments can establish causal relationships.
AP exam tip
When interpreting slope on the AP exam, use this template: "For each additional [one unit of x], the predicted [y] changes by [b] [units of y]." Avoid saying "for every" — say "for each additional."
Connections to other units
- Unit 1: Regression builds on understanding center and spread of individual variables.
- Unit 9: Inference for slopes uses the regression framework to test whether the true slope is zero.
- Unit 3: Data collection methods determine whether regression results support causal conclusions.
