Representing a Categorical Variable with Tables

Counting categories

• The simplest way to summarize a categorical variable is to count how many fall in each group.
• A frequency table 频数表 lists each category with its count (frequency).
• It turns a long list of labels into a compact, readable summary.
• From counts, you can compute proportions and compare groups.

Relative frequencies

• Raw counts don't show share of the whole — for that, use relative frequencies 相对频数.
• Divide each count by the total to get a proportion (or ×100 for a percentage).
• $30$ out of $120$ students prefer maths → relative frequency $\tfrac{30}{120}=0.25=25\%$.
• Relative frequencies let you compare data sets of different sizes fairly.

A frequency table lists counts; relative frequency is each count divided by the total

Two-way tables

• To show two categorical variables at once, use a two-way table 双向表 (contingency table).
• Rows are one variable's categories, columns the other's; each cell is a joint frequency 联合频数 (count in both).
• Example: rows = grade level, columns = favorite subject; each cell counts students in that combination.
• It reveals how the two variables relate.

Marginal vs. joint frequencies

• Joint frequency: a single inner cell — individuals in both categories.
• Marginal frequency 边际频数: a row or column total — the count for one variable, found in the margins.
• The margins recover each variable's own distribution; the inner cells show the combinations.
• Reading which is which is a common exam skill.

Write from a relative frequency of 0.2

• 8 of 40 students prefer tea.
• The lesson relative frequency 0.25 and the percent 33 come from other counts.