Comparing Categorical Groups

Comparing groups fairly

• To see if two categorical variables are related, compare conditional distributions 条件分布 across groups.
• A conditional distribution is the set of proportions for one variable within a single group.
• "What does the subject breakdown look like for girls vs for boys?"
• Using proportions (not raw counts) makes the comparison fair when groups differ in size.

Association vs. independence

• If the conditional distributions are the same across groups, the variables appear independent 独立.
• If they differ, the variables show an association — knowing one tells you something about the other.
• The bigger the difference between groups, the stronger the association.
• "Same proportions everywhere" is the picture of no relationship.

A segmented bar chart compares the conditional distribution in each group

Differences in proportions

• Quantify the association with a difference in proportions: e.g. $60\%$ of girls vs $40\%$ of boys chose science.
• That $20$-percentage-point gap is the evidence of a relationship, stated in context.
• Always report which group is higher and by how much.
• A tiny difference may just be sampling noise; a large one signals real association.

Association is still not causation

• Even a strong association between two categorical variables does not prove causation 因果关系.
• A hidden confounding variable 混杂变量 could drive both.
• Observational data can reveal association but rarely establishes cause.
• Only a well-designed experiment (Unit 3) can support a causal claim.

Write from age as a confounder

• A treatment group is mostly older than the comparison group.
• The success rates differ by 15 percentage points. The lesson gap of 25 points is another study.