Categorical Data — Frequency Tables, Proportions, and Charts

🎯 Learning Objectives

• Construct frequency tables and relative frequency tables from raw categorical data
• Compute and interpret proportions, percentages, and ratios
• Create and interpret bar charts, pie charts, and Pareto charts
• Choose the appropriate chart type for a given dataset
• Recognize common visualization pitfalls and misleading charts

📋 Prerequisites

• Data Types & Scales (01-data-types-scales) — categorical data comes in nominal or ordinal scales
• Intro to Statistics (00-intro-statistics) — understand population vs sample
• Basic arithmetic: counting, division, percentages

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📖 Core Content

3.1 Intuition: Making Sense of Categories

Chocolate, Vanilla, Strawberry, Chocolate, Vanilla, Chocolate, Mint Chip, Vanilla, Chocolate, Strawberry, ... Reading this raw list tells you almost nothing useful. You need to organize it — count how many people chose each flavor. Once you do that, patterns emerge:

🔑 Key Insight: With categorical data, we can't compute means or standard deviations. Our main tool is counting — frequencies, proportions, and the comparisons between them.

3.2 Frequency Tables

3.2.1 Simple Frequency

3.2.2 Relative Frequency

$$\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total}} = \frac{f}{n}$$ $$\text{Percentage} = \text{Relative Frequency} \times 100\%$$

Check: Relative frequencies always sum to 1. Percentages always sum to 100% (barring rounding).

3.2.3 Cumulative Frequency (for Ordinal Data)

🔑 Note: Cumulative frequency is only meaningful when categories have a natural order. For nominal data (like ice cream flavors), cumulative frequency doesn't make sense — what does "at most Chocolate" mean?

3.3 Bar Charts

Anatomy of a Bar Chart

1. Bars should have equal width — only height varies
2. Bars should have gaps between them (unlike histograms for continuous data)
3. Bars can be vertical (column chart) or horizontal (bar chart)
4. Order bars by natural order (ordinal) or by frequency (nominal)

When to use a bar chart:

• Comparing frequencies across categories
• Showing the distribution of a categorical variable
• Making quick visual comparisons

Example: Bar Chart of Ice Cream Preferences

• X-axis: Chocolate, Vanilla, Strawberry, Mint Chip, Other
• Y-axis: Frequency (0 to 40)
• Chocolate bar reaches 35, Vanilla reaches 28, etc. The viewer instantly sees: chocolate is most popular; Mint Chip and Other are less common.

3.4 Pie Charts

• Showing parts of a whole (all categories together form 100%)
• When there are few categories (ideally 2-5)
• When you want to emphasize relative proportions When NOT to use a pie chart:
• With many categories (slices become too small)
• When comparing small differences (humans are bad at comparing angles)
• When precise values matter (bar charts show values more clearly)

Rule of thumb: If you have more than 5 categories, use a bar chart instead of a pie chart.

3.5 Pareto Charts

When to use a Pareto chart:

• Identifying the "vital few" categories that dominate
• Quality control and problem-solving (focus on the biggest issues)
• Resource allocation (prioritize the most frequent problems)

Example: Customer Complaints at a Call Center

• Bars (from tallest to shortest): Long wait, Unresolved, Rude staff, Billing, Wrong dept., Other
• A line graph showing cumulative percentage Interpretation: The first two categories (Long wait + Unresolved) account for 60% of complaints. Fixing the top three could address nearly 75% of issues. This is the Pareto principle in action — focus on the few categories that give the most impact. (Diagram)

3.6 Comparing Distributions

Side-by-Side Bar Charts

Stacked Bar Charts

• Comparing proportions across groups
• Showing how the "mix" of categories changes

Grouped (Clustered) Bar Charts vs. Stacked Bar Charts

3.7 Worked Examples

Example 1: Building a Frequency Table (Easy)

Math, Science, Math, English, Science, Math, History, Math, Science, English, Math, Math, Science, English, Math, Science, History, Math, Science, Math, English, Math, Science, Math, History, Math, Science, English, Math, Science Task: Create a frequency table with frequencies, relative frequencies, and percentages. Solution:

Example 2: Pie Chart Construction (Medium)

