Putting Two-Way Table Skills Together
A two-way table can answer several different kinds of questions. It can describe a category across the whole group, a combination of two categories, or the share within a specified group. The arithmetic may look similar in each case, but the denominator changes with the question.
In Marginal Distributions From a Two-Way Table, Joint Relative Frequencies, and Conditional Distributions by Row and Conditional Distributions by Column, you learned these calculations separately. This tutorial brings them together in mixed practice. As in Choosing the Correct Denominator for a Percentage, start by identifying the group named in the question. Then select the total that represents that group.
A useful new exam technique is to make a quick denominator map before calculating. Beside each requested quantity, note whether its denominator is the grand total, a row total, or a column total. This small step separates questions that sound alike but ask for different proportions.
Underline the category or combination the question asks about.
“Of all” points to the grand total. “Of those in this row” or “among this column” points to that row or column total.
Use the cell count for a joint or conditional category, or a margin total for a marginal category.
Name the group and category the proportion describes. If comparing groups, compare their conditional percentages.
A percentage without its denominator is difficult to interpret. For example, “60% completed the quiz” could describe everyone in a table or only one row group. State what the percentage is “of,” and include the count and denominator when that makes the meaning clearer.
Worked Example: Marginal, Joint, and Conditional Proportions
Worked Example: Marginal, Joint, and Conditional Proportions
A fictional survey at a community learning center records whether each attendee joined a class in person or online and whether the attendee completed a practice quiz. Use the table to answer the questions below. Treat a randomly selected attendee as one of the 180 surveyed attendees.
| Class format | Completed quiz | Did not complete quiz | Total |
|---|---|---|---|
| In person | 72 | 18 | 90 |
| Online | 54 | 36 | 90 |
| Total | 126 | 54 | 180 |
First check the margins: \(72+18=90\), \(54+36=90\), and \(90+90=180\). The column totals also check: \(72+54=126\) completed quizzes and \(18+36=54\) not completed. These checks help catch table-reading or arithmetic errors before calculating proportions.
1. What proportion of all surveyed attendees joined online? This is marginal: the group is everyone in the table, so use the grand total.
Thus, 50% of the surveyed attendees joined online. The denominator is not the online row total; that total would describe a percentage within the online group.
2. What proportion of all surveyed attendees joined online and completed the quiz? This is joint: both categories must be true, so use the inside cell and divide by the grand total.
Thirty percent of all surveyed attendees both joined online and completed the quiz. This describes a share of the entire group, not the share of online attendees who completed the quiz.
3. Among in-person attendees, what percentage completed the quiz? “Among in-person attendees” names the row group, so use the in-person row total.
Eighty percent of the in-person attendees completed the quiz. Among online attendees, the corresponding percentage is:
The difference is \(80\%-60\%=20\) percentage points. In this survey, quiz completion was more common among in-person attendees than among online attendees. The conditional distributions differ, so the table shows an association between class format and quiz completion among these surveyed attendees. This descriptive comparison does not establish that class format caused the difference.
4. Among attendees who completed the quiz, what percentage joined online? The group is now the completed-quiz column, not the online row.
About 42.9% of attendees who completed the quiz joined online. Notice the change from the earlier conditional question: “percentage of online attendees who completed” was \(54/90=60\%\), while “percentage of quiz completers who joined online” is \(54/126\approx42.9\%\). The same cell count appears in both calculations, but the group in the wording determines the denominator.
Worked Example: Do Not Reverse the Condition
Worked Example: Do Not Reverse the Condition
A fictional community survey records dwelling type and whether a household composts food scraps. Answer each question, then explain why two conditional proportions that use the same cell can differ.
| Dwelling type | Composts | Does not compost | Total |
|---|---|---|---|
| Apartment | 24 | 56 | 80 |
| Single-family house | 96 | 64 | 160 |
| Total | 120 | 120 | 240 |
What proportion of all households are apartments and compost? Both categories are specified, so this is a joint proportion.
Ten percent of all surveyed households were apartments that compost. For the marginal proportion that composts, ignore dwelling type and use the composting column total:
Now compare two different conditional questions. Among apartment households, the percentage that composts is:
Among households that compost, the percentage that are apartments is:
The first percentage is conditional on dwelling type; the second is conditional on composting. Both use the apartment-and-compost cell, but the reference groups—and therefore the denominators—are different. A useful way to check your reading is to finish the phrase “___ percent of ___.” Here, the answers are “30% of apartment households” and “20% of composting households.”
