From Conditional Percentages to Clear Comparisons
In the previous tutorial, Is There an Association Between Two Categorical Variables, we compared conditional distributions to see whether they differed across groups. This tutorial focuses on the next step: writing a precise sentence about a particular percentage in each group. A good comparison tells readers which groups are being compared, which outcome or category the percentages describe, and how the percentages differ.
As in the earlier tutorials on Conditional Distributions by Row and Choosing the Correct Denominator for a Percentage, the denominator comes from the group named in the comparison. If the question is “What percentage of apartment residents use this service?”, divide the count of apartment residents who use it by the total number of apartment residents. Then compare that percentage with the corresponding percentage for the other group.
A useful comparison sentence has a reliable structure: “In [group A], [percentage] of individuals were in [outcome category], compared with [percentage] in [group B]; the percentage was [difference] percentage points higher in [group].” This wording makes the denominator groups visible and states the direction of the difference.
The order of subtraction determines the sign. A positive result means the percentage is higher in group A; a negative result means it is lower in group A. You can also state the size of the difference as a positive number and then name which group has the higher percentage.
For example, a difference between 39% and 35% is 4 percentage points, not 4%. Percentage points describe the distance between two percentages. Saying “4% higher” can be ambiguous, so for AP Statistics comparisons, report the difference in percentage points when subtracting percentages.
Build the Sentence From the Table
Before drafting a comparison, decide what category you want to describe and which groups you are comparing. Check that both percentages refer to the same category and that each denominator is the total for the group named. Then find the difference and state it in context.
Identify the two group categories and the specific response category whose percentages you will compare.
For each group, divide the count in the outcome category by that group’s total and convert the proportion to a percentage.
Subtract in a stated order, or report the absolute difference and identify which group has the higher percentage.
Name the group associated with each percentage, describe the outcome category, and avoid claiming that group membership caused the difference.
Keep the comparison focused. If the outcome has two categories, the percentage for the other category will provide the same difference in the opposite direction. You usually do not need to list every percentage in both conditional distributions when the question asks about one category. If the question asks about the full distributions, describe the other categories too.
Worked Examples: Writing Accurate Comparison Sentences
Worked Example: Comparing Home Energy-Service Sign-Ups
A fictional community survey records whether households signed up for a home energy-update service and whether they live in an apartment or a detached house. Compare the percentage who signed up. Treat the figures as a description of the surveyed households, not as evidence about what caused sign-up.
| Housing type | Signed up | Did not sign up | Total |
|---|---|---|---|
| Apartment | 42 | 78 | 120 |
| Detached house | 78 | 122 | 200 |
The variables are housing type and sign-up status. The comparison is the percentage of households that signed up in apartments versus detached houses.
Find the conditional percentage who signed up within each housing group. Use 120 as the apartment denominator and 200 as the detached-house denominator.
For apartments, \(42/120\times100\%=35\%\). Check: \(42\div120=0.35\), and \(0.35\times100\%=35\%\). For detached houses, \(78/200\times100\%=39\%\). Check: \(78\div200=0.39\), and \(0.39\times100\%=39\%\). The difference, apartment minus detached house, is \(35\%-39\%=-4\) percentage points. Equivalently, the detached-house percentage is \(39\%-35\%=4\) percentage points higher.
Among the households surveyed, 35% of apartment households signed up, compared with 39% of detached-house households; the sign-up percentage was 4 percentage points higher for detached-house households.
Notice how the final sentence connects each percentage to its own group. It does not say that detached-house households signed up more often because of their housing type. The survey table describes a difference; it does not by itself establish a cause.
Worked Example: Comparing E-Book Checkout Percentages by Grade
A fictional school survey records whether students checked out an e-book from the school library during a particular month. Compare the percentages for grade 9 and grade 10 students.
| Grade | Checked out an e-book | Did not check one out | Total |
|---|---|---|---|
| Grade 9 | 54 | 36 | 90 |
| Grade 10 | 45 | 55 | 100 |
Among grade 9 students, the percentage who checked out an e-book is \(54/90\times100\%=60\%\). Checking the decimal, \(54\div90=0.60\), which is 60%. Among grade 10 students, the percentage is \(45/100\times100\%=45\%\); \(45\div100=0.45\), or 45%.
The grade 9 percentage minus the grade 10 percentage is \(60\%-45\%=15\) percentage points. Thus, a suitable sentence is: “In this survey, 60% of grade 9 students checked out an e-book during the month, compared with 45% of grade 10 students; the percentage was 15 percentage points higher among grade 9 students.”
