Topic 2.1 Introducing Statistics - Are Variables Related?: identify questions about the association between two variables, distinguish association from causation, and recognize what two-variable data can answer.

What this topic is asking

The College Board (Topic 2.1) wants you to frame statistical questions about the relationship between two variables, to identify the explanatory and response variables, and above all to distinguish association from causation, recognizing that observational two-variable data can show a relationship but not prove that one variable causes the other.

Explanatory and response variables

Choosing which is which is a modelling decision driven by the question, not by the data alone. "Does studying time explain exam score?" makes study time explanatory and score the response. The labels matter because they fix the axes of a scatterplot and the direction of any prediction you later make.

What "associated" means

The form the analysis takes depends entirely on the variable types, which is why Topic 1.2's classification returns here: two categorical variables call for two-way tables (Topics 2.2 to 2.3), while two quantitative variables call for scatterplots, correlation, and regression (Topics 2.4 to 2.9).

Association is not causation

The defining lesson of this topic, and one of the most important in the whole course, is that an association does not by itself mean one variable causes the other. There are several reasons an association can appear without a causal link. A lurking (confounding) variable may influence both: ice-cream sales and drowning deaths rise together, but hot weather drives both, with no causal link between ice cream and drowning. Reverse causation is possible: maybe the response actually affects the explanatory variable. And the association could even be coincidence in a small sample. Because observational data cannot rule these out, the only design that supports a causal claim is a randomised experiment, in which random assignment to treatment groups balances out lurking variables. So when a question shows an observational study, the exam expects you to describe the association ("students who slept more tended to score higher") and then explicitly decline to claim cause, naming a plausible lurking variable or noting the lack of random assignment. Writing "this proves that X causes Y" from observational data is the single error most reliably punished in Unit 2 and beyond.

What two-variable data can and cannot answer

Two-variable data extend what you can ask beyond Unit 1's single distributions: you can now ask whether and how two variables move together, predict one from the other (with regression), and quantify the strength of a linear relationship (with correlation). What they still cannot do, in an observational setting, is establish cause, and they cannot generalize beyond the individuals studied unless those individuals were a random sample of a defined population. Keeping these limits in mind frames the rest of the unit honestly: correlation and regression are tools for describing and predicting an association, not for proving that manipulating one variable would change the other. The most sophisticated exam answers therefore pair a confident description of the relationship with an equally confident statement of its limits, which is exactly the balance the College Board is looking to assess from the very first topic of the unit.

Try this

Q1. In "does fertilizer amount explain crop yield?", identify the explanatory and response variables. [1 point]

• Cue. Fertilizer amount is explanatory; crop yield is the response.

Q2. A study finds taller children have larger vocabularies. Explain why this does not mean height causes vocabulary. [2 points]

• Cue. Age is a lurking variable: older children are both taller and have larger vocabularies, so age drives both and there is no direct causal link.