To consider whether dwelling type and composting are associated in the table, compare the percentage that composts within each dwelling group. It is 30% for apartment households and:
for households in single-family houses. The conditional percentages differ by \(60\%-30\%=30\) percentage points. Thus, composting was more common among the surveyed single-family-house households. This is an observed association in the survey, not evidence that dwelling type causes composting behavior.
Worked Example: Margins and a Multi-Category Distribution
Worked Example: Margins and a Multi-Category Distribution
A fictional survey of 150 visitors to a community workshop records whether each visitor is a member and the visitor’s rating of the workshop. Find the overall proportion who were satisfied, the joint proportion who were members and satisfied, and the conditional rating distributions for members and nonmembers. Then write a conclusion about the pattern.
| Membership | Satisfied | Neutral | Dissatisfied | Total |
|---|---|---|---|---|
| Member | 40 | 20 | 10 | 70 |
| Nonmember | 24 | 36 | 20 | 80 |
| Total | 64 | 56 | 30 | 150 |
The overall proportion satisfied is marginal because it describes the entire group:
The joint proportion who were members and satisfied is:
For the conditional distribution among members, divide each member count by the member row total of 70:
The percentages describe, respectively, satisfied, neutral, and dissatisfied members. They add to 100% after rounding. Among nonmembers, use the nonmember row total of 80:
These conditional distributions differ. For example, 57.1% of members were satisfied, compared with 30% of nonmembers, a difference of about 27.1 percentage points. A complete interpretation is: “Among the surveyed workshop visitors, membership and workshop rating appear to be associated because the conditional distributions differ. About 57.1% of members were satisfied, compared with 30% of nonmembers; neutral and dissatisfied ratings were also more common among nonmembers. This table describes an association among these visitors, but does not show that membership caused the ratings.”
When an outcome has more than two categories, compare the whole conditional distribution when possible. A single percentage can highlight an important difference, but the remaining percentages give a fuller account of how the groups’ responses are distributed.
Common Mistakes and AP Exam Tips
- Using the grand total for every calculation. The grand total is right for marginal and joint proportions, but a conditional proportion needs the total for its stated group.
- Using a row total when the question names a column group, or vice versa. Mark the group named after “among,” “of those,” or “for.” Then use that group’s total.
- Reversing a conditional statement. “Of apartment households, what proportion composts?” and “of households that compost, what proportion are apartments?” are not the same question.
- Confusing a joint proportion with a conditional proportion. Both use a cell count, but the joint proportion divides by the grand total; the conditional proportion divides by the total for the specified group.
- Reporting a number without its meaning. State the category, group, and denominator in words. “30%” is incomplete; “30% of apartment households compost” identifies what the percentage describes.
- Comparing counts when group sizes differ. Use conditional percentages to compare shares across groups, and state the percentage-point difference if it helps describe the size of the gap.
- Claiming causation from a descriptive table. Say that the variables are associated in the displayed data when conditional distributions differ. Do not claim that one variable caused the other.
For a full-credit exam response, show the count divided by the correct total, give the resulting proportion or percentage, and interpret it in context. For a comparison, identify both groups, report the relevant conditional percentages, and describe what differs. Check that marginal totals match the grand total and that each conditional distribution adds to 100%, allowing for rounding.
Check Your Understanding
For each question, identify the type of proportion, choose the denominator, and interpret the result in context.
- A table has 45 ninth-graders who use a school app, 30 ninth-graders who do not, and 75 ninth-graders total. What proportion of ninth-graders use the app? What kind of proportion is this?
- In a survey of 200 residents, 28 live in apartments and use a bike-share program. What is the joint proportion of residents who are apartment dwellers and bike-share users?
- Of 80 online class attendees, 52 completed an assignment. What percentage of online attendees completed it? Which total belongs in the denominator?
- In a table, 36 of 90 people who attended a workshop were members. What percentage of workshop attendees were members? How would the denominator change if the question instead asked what percentage of members attended the workshop?
- Two groups have conditional percentages of 42% and 57% in a specified category. Find the difference in percentage points and write a cautious sentence describing the comparison.