Example 3: Choosing the Right Chart (Harder)

• Age groups are ordinal (natural order)
• 5 categories — manageable number
• We want to show the distribution across groups Recommended charts:

3.8 Edge Cases & Gotchas

The "Other" Category

• "Other" should not be the largest category
• If "Other" becomes large, you need a better categorization scheme
• Always define what "Other" includes

Misleading Pie Charts

1. 3D effects: Make slices look bigger than they are
2. Exploded slices: Emphasize one slice disproportionately
3. Too many slices: Creates a cluttered, unreadable chart
4. Non-additive percentages: If percentages don't sum to 100%, the chart is wrong

Zero Frequencies

• Yes — if the category is logically part of the variable and might have observations in other contexts
• No — if including it creates unnecessary visual clutter

3.9 Why This Matters

• Market research: What brand do customers prefer?
• Medicine: What blood type is most common?
• Quality control: What defect types occur most often?
• Politics: How do different demographics vote? The tools in this topic — frequency tables, bar charts, pie charts, Pareto charts — are used daily by analysts worldwide. They form the foundation for:
• Week 4 (Contingency Tables): Extending frequency analysis to two categorical variables
• BSMA1004 (Stats 2): Chi-square tests for categorical data
• BSMS2002 (Business Analytics): Customer segmentation and market analysis

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📐 Key Formulas / Concepts

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⚠️ Common Pitfalls

Pitfall 1: Using a Pie Chart with Too Many Categories

Pitfall 2: Misleading Y-Axis on Bar Charts

Pitfall 3: Confusing Bar Charts with Histograms

Pitfall 4: Cumulative Frequency on Nominal Data

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📝 Practice Questions

Q1: Frequency Table Construction

The following data shows the blood types of 40 patients:

A, B, O, A, AB, A, O, B, O, A, A, B, O, A, AB, O, A, O, B, A, O, A, B, O, A, AB, O, A, O, B, A, O, A, B, O, A, AB, O, A, B

Construct a frequency table with frequencies, relative frequencies, and percentages.

<details> <strong>Solution</strong>

Step 1: Count each blood type

Step 2: Verify total: 15 + 8 + 4 + 13 = 40 ✅

Step 3: Compute relative frequencies and percentages

Interpretation: Blood type A is most common (37.5%), followed by O (32.5%). AB is rarest (10%).

</details> > **Q2: Pie Chart Angles** > > Using the blood type data from Q1, calculate the central angles for a pie chart. > > <details> <strong>Solution</strong> > >

Check: 135 + 72 + 36 + 117 = 360° ✅

</details> > **Q3: Chart Selection** > > For each scenario, recommend the best chart type and explain why: > > a) Showing the market share of 5 smartphone brands b) Identifying the most common types of defects in a manufacturing process c) Comparing the education levels (HS, Bachelor's, Master's, PhD) across two companies > > <details> <strong>Solution</strong> > > **a) Market share of 5 brands:** **Pie chart** or **bar chart** > > - Pie chart works because there are few categories and we want to show "parts of a whole" (market share) > - Bar chart also works if we want precise comparisons between brands > > **b) Most common defects:** **Pareto chart** > > - The goal is to identify which defects occur most frequently > - Sorting by frequency and adding a cumulative % line shows where to focus quality improvement efforts > - The Pareto principle likely applies (a few defect types account for most problems) > > **c) Education levels across 2 companies:** **Side-by-side bar chart** or **stacked bar chart** > > - We need to compare distributions across groups > - Side-by-side bars for each education level allow direct comparison > - Stacked bars show the composition of each company's workforce </details> > **Q4: Cumulative Frequency** > > A survey asks 500 people their highest education level: > >

Compute the cumulative frequencies and cumulative percentages. What percentage have at most a Bachelor's degree?

<details> <strong>Solution</strong>

Interpretation: 64% of respondents have at most a Bachelor's degree. In other words, 64% have a Bachelor's or less (HS + Bachelor's).

Alternative phrasing: 36% have a Master's or higher (100% - 64%).

