Let the Wording Choose the Denominator
In Conditional Distributions by Row and Conditional Distributions by Column, you divided a cell count by the total for the group named in the question. The important choice is not simply whether a number appears in a row or column. First identify which individuals the question is about. Phrases such as “of those who,” “among,” and “for people who” often name that group directly.
For example, “Of those who use the school library, what percentage are ninth graders?” asks about library users. The denominator must include all library users, regardless of grade. But “Of ninth graders, what percentage use the library?” asks about ninth graders. The two questions may involve the same cell count, yet their denominators—and answers—can differ.
This is the same denominator logic used in the earlier tutorials on conditional distributions. When the named group is a row category, use that row total; when it is a column category, use that column total. The words in the question tell you which group to find. The table’s layout tells you where that group’s total is.
A Quick Translation From Words to Totals
Before calculating, restate the question as: “Out of all the individuals who are [named group], what percentage are [category of interest]?” The phrase after “out of all” identifies the denominator. The count that satisfies both descriptions goes in the numerator.
Look for wording such as “of ninth graders,” “among customers who chose pickup,” or “of those who compost.”
Use its row total or column total, according to how the two-way table is arranged.
The numerator counts individuals who are both in the named group and in the category being asked about.
Divide the cell count by the group total, multiply by 100%, and name the group and category in your answer.
A useful check is to ask whether the denominator includes everyone described by the group phrase. If it includes only some of that group, it is too small; if it includes people outside that group, it is too large for a conditional percentage. The grand total is right only when the question asks for a percentage of everyone represented in the table.
Worked Examples: The Same Cell, Different Denominators
Worked Example: “Of Ninth Graders” or “Of Library Users”?
A fictional survey records whether students used a school library during a month and their grade. Find the percentage of ninth graders who used the library, the percentage of library users who were ninth graders, and the percentage of all surveyed students who were both ninth graders and library users.
| Library use | Ninth grade | Tenth grade | Row total |
|---|---|---|---|
| Used library | 42 | 18 | 60 |
| Did not use library | 28 | 32 | 60 |
| Column total | 70 | 50 | 120 |
Question 1: “Of ninth graders, what percentage used the library?” The group phrase is “ninth graders,” so the denominator is the Ninth grade column total, 70. Of those 70 students, 42 used the library.
In this survey, 60% of ninth graders used the library. This is a column percentage because the group named in the question is a column category.
Question 2: “Of library users, what percentage were ninth graders?” Now the group phrase is “library users,” so the denominator is the Used library row total, 60. The count who were both library users and ninth graders remains 42.
In this survey, 70% of students who used the library were ninth graders. This is a row percentage. It is not a different calculation of the same question: it answers a different question about a different group.
Question 3: “What percentage of all surveyed students were ninth graders who used the library?” The wording asks about all surveyed students, so the denominator is the grand total, 120.
In this survey, 35% of all surveyed students were ninth graders who used the library. This joint percentage is a share of the whole sample, not a percentage within the ninth-grade group or within the library-user group.
Side by side, \(42/70\), \(42/60\), and \(42/120\) all use the same cell count. The different denominators follow the different groups named in the questions. The phrases “of ninth graders,” “of library users,” and “of all surveyed students” are not interchangeable.
Worked Example: A Denominator That Follows the Question, Not the Column
A fictional consumer survey records how customers placed an order and whether they selected pickup or delivery. Orders are classified by fulfillment method in rows and ordering method in columns. Find the percentage of app orders that were for pickup and the percentage of pickup orders that came through the app.
| Fulfillment method | App | Website | Row total |
|---|---|---|---|
| Pickup | 54 | 30 | 84 |
| Delivery | 36 | 60 | 96 |
| Column total | 90 | 90 | 180 |
“Of customers who ordered through the app, what percentage chose pickup?” The named group is app customers. Its total is 90, and 54 of those customers chose pickup.
In this survey, 60% of customers who ordered through the app chose pickup. The denominator is the App column total, even though Pickup is the row containing the cell.
“Of customers who chose pickup, what percentage ordered through the app?” This wording names pickup customers as the group. The Pickup row total is 84, of whom 54 ordered through the app.
In this survey, about 64.3% of customers who chose pickup ordered through the app. The cell count is still 54, but the denominator has changed from the App column total to the Pickup row total. Dividing by the grand total instead would answer a third question: \(54/180\times100\%=30\%\) of all surveyed orders were app orders for pickup.
