Summarizing One Variable From a Two-Way Table
In Reading a Two-Way Table: Rows, Columns, and Totals, you learned to identify cell counts, row totals, column totals, and the grand total. A two-way table can also answer a question about just one of its variables. To do that, use the totals along the table’s margins: the ends of the rows or the bottoms of the columns.
For example, if a table classifies households by both home type and heating system, a question about home types across all households does not require separating the households by heating system. Add across the rows to get each home-type count, then compare those counts with the number of households in the entire table. This summary is called a marginal distribution.
A marginal distribution turns attention to one variable at a time. Its count for a category is a margin total, and its percentage is that count divided by the table’s grand total. The percentage answers what fraction of all individuals represented in the table belong to that category, regardless of their category in the other variable.
Calculate a Marginal Count and Percentage
Consider a fictional survey of 240 households. Each household is classified by home type and primary heating system. Home type is displayed in rows; heating system is displayed in columns.
| Home type | Electric | Gas | Other | Row total |
|---|---|---|---|---|
| Apartment | 30 | 24 | 18 | 72 |
| Detached house | 36 | 54 | 18 | 108 |
| Townhouse | 24 | 18 | 18 | 60 |
| Column total | 90 | 96 | 54 | 240 |
The row totals summarize home type across all heating systems. For instance, the Apartment row total is \(30+24+18=72\). The column totals summarize heating system across all home types: \(30+36+24=90\) households use electric heat. Both are marginal counts. The grand total is 240 households.
Use the grand total as the denominator for a marginal percentage, whether the category is a row category or a column category. The denominator represents the full group in the table. In the household example, each marginal percentage is calculated using 240 households.
For a home-type marginal distribution, first use the row totals and then divide each by 240. For a heating-system marginal distribution, use the column totals and divide each by 240. Each distribution describes one variable across all households.
Worked Examples: Marginal Distributions
Worked Example: Find the Home-Type Marginal Distribution
Use the household table. Find the marginal count and percentage for each home type, and interpret the result for detached houses.
Find the marginal counts: The home-type counts are the row totals: 72 apartments, 108 detached houses, and 60 townhouses. Check that they include every household:
Calculate the percentages: Divide each row total by the grand total, 240, and multiply by 100%.
The home-type marginal distribution is 72 apartments (30%), 108 detached houses (45%), and 60 townhouses (25%). In context, 45% of the 240 surveyed households were detached houses, regardless of which heating system they used. The percentages add to \(30\%+45\%+25\%=100\%\), as expected for a distribution covering all households.
Worked Example: Find the Heating-System Marginal Distribution
Using the same table, find the marginal count and percentage for each heating system. Which heating system was most common overall among the households represented?
Use the column totals: The counts are 90 households using electric heat, 96 using gas heat, and 54 using another heating system. Verify that the column totals add to the grand total:
Convert each count to a percentage of all 240 households:
The heating-system marginal distribution is 90 households (37.5%) using electric heat, 96 (40%) using gas heat, and 54 (22.5%) using another system. Gas heat was most common overall in this group because its marginal count, 96, is the largest. The percentages add to \(37.5\%+40\%+22.5\%=100\%\).
Notice what the percentages describe: for example, 40% is the percentage of all 240 households that used gas heat. It is not calculated using only one home-type row. The question is about heating system across the whole group, so the column total is compared with the grand total.
Worked Example: Read Both Margins in a Different Context
A fictional survey records how 160 residents access community alerts and groups them by age. Each resident appears once in the table. Calculate both marginal distributions and interpret one percentage from each.
| Age group | Phone | Computer | Other | Row total |
|---|---|---|---|---|
| 18–39 | 52 | 18 | 10 | 80 |
| 40 and older | 36 | 24 | 20 | 80 |
| Column total | 88 | 42 | 30 | 160 |
Age-group margin: Divide each row total by 160:
The age-group marginal distribution is 80 residents aged 18–39 (50%) and 80 residents aged 40 and older (50%). In context, half of the surveyed residents were in each age group, without separating them by how they accessed alerts.
Access-method margin: The column totals are 88 for phone, 42 for computer, and 30 for other. Divide each by the grand total, 160:
The access-method marginal distribution is 88 residents (55%) using a phone, 42 (26.25%) using a computer, and 30 (18.75%) using another method. In context, 55% of all 160 surveyed residents accessed community alerts by phone. The percentages add to \(55\%+26.25\%+18.75\%=100\%\). The row totals and column totals each add to 160, which checks both margins against the grand total.
How to Choose the Right Margin
Start by identifying the variable named in the question. If it is the row variable, its categories are summarized by row totals. If it is the column variable, its categories are summarized by column totals. Then check whether the question asks for a count or a percentage. A count is the margin total itself; a percentage requires dividing that count by the grand total.
Read the row and column headings. Decide which one the question asks you to summarize.
Use the row totals for a row variable or the column totals for a column variable.
For each category, divide its marginal count by the grand total and multiply by 100%.
Confirm that marginal counts add to the grand total and percentages add to 100%, allowing for small rounding differences. State what the result means in context.
The row and column margins each give a separate summary of one variable. They are not two pieces of a single distribution: one summarizes the row variable, and the other summarizes the column variable. In the household table, the home-type percentages add to 100%, and the heating-system percentages also add to 100%. Each set is a complete summary of its own variable.
As in Frequency Tables and Relative Frequency, percentages express category counts relative to a total. The important choice here is which total to use. For a marginal distribution, the denominator is the grand total, because the percentage is about a category among all individuals in the table.
Common Mistakes and AP Exam Tips
- Using a cell count instead of a marginal count. A cell describes two categories together. A marginal count is a row or column total. If asked how many households used gas heat overall, use the gas column total, not one cell in that column.
- Dividing by a row or column total. For a marginal percentage, use the grand total. Dividing by a row total or column total changes the group being described and does not give the marginal percentage requested.
- Choosing the wrong margin. Follow the variable named in the question. Row totals summarize the row variable; column totals summarize the column variable. Do not assume that one kind of variable is always in the rows.
- Leaving out the context. A full-credit interpretation names the category, percentage, and group. For example: “Among the 240 households surveyed, 40% used gas heat,” rather than just “40%.”
- Assuming a rounded total must be exactly 100%. If percentages are rounded, their displayed sum can differ slightly from 100%. Check the unrounded fractions if needed and state that a small discrepancy is due to rounding.
- Reporting percentages without counts when both are requested. Give the marginal count and the percentage, and identify the denominator used. This makes clear how the percentage was calculated.
Check Your Understanding
Use the 240-household table to answer the questions. Show the division used for each requested percentage and describe the result in context.
- What is the marginal count and percentage for apartments?
- What is the marginal count and percentage for gas heat? What group is the denominator?
- Which margin would you use to summarize home type: row totals or column totals? Explain.
- Use the 240-household table to find the percentage of households using another heating system.
- Why should the home-type percentages add to 100%, and what does that total represent?