Statistics I · Week 2 — Categorical data

Week map

Frequency notation

• Frequency $f$ → count in category → 23 students chose tea.
• Relative frequency → $f / n$ → proportion of whole → 0.23 if $n=100$.
• Percent → relative × 100 → 23%.
• Complement → all categories not A → if A is 30%, complement is 70%.

• Rel freq: 0.40, 0.35, 0.25
• Percents: 40%, 35%, 25%

Charts

Bar chart

• Categories on axis (nominal or ordinal).
• Bar height = count or percent.
• Good for compare across categories.

Pie chart

• Wedges show part-whole.
• Works for few categories; weak for many thin slices.
• Angles proportional to relative frequency.

Pattern families

Easy — Build frequency table

• Tally categories from raw list.
• Compute $n$ as sum of counts.
• Relative freq = count/$n$.

Medium — Chart choice and read

• Pick bar vs pie from question goal.
• Read missing count from percent and $n$: count = percent/100 × $n$.
• Compare two categories via difference in counts or percents.

Hard — Rounding and complements

• Percents sum to 99% or 101% due to rounding — know largest category still identifiable often.
• Given two category percents, find third when three categories total 100%.
• Two-way categorical preview: row percent vs overall percent (setup for week 4).

Worked mini-examples

Traps

• Confusing frequency with relative frequency.
• Pie chart with dozens of categories.
• Percents not summing to 100 — rounding, not always error.
• Bar chart ordered as if ordinal when nominal — order arbitrary unless by count.
• Using pie when categories are not parts of one group total.

Diagnostic (try yourself)

1. Raw: [R,R,B,G,R,B,G,G,R] — build frequency table and $n$.
2. $n=80$, counts A=32, B=20, C=28. Find relative frequencies and percents.
3. In a survey, 35% preferred online and 45% preferred hybrid. The only other option is in-person. What percent chose in-person?
4. $n=200$, 12.5% chose “other”. How many cases?
5. When would a bar chart be better than a pie chart for the same categorical data? One sentence.