The table’s orientation alone does not determine the denominator. For the first question, use a column total because the group is in a column. For the second, use a row total because the group is in a row. Always follow the wording.
Worked Example: Compare Percentages Within Two Groups
A fictional neighborhood survey asks residents whether they compost and whether they live in an apartment or a house. Compare the percentage who compost in the two housing groups. Then distinguish that comparison from the percentage of composters who live in a house.
| Composting | Apartment | House | Row total |
|---|---|---|---|
| Compost | 42 | 72 | 114 |
| Do not compost | 58 | 48 | 106 |
| Column total | 100 | 120 | 220 |
We want to compare the percentage who compost among apartment residents with the percentage among residents who live in a house.
The group phrases are “apartment residents” and “residents who live in a house.” Each is a column group, so divide the Compost count in each column by that column’s total.
Apartment residents: \(42/100\times100\%=42\%\). House residents: \(72/120\times100\%=60\%\). The difference is \(60\%-42\%=18\) percentage points.
In this survey, 42% of apartment residents composted, compared with 60% of residents who lived in a house. The percentage was 18 percentage points higher among house residents.
The group in the question controls each denominator: 100 apartment residents and 120 house residents. If the question instead asks “Of those who compost, what percentage live in a house?”, the group is all composters, so use the Compost row total, 114.
Thus, about 63.2% of the residents who composted lived in a house. This answers a different question from “What percentage of house residents compost?” The first describes the housing categories among composters; the second describes composting within a housing group. A percentage-point comparison is appropriate for the two within-group percentages, 42% and 60%; it does not compare the 63.2% answer with either of those as though they described the same distribution.
When the Denominator Is the Grand Total
The grand total is not a “wrong” denominator by itself. It is the correct denominator when the question asks what percentage of the entire sample falls into a combination of categories. In the library example, 35% of all surveyed students were both ninth graders and library users. That describes the joint category represented by one cell.
By contrast, “Of ninth graders, what percentage used the library?” asks about a subset defined by grade. Dividing by the grand total would include tenth graders, who are outside the group named by the question. Similarly, dividing by the library-user total would answer “Of library users, what percentage were ninth graders?” rather than the question about ninth graders.
Use the following restatement to tell these questions apart:
- Conditional: “Out of the people who are in this named group, what percentage are in the category of interest?” Use the named group’s total.
- Joint: “What percentage of everyone is in both categories?” Use the grand total.
If a question says only “What percentage are ninth graders who used the library?” and does not specify “of ninth graders,” “of library users,” or “of everyone,” pause before calculating. The wording may be incomplete: ask which overall group is the reference group. A cell count alone does not identify which of several legitimate percentages is wanted.
Common Mistakes and AP Exam Tips
- Dividing by the total of the category named after “what percentage are.” In “Of ninth graders, what percentage used the library?”, ninth grade is the denominator group. Library users are the outcome category in the numerator; the question is not asking what share of library users were ninth graders.
- Using the column total just because the cell is in that column. The wording may name the row group instead. In “Of pickup customers, what percentage ordered through the app?”, use the Pickup row total.
- Using the grand total for a within-group question. This changes the reference group to everyone in the sample and gives a joint percentage instead of the requested conditional percentage.
- Reporting a number without naming its group. “60% composted” is unclear if the table separates apartments and houses. A complete interpretation names the group: “60% of surveyed residents who lived in a house composted.”
- Assuming two percentages must match because they use the same cell. A cell can be divided by its row total, column total, or grand total, depending on the question. The denominator determines what the resulting percentage describes.
- Calling a percentage-point difference a percent difference. When subtracting two percentages, describe the result in percentage points. For example, 60% compared with 42% is a difference of 18 percentage points.
For full credit, show the cell count divided by the total for the group named in the question, convert to a percentage, and interpret the result in context. If two groups are compared, calculate each percentage with its own group total, state both percentages, and report their difference in percentage points.
Check Your Understanding
Use the tables and contexts in this tutorial. For each question, name the group in the denominator before calculating.
- In the school survey, what percentage of tenth graders did not use the library? Give the cell count, denominator, and interpretation.
- In the consumer survey, what percentage of website orders were for delivery? Which total is the denominator?
- In the neighborhood survey, what percentage of apartment residents did not compost? Explain why the Compost row total is not the denominator.
- In the neighborhood survey, what percentage of all surveyed residents both lived in an apartment and composted? How does this differ from the answer to question 3?
- For the app-and-pickup cell, explain what question is answered by dividing 54 by 90 and what question is answered by dividing 54 by 84.