AP Statistics 2.1 Introducing Statistics: Are Variables Related? Study Notes
AP Statistics 2.1 Introducing Statistics: Are Variables Related? Study Notes- New syllabus
AP Statistics 2.1 Introducing Statistics: Are Variables Related? Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- Given that variation may be random or not, conclusions are uncertain.
Key Concepts:
- Introducing Statistics: Are Variables Related?
Introducing Statistics: Are Variables Related?
Introducing Statistics: Are Variables Related?
A major goal of statistics is to determine whether two variables are related. Relationships can be described in terms of association, meaning the value of one variable gives us information about the value of another.
1. Types of Relationships
- Categorical ↔ Categorical: Use two-way tables or segmented bar graphs to examine association (e.g., gender vs. preferred subject).
- Categorical ↔ Quantitative: Compare distributions of the quantitative variable across categories (e.g., mean income for males vs. females).
- Quantitative ↔ Quantitative: Use scatterplots and correlation to study linear or non-linear relationships (e.g., height vs. weight).
2. Association vs. Causation
- Two variables may show an association (a pattern of relationship).
- But association does not imply causation — the relationship may be due to a lurking variable or chance.
3. How We Test Relationships
- Categorical data: Chi-square test of independence.
- Quantitative data: Correlation coefficient, regression, or scatterplots.
- Mixed data: Compare group means (e.g., t-tests, ANOVA).
Example
A school surveys students about their gender (male/female) and preferred subject (math/science/arts). The two-way table shows more males prefer science while more females prefer arts.
Is there an association between gender and subject preference?
▶️ Answer / Explanation
Yes, because the distribution of preferences differs across genders. If the variables were unrelated, the proportions would be similar. This suggests an association, though not causation.
Example
A scatterplot of hours studied vs. exam score shows a positive linear trend.
What can you conclude about the relationship?
▶️ Answer / Explanation
The variables are positively associated: more hours studied generally leads to higher exam scores. However, we cannot claim causation without experimental control.
Example
Researchers compare the average sleep hours for students who play sports vs. those who do not.
How can this relationship be examined?
▶️ Answer / Explanation
We can compare the means of the two groups, or use side-by-side boxplots to visualize differences. If athletes sleep more on average, there may be an association between sports participation and sleep, but again, not necessarily causation.