Exploring Two-Variable Data

What this unit covers

The topics below follow the published Statistics course framework for Unit 2. This unit is worth 5–7% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Scatterplots & Association13 min · 3 objectivesIdentify explanatory and response variables and plot them correctly · Describe a scatterplot by direction, form, strength, and unusual features · Distinguish association from causation
• Correlation13 min · 3 objectivesInterpret the correlation coefficient r as a measure of linear association · State the properties and limitations of correlation · Explain why a strong correlation does not confirm a linear relationship
• Least-Squares Regression15 min · 3 objectivesInterpret the slope and y-intercept of a least-squares regression line in context · Use a regression equation to make predictions and recognize extrapolation · Interpret the coefficient of determination r²
• Residuals & Model Fit13 min · 3 objectivesCompute and interpret a residual in context · Use a residual plot to assess whether a linear model is appropriate · Explain the danger of extrapolation and the effect of influential points

Formulas in Unit 2

Every term in Unit 2

All 19 terms we publish for Exploring Two-Variable Data, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Residual

Observed minus predicted, y − ŷ. A positive residual means the model underestimated that observation.

Explanatory vs response variable

The explanatory variable goes on the x-axis and is thought to explain changes in the response on the y-axis.

Describing a scatterplot

Direction, form, strength and unusual features, in context. All four are required for full credit.

Correlation coefficient r

Measures the strength and direction of a LINEAR relationship, between −1 and 1. It has no units and is unaffected by which variable is x.

What r does not tell you

It does not establish causation, does not detect curved relationships, and is not resistant to outliers.

Least-squares regression line

ŷ = a + bx, minimizing the sum of squared residuals. It always passes through the point (x̄, ȳ).

Interpreting the slope

"For each additional one-unit increase in x, the predicted y increases by b units" — predicted, in context, with units.

Interpreting the y-intercept

The predicted response when x = 0. Often meaningless in context, and worth saying so.

Coefficient of determination r²

The percentage of variation in y explained by the linear relationship with x. Always report it as a percentage of variation.

Residual plot

A random scatter about zero supports a linear model; a curved pattern means the relationship is not linear and a different model is needed.

Influential point vs outlier

An outlier lies far from the pattern; an influential point substantially changes the regression line, usually because its x-value is extreme.

Extrapolation

Predicting outside the range of the observed x-values. Unreliable, because there is no evidence the pattern continues.

Transforming to achieve linearity

Taking logs of y linearises exponential relationships; logs of both linearises power relationships.

Interpreting r in context

State the direction, the strength and that the relationship is linear, naming both variables. r alone is not an interpretation.

Predicting with a regression equation

Substitute and label the answer as a predicted value. Saying it will happen overstates what a model provides.

Residual for a specific point

Observed minus predicted at that x. A positive residual means the model underpredicted that observation.

Why r² is not r squared conceptually

r² is the proportion of variation in the response explained by the model, which is a different statement from the strength of the linear association.

Effect of removing an influential point

Recompute and compare. An influential point that pulls the line toward itself will change the slope noticeably when removed.

Choosing a transformation

A curved residual plot means the linear model is wrong. Try log(y) for exponential growth and log-log for a power relationship.

What examiners penalize here

• Slope interpretations need three parts: the words "predicted" or "on average," the units of y, and the units of x. "For each additional *hour*, the *predicted score* increases by 6 *points*" scores; "the score goes up 6" does not.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 9 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data
2. Unit 2 · Exploring Two-Variable Data
3. Unit 3 · Collecting Data
4. Unit 4 · Probability & Random Variables
5. Unit 5 · Sampling Distributions
6. Unit 6 · Inference for Proportions
7. Unit 7 · Inference for Means
8. Unit 8 · Inference for Categorical Data: Chi-Square
9. Unit 9 · Inference for Quantitative Data: Slopes