Representing a Categorical Variable with Tables
Introduced| English |
|---|
| frequency table/ˈfriːkwənsi ˈteɪbl/ |
| relative frequencies/ˈrelətɪv ˈfriːkwənsiz/ |
| two-way table/tuː weɪ ˈteɪbl/ |
| joint frequency/dʒɔɪnt ˈfriːkwənsi/ |
| marginal/ˈmɑːdʒɪnl/ |
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.
Shares of the whole
Relative frequencies are each category's share of the total — exactly what a pie chart of the frequency table shows.
Of $120$ students, $30$ prefer maths. What is the relative frequency (as a decimal)?
$30/120=0.25$.
Of $120$ students, $40$ prefer arts. What percent is that (to the nearest whole)?
$40/120\approx33\%$.
Relative frequencies let you fairly compare groups of different sizes.
Proportions remove the effect of group size.
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.
A two-way table is used to display...
It cross-tabulates two categorical variables.
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.
In a two-way table, a single inner cell count (both categories at once) is a...
Inner cell = joint frequency.
A row or column total in a two-way table is a ____ frequency.
Marginal = the total for one variable.
Keep marginal and joint frequencies distinct: a joint frequency is one inner cell (both categories at once), while a marginal frequency is a row/column total (one variable only). And relative frequencies are counts ÷ total — decide whether "the total" is the grand total, a row total, or a column total, because each gives a different proportion.
$120$ students: $30$ prefer maths, $50$ science, $40$ arts.
- Frequency table: maths $30$, science $50$, arts $40$.
- Relative frequencies: $25\%$, $\approx41.7\%$, $\approx33.3\%$ (each count ÷ $120$).
- If also split by grade, a two-way table's cell "Grade 11 ∩ science" would be a joint frequency.
Summarize a categorical variable with a frequency table of counts, then compute relative frequencies (count ÷ total) as proportions/percentages. A two-way table shows two categorical variables: inner cells are joint frequencies, and row/column totals are marginal frequencies.