5.1 Introducing Statistics: Do Two Variables Have a Meaningful Relationship?

Why explore relationships between variables?

Statistics helps us ask whether two quantitative variables are meaningfully related — like study time and test scores. We look for association, but must remember: a relationship, even a strong one, does not automatically imply causation. Confounding variables or coincidence could explain patterns instead.

Explanatory vs. response variables

The explanatory variable is the one we think might explain or predict changes; the response variable is the outcome we measure. In statistics, we often label the explanatory variable x and response variable y, then investigate how changes in x relate to changes in y.

Setting up the investigation

Before running numbers, ask: are both variables quantitative? Is there a plausible reason one might influence the other? Recognizing the roles of variables and being cautious about causal claims sets the foundation for regression analysis, which formally models relationships between two quantitative variables.

Key terms

Explanatory variable

The variable thought to explain or predict changes in another variable; often labeled x.

Response variable

The outcome variable being measured or predicted; often labeled y.

Association

A relationship between two variables where changes in one tend to correspond with changes in the other.

Correlation vs. causation

A relationship between variables does not automatically mean one causes the other — confounding variables or chance may be responsible.

Requirement for regression analysis

Both variables being studied must be quantitative.

Try an AP-style question

Which four features should you describe in a scatterplot?

1. Direction, form, strength, and unusual features
2. Slope, intercept, r, and residuals
3. Mean, median, mode, and range
4. Shape, center, spread, and outliers only