From a Table to a Supported Conclusion
A two-way table can show whether two categorical variables are associated in the data. Writing a conclusion takes one more step: use relevant conditional percentages to explain what pattern you see and support your statement with numbers. A conclusion that says only “the variables are associated” does not show which groups differ or how the table supports that claim.
In Is There an Association Between Two Categorical Variables and Comparing Conditional Percentages in Context, you learned to look for differences between conditional distributions and compare percentages within their own groups. Here, the focus is on turning that comparison into a concise written response. As emphasized in Common Errors Reading Two-Way Tables, first identify the group named by each percentage and use that group’s total as the denominator.
For a clear response, choose one variable as the grouping variable and compare the conditional distribution of the other variable across its categories. If the outcome has several categories, compare the distribution rather than relying on a single cell. Name the category or outcome of interest, report the percentages with their group labels, and explain whether the distributions differ.
When comparing two percentages, subtract them to find the difference in percentage points. For example, a change from 30% to 50% is a difference of 20 percentage points. This helps make the size and direction of an observed difference explicit. It is not enough to list two values without saying which group has the higher percentage or what the values describe.
A Reliable Structure for the Written Response
Before writing, identify the claim and the conditional percentages that address it. If the claim concerns whether neighborhood is associated with commuting method, compare the commuting distributions within the neighborhoods. Do not substitute a marginal percentage for a within-group percentage: the marginal describes everyone combined, not the conditional distribution within each neighborhood.
Say which two categorical variables are being compared and identify the groups whose conditional percentages you will compare.
Give the percentage in the outcome category for each group. Include the group names so the reader can tell what each percentage describes.
State which group has the higher percentage and, when useful, give the difference in percentage points.
Say whether the conditional distributions differ in the table, and describe that as an observed association. Do not claim that one variable caused the other.
This structure makes the evidence visible. It also helps avoid overstating the conclusion. A table can show an association among the individuals represented, but a descriptive comparison by itself does not establish causation or justify generalizing to people who were not represented.
Worked Example: Comparing Two Neighborhoods
Worked Example: Comparing Two Neighborhoods
A fictional survey of 240 residents records their neighborhood and whether they usually commute by bicycle. Does the table support the claim that neighborhood and usual commuting method are associated among the surveyed residents? Support your answer with conditional percentages.
| Neighborhood | Usually bikes | Does not usually bike | Total |
|---|---|---|---|
| Near transit | 54 | 36 | 90 |
| Farther from transit | 45 | 105 | 150 |
| Total | 99 | 141 | 240 |
The explanatory grouping variable for this comparison is neighborhood. To compare commuting patterns, find the percentage who usually bike within each neighborhood. The denominators are the neighborhood row totals, not the grand total.
The difference is \(60\%-30\%=30\) percentage points. The conditional distributions also show the complementary percentages who do not usually bike: \(36/90=40\%\) near transit and \(105/150=70\%\) farther from transit. Each pair sums to 100% within its neighborhood.
A complete conclusion could be: “Among the surveyed residents, neighborhood and usual commuting method appear to be associated. Sixty percent of residents who live near transit usually bike, compared with 30% of residents who live farther from transit, a difference of 30 percentage points. The conditional distributions of commuting method differ between the two neighborhoods.”
This response makes a claim, cites the relevant percentages, identifies the groups, and describes the difference. It does not say that living near transit causes residents to bike. The survey table describes a pattern among these respondents; it does not, by itself, explain why the pattern occurs.
Worked Example: Describe the Size of a Difference Carefully
Worked Example: Describe the Size of a Difference Carefully
In a fictional survey, students report whether their household has a usual screen-free time before bed and whether they meet a stated sleep goal. Does the table show an association in the surveyed students? Give evidence and avoid overstating what the difference means.
| Screen-free time before bed | Meets sleep goal | Does not meet sleep goal | Total |
|---|---|---|---|
| Usually has screen-free time | 52 | 28 | 80 |
| Does not usually have screen-free time | 66 | 54 | 120 |
| Total | 118 | 82 | 200 |
Compare the percentage meeting the sleep goal within each screen-time group. The group totals are 80 and 120:
The difference is \(65\%-55\%=10\) percentage points. The full conditional distributions are 65% meeting and 35% not meeting the goal among students who usually have screen-free time, compared with 55% meeting and 45% not meeting among students who do not usually have screen-free time.
A supported conclusion is: “In this survey, the conditional distributions of meeting the sleep goal differ between students who usually have screen-free time before bed and those who do not. Sixty-five percent of the first group met the goal, compared with 55% of the second group, a difference of 10 percentage points. This is an observed association in the surveyed students.”
The difference is smaller than in the neighborhood example, but the conclusion should still describe what the table shows: the conditional percentages are not equal. Do not turn the comparison into “screen-free time improves sleep” or claim that the table proves an effect. Other differences between the students could help explain the pattern, and this descriptive table does not isolate a cause.
Worked Example: Comparing a Distribution With Several Categories
Worked Example: Comparing a Distribution With Several Categories
A fictional survey of library visitors records whether each person uses a library app and their rating of a new reservation system. Use the conditional distributions to assess the claim that app use and rating are associated among the surveyed visitors.
| Uses library app | Satisfied | Neutral | Dissatisfied | Total |
|---|---|---|---|---|
| Yes | 