The sentence identifies the month, the outcome, both groups, and the direction and size of the difference. It does not say that being in grade 9 made students check out e-books. Students were surveyed, not randomly assigned to grades, and the table alone cannot explain why the percentages differ.
Worked Example: When the Larger Count Has the Lower Percentage
A fictional commuter survey records whether people arrived before 8:00 a.m. and their usual travel mode. Compare the percentage arriving before 8:00 for shuttle riders and cyclists. This table illustrates why comparing counts alone can give the wrong impression.
| Usual travel mode | Arrived before 8:00 a.m. | Arrived at or after 8:00 a.m. | Total |
|---|---|---|---|
| Shuttle | 32 | 48 | 80 |
| Bicycle | 54 | 126 | 180 |
For shuttle riders, \(32/80\times100\%=40\%\); the check \(32\div80=0.40\) also gives 40%. For cyclists, \(54/180\times100\%=30\%\); the check \(54\div180=0.30\) gives 30%. The bicycle group has more people arriving before 8:00 in raw counts, 54 compared with 32, but it also has a larger total.
The conditional percentages show the comparison more clearly. Shuttle riders’ percentage is \(40\%-30\%=10\) percentage points higher than cyclists’ percentage. A complete sentence is: “Among the commuters surveyed, 40% of shuttle riders arrived before 8:00 a.m., compared with 30% of cyclists; the percentage was 10 percentage points higher for shuttle riders.”
This is a descriptive comparison of the surveyed commuters. It would overstate what the table shows to write “taking the shuttle caused commuters to arrive earlier.” Other factors could be related to both travel mode and arrival time, and the table by itself does not establish a cause.
Describe the Difference, Not an Unproven Cause
A comparison sentence should report what the percentages in the data show. Words such as “higher,” “lower,” and “compared with” make a descriptive comparison. Words such as “caused,” “led to,” or “resulted in” make a causal claim. Do not use causal language just because the table displays a difference.
The same caution applies to phrases like “the group was more likely to.” Such phrasing can be understood as a comparison of the observed percentages, but “a larger percentage of the group” is often clearer and less likely to sound like a causal explanation. In any case, state the actual percentages and the groups so the reader can see what you mean.
Also take care with the word “increase.” Saying one percentage “increased” can suggest a change over time or an effect. If the table compares two different groups at one time, say that one group’s percentage was higher than the other’s. Use “increased” only when the context actually describes a change and the data support that description.
A percentage-point difference is descriptive, not a verdict about whether an association is “important.” Context matters: a difference of 10 percentage points may be noteworthy in one setting and modest in another. For this tutorial, the task is to report the observed comparison accurately, not to apply a universal cutoff or infer a population effect.
Common Mistakes and AP Exam Tips
- Using the wrong denominator. To report the percentage of grade 9 students who checked out an e-book, divide by all surveyed grade 9 students, not by everyone who checked out an e-book. State what group the percentage is “of.”
- Comparing counts as if they were percentages. A group with more individuals may have a larger count even when a smaller share belongs to the outcome category. Divide by each group total before comparing.
- Writing “15% higher” when the subtraction gives 15 percentage points. If you subtract 60% and 45%, report a 15-percentage-point difference. This says how far apart the two percentages are.
- Leaving the direction unclear. “The difference was 10 percentage points” gives the size but not which group had the higher percentage. Name the group with the higher percentage.
- Leaving the context out. “40% versus 30%” does not identify the groups or the outcome. A full-credit sentence says 40% of whom, 30% of whom, and what each percentage describes.
- Implying causation. “Shuttle travel made commuters arrive earlier” goes beyond a descriptive table. Say that the observed percentage was higher among shuttle riders, and avoid claiming why.
- Reversing the conditional statement. “40% of shuttle riders arrived early” is not equivalent to “40% of early arrivals rode the shuttle.” Those statements use different denominator groups.
For a strong AP response, use the correct conditional percentages, identify the group each percentage describes, give the difference in percentage points when appropriate, and write the comparison in context. Report what the data show without treating an association as proof of cause and effect.
Check Your Understanding
Use the group totals as denominators and write comparisons that describe the data without making causal claims.
- In the energy-service example, what percentage of apartment households did not sign up? Show the numerator and denominator.
- In the e-book example, by how many percentage points did the grade 9 and grade 10 checkout percentages differ? Which grade had the higher percentage?
- In the commuter example, why is it misleading to conclude that cyclists had the larger early-arrival percentage from the counts alone?
- A table shows that 28% of Group A and 36% of Group B selected an option. Write a comparison sentence, including the difference in percentage points.
- What is the problem with the sentence “Living in a detached house caused a higher sign-up rate” when the only evidence is the survey table?