Two Categorical Variables

Two categories at once

• Sometimes both variables are categorical 分类 — like gender and favorite subject.
• Organize them in a two-way table 双向表 (also called a contingency table 列联表).
• Rows are the categories of one variable; columns are the categories of the other.
• Each cell holds the count of individuals in that row-and-column combination.

Marginal and joint

• The margins (row and column totals) give marginal frequencies — one variable, ignoring the other.
• A single inner cell gives a joint frequency — both categories at once.
• Divide by the grand total to get relative frequencies (proportions).
• Marginal relative frequency = a total ÷ grand total; joint = a cell ÷ grand total.

A segmented bar chart 分段条形图 compares the conditional distribution in each group

Conditional relative frequency

• A conditional relative frequency 条件相对频率 restricts attention to one group, then finds a proportion inside it.
• "Of the girls, what fraction chose science?" — divide that cell by the girls' row total.
• The key is the denominator: a row total (or a column total), not the grand total.
• Conditionals are how we compare groups fairly, even when the groups differ in size.

Showing it as bars

• A side-by-side bar chart puts each group's bars next to each other for comparison.
• A segmented bar chart stacks the proportions within each group into one full bar.
• Segmented bars are ideal for comparing conditional distributions — each bar sums to $100\%$.
• If the segments line up the same in every bar, the variables look unrelated.

Write from a conditional relative frequency of 0.4

• In one row of a two-way table, 12 of 30 students said yes.
• The lesson conditional relative frequency 0.6 is another row.