What it covers: Describing and modeling the relationship between two quantitative variables — scatterplots, correlation, the least-squares regression line, residuals, and r².
Exam weight: About 10–20% of the AP Statistics exam.
The big question: How do we model the linear relationship between two quantitative variables, and how well does that model fit?
Statistical practices: Formulate Questions (P1), Collect Data (P2), Analyze Data (P3), Interpret Results (P4).
Key topics at a glance
Scatterplots
Describe two quantitative variables by Direction (positive/negative), Form (linear/curved), Strength, and Unusual features — in context. Explanatory = x, response = y.
Correlation r
Measures the strength and direction of a linear relationship, from −1 to 1. No units, symmetric in x and y, only linear, and not resistant to outliers.
Correlation vs. Causation
A strong r does not prove x causes y — beware lurking variables. Only a randomized experiment supports causation.
The Regression Line
ŷ = a + bx, the least-squares line (minimizes squared residuals). Slope b = r·(s_y/s_x); the line passes through (x̄, ȳ).
Interpreting Slope & Intercept
Slope: for each 1-unit increase in x, predicted y changes by b (in context). Intercept: predicted y when x = 0 (meaningful only near the data).
Prediction & Extrapolation
Plug an x into the equation for ŷ. Extrapolation (predicting outside the data's x-range) is unreliable.
Residuals
residual = observed − predicted = y − ŷ. A residual plot with no pattern supports a linear model; a curve or fan means it does not fit.
r², s & Influential Points
r² = proportion of variation in y explained by the line. s = typical residual size. Influential points (often high-leverage) can pull the line.
The key terms you must know
Explanatory / response variable — the predictor x and the outcome y.
Scatterplot (DUFS) — describe Direction, Form, Strength, and Unusual features.
Correlation r — strength and direction of a linear relationship, from −1 to 1; no units, not resistant.
Least-squares regression line — ŷ = a + bx, minimizing the sum of squared residuals.
Slope b — predicted change in y per one-unit increase in x; b = r·(s_y/s_x).
y-intercept a — predicted y when x = 0.
Predicted value (ŷ) — the model's estimate of y for a given x.
Extrapolation — predicting outside the range of the data; unreliable.
Residual — observed minus predicted, y − ŷ.
Residual plot — residuals vs. x; no pattern supports a linear model.
Coefficient of determination r² — proportion of variation in y explained by the line.
Influential / high-leverage point — a point that strongly affects the line, often with extreme x.
Key themes to remember
Always look at the scatterplot. r and the LSRL only make sense for a linear pattern.
Correlation is not causation. A relationship can be explained by a lurking variable.
Interpret slope and intercept in context. Include the variables and their units.
Residual plots judge the fit. Random scatter is good; a pattern means the linear model is wrong.
r² measures explained variation. A higher r² means the line accounts for more of the spread in y.
Common exam traps
r near 0 does not mean 'no relationship' — only no linear relationship; a curve can still exist.
r and r² are not the same. r² is the square; take the sign of r from the scatterplot's direction.
Residual = observed − predicted, not predicted − observed. Watch the order.
Don't extrapolate. Predictions far outside the x-range are not trustworthy.
The intercept is often not meaningful when x = 0 is far from the data.
r is not resistant. One outlier can dramatically change it, so examine the plot.
A high r² does not prove causation or that a linear model is appropriate — check the residual plot.