Cheatsheet: Two-Way Tables

1. Two-Way Table Fundamentals

1.1 Definition and Structure

1.2 Table Components

• Joint Frequencies: Counts in individual cells representing both variables simultaneously
• Row Totals: Sum of all frequencies in each row (marginal totals for row variable)
• Column Totals: Sum of all frequencies in each column (marginal totals for column variable)
• Grand Total: Sum of all frequencies; equals sum of row totals or sum of column totals
• Marginal Distribution: Distribution of one variable alone, shown in row/column totals

2. Frequencies and Counts

2.1 Types of Frequencies

2.2 Reading Two-Way Tables

• Identify row and column variables and their categories
• Locate joint frequencies in interior cells
• Find marginal frequencies in row/column totals
• Verify grand total = sum of any complete row/column of marginal totals

3. Relative Frequencies

3.1 Types of Relative Frequencies

3.2 Converting to Percentages

• Multiply any relative frequency by 100 to get percentage
• Joint relative frequencies sum to 1 (or 100%)
• All marginal relative frequencies for one variable sum to 1 (or 100%)
• Conditional relative frequencies within one row/column sum to 1 (or 100%)

4. Conditional Distributions

4.1 Definition and Purpose

4.2 Calculating Conditional Distributions

4.2.1 Conditional Distribution Given Row

• Fix a specific row
• Divide each cell in that row by the row total
• Results show distribution of column variable for that row category

4.2.2 Conditional Distribution Given Column

• Fix a specific column
• Divide each cell in that column by the column total
• Results show distribution of row variable for that column category

4.3 Interpreting Conditional Distributions

• Compare conditional distributions across different rows or columns to identify patterns
• If conditional distributions differ substantially, variables may be associated
• If conditional distributions are similar across all rows/columns, variables may be independent

5. Association and Independence

5.1 Association Between Variables

5.2 Identifying Association

• Compare conditional distributions across rows or columns
• If conditional distributions differ, association exists
• If conditional distributions are the same, no association exists
• Small differences may occur due to random variation in samples

5.3 Independence

5.4 Expected Frequencies Under Independence

• Expected frequency = (Row total × Column total) ÷ Grand total
• Compare observed frequencies to expected frequencies
• Large differences suggest association exists
• Small differences suggest variables may be independent

6. Segmented Bar Charts

6.1 Definition and Purpose

6.2 Construction

• Create one bar for each category of one variable (conditioning variable)
• Divide each bar into segments representing categories of the other variable
• Segment heights represent conditional relative frequencies
• All bars have same total height (100% or 1.0)

6.3 Interpretation

• Compare segment proportions across bars to assess association
• Similar segment patterns across bars suggest no association
• Different segment patterns suggest association exists
• Easier visual comparison than examining numerical tables

7. Common Applications and Calculations

7.1 Key Formulas

7.2 Problem-Solving Steps

• Step 1: Identify the two categorical variables and their categories
• Step 2: Locate or calculate grand total
• Step 3: Find appropriate row/column totals for the question
• Step 4: Calculate requested frequency (joint, marginal, or conditional)
• Step 5: Convert to percentage if required (multiply by 100)

7.3 Common Question Types

8. Important Distinctions

8.1 Conditional vs. Joint Probabilities

8.2 Row vs. Column Conditional Distributions

• Row conditional: Fix a row, divide by row total; shows column variable distribution
• Column conditional: Fix a column, divide by column total; shows row variable distribution
• Choice depends on which variable you are "given" or conditioning on

8.3 Association vs. Causation

• Two-way tables can reveal association between variables
• Association does not imply causation
• Confounding variables may explain observed associations
• Additional study designs needed to establish causation

9. Common Mistakes to Avoid

• Using wrong denominator: grand total vs. row total vs. column total
• Confusing joint and conditional relative frequencies
• Failing to verify calculations sum to 1.0 or 100%
• Comparing joint frequencies instead of conditional distributions for association
• Forgetting to convert relative frequencies to percentages when requested
• Misidentifying which variable is the condition in conditional probability questions
• Reversing row and column when calculating conditional distributions
• Assuming association implies causation

10. Quick Reference

10.1 Calculation Checklist

10.2 Verification Checks

• All joint relative frequencies sum to 1
• Marginal relative frequencies for one variable sum to 1
• Conditional relative frequencies in one row/column sum to 1
• Row totals sum to grand total
• Column totals sum to grand total