</details> > **Q5: Pareto Analysis** > > A hospital tracks reasons for readmission within 30 days: > >

Create a Pareto analysis (sorted frequencies + cumulative percentages). Which reasons should be addressed first?

<details> <strong>Solution</strong>

Step 1: Sort by frequency (descending)

Total readmissions: 85 + 42 + 38 + 25 + 20 + 15 = 225

Step 2: Identify the vital few

• The top 2 reasons (Infection + Medication error) account for 56.4% of readmissions
• The top 3 reasons account for 73.3%
• The top 4 account for 84.4%

Recommendation: Focus on Infection and Medication error first — these two categories alone account for over half of all readmissions. Addressing the top 3 categories would address nearly 75% of the problem.

</details> > **Q6: True or False** > > For each statement, determine if it's true or false. If false, explain why. > > a) Bar charts should have gaps between bars. b) Pie charts are ideal for comparing small differences between categories. c) Cumulative frequency can be computed for nominal data. d) A Pareto chart combines bars and a line graph. > > <details> <strong>Solution</strong> > > a) **True** — Gaps distinguish bar charts (categorical data) from histograms (continuous data). > > b) **False** — Humans are bad at comparing angles and areas. Small differences are better shown with bar charts where height differences are easier to perceive. > > c) **False** — Cumulative frequency requires ordered categories. Nominal data has no order, so "cumulative" makes no sense. > > d) **True** — A Pareto chart has bars (sorted by frequency) AND a line showing cumulative percentage. </details> > **Q7: Chart Criticism** > > A magazine shows a pie chart with 12 categories, some with 3D effect, and the percentages sum to 97%. Critique this visualization. > > <details> <strong>Solution</strong> > > **Problems identified:** > > 1. **Too many categories (12):** Pie slices will be too thin to distinguish. Bar chart would be better. > > 2. **3D effect distorts perception:** The 3D perspective makes slices at the front look larger than they are. This is a well-known visual bias. > > 3. **Percentages don't sum to 100%:** They sum to 97%, which means 3% of the data is unaccounted for. Either there's a rounding error, or some data was omitted. > > 4. **Possible missing "Other" category:** If some categories were combined into "Other," it's not shown. > > > **Better approach:** A bar chart (or horizontal bar chart) with categories sorted by frequency, with a clear Y-axis starting at 0, and percentages that sum to 100%. </details> > **Q8: Application — Survey Analysis** > > You work for a streaming service. A survey of 1,000 users asks: "What's your primary reason for subscribing?" Results: > >

a) Compute relative frequencies and percentages. b) What chart would you use to present this to management? c) Write a one-paragraph summary of the findings.

<details> <strong>Solution</strong>

a) Relative frequencies and percentages:

b) Recommended chart: A Pareto chart would be excellent here because:

• It shows the relative importance of each reason
• The cumulative line demonstrates that Content library + Price account for 70% of reasons
• Management can see where to focus retention efforts

Alternatively, a bar chart sorted by frequency works well.

c) Summary paragraph: "The primary driver of subscriptions is content — 42% of users subscribe for the content library. Price is the second most important factor (28%), followed by original shows (18%). User interface and customer service together account for only 12% of subscriptions. This suggests that investment in content acquisition should be the top priority for retention and growth, while pricing strategy and original content production are secondary levers."

</details> * * * ## 🔗 Cross-References - **Next topic:** [Central Tendency — Mean, Median, and Mode](/notes/01-foundation-bsma1002-stats-1-week03-03-central-tendency) — now that we can summarize categorical data, we move to numerical data - **Previous topic:** [Data Types & Scales](/notes/01-foundation-bsma1002-stats-1-week01-01-data-types-scales) — categorical data definitions - **Week 4 (Contingency Tables):** Extending frequency analysis to two categorical variables - **BSMA1004 (Stats 2):** Chi-square tests for categorical data - **BSMS2002 (Business Analytics):** Market segmentation using categorical analysis [Join Discord](https://discord.gg/gE2m4Qrdqv) [Previous**Sampling Methods**](/notes/01-foundation-bsma1002-stats-1-week01-00a-sampling-methods)[Next**Central Tendency**](/notes/01-foundation-bsma1002-stats-1-week03-03-central-tendency)