Cheatsheet: Analyzing One Categorical Variable

1. Categorical Variables

1.1 Definition and Types

1.2 Data Collection

• Record each observation as a category label
• Ensure categories are mutually exclusive (no overlap)
• Ensure categories are exhaustive (cover all possibilities)
• May include an "Other" category for rare responses

2. Frequency Distributions

2.1 Frequency Tables

2.2 Key Formulas

3. Graphical Displays

3.1 Bar Chart

• Vertical or horizontal bars representing frequency or relative frequency
• Categories on one axis, frequency/percentage on the other
• Bars are separated by gaps (not touching)
• Bar height corresponds to frequency of category
• Can be ordered by frequency (descending) or alphabetically

3.2 Pie Chart

• Circle divided into slices representing proportions
• Angle of each slice = (relative frequency) × 360°
• Best used when categories sum to meaningful whole
• Most effective with 2-5 categories
• Difficult to compare similar proportions

3.3 Pareto Chart

• Bar chart with categories ordered by decreasing frequency
• Often includes cumulative percentage line
• Used to identify most important categories
• Helps apply 80/20 rule (80% of effects from 20% of causes)

3.4 Comparison of Graphs

4. Measures of Center

4.1 Mode

4.2 Properties

• Mode is the only measure of center for categorical variables
• Mean and median cannot be calculated for nominal variables
• For ordinal variables, median can sometimes be identified
• Mode can be used for any type of data (categorical or numerical)

5. Proportions and Percentages

5.1 Sample Proportion

5.2 Key Formula

• Sample proportion: p̂ = x/n
• 0 ≤ p̂ ≤ 1
• Sum of all category proportions equals 1

6. Two-Way Tables (Contingency Tables)

6.1 Structure

6.2 Types of Distributions

6.3 Analysis Tips

• Marginal distribution examines one variable while ignoring the other
• Conditional distribution examines one variable given a specific category of the other
• Compare conditional distributions to assess relationship between variables
• Row percentages: divide each cell by its row total
• Column percentages: divide each cell by its column total

7. Data Presentation Guidelines

7.1 Creating Frequency Tables

• List all categories in first column
• Include frequency, relative frequency, and/or percentage columns
• Add totals at bottom to verify calculations
• Label all columns clearly with units
• Round percentages to 1 decimal place unless specified otherwise

7.2 Graph Design Principles

• Include clear, descriptive title
• Label both axes with variable name and units
• Use consistent scale intervals
• Start vertical axis at zero for bar charts
• Order categories logically (frequency, alphabetical, or natural order)
• Avoid 3D effects that distort comparisons
• Use color/shading consistently

8. Common Applications

8.1 Survey Analysis

• Summarize responses to categorical survey questions
• Identify most common response (mode)
• Calculate percentage of respondents in each category
• Display results with bar or pie charts

8.2 Quality Control

• Classify defects by type or cause
• Use Pareto charts to prioritize improvement efforts
• Track frequency of nonconforming items
• Monitor proportions over time

8.3 Demographic Analysis

• Classify populations by gender, race, education level, etc.
• Calculate proportions in each demographic category
• Use two-way tables to examine relationships between demographics
• Compare conditional distributions across groups

9. Common Errors to Avoid

9.1 Data Collection

• Overlapping categories (not mutually exclusive)
• Missing categories (not exhaustive)
• Ambiguous category definitions
• Forcing ordinal structure on nominal variables

9.2 Analysis and Display

• Using mean or median for nominal variables
• Creating pie charts with too many categories (>5-7)
• Forgetting to label axes and provide titles
• Not verifying that frequencies sum to total
• Not verifying that proportions sum to 1.0
• Starting bar chart axis at non-zero value
• Using touching bars (histogram style) for categorical data

10. Summary Checklist

10.1 Analyzing One Categorical Variable

• Identify variable type (nominal or ordinal)
• Create frequency distribution table with counts, proportions, percentages
• Verify totals: Σf = n and Σp = 1.0
• Determine mode (most frequent category)
• Select appropriate graph (bar chart, pie chart, or Pareto chart)
• Label all displays clearly and completely
• Interpret results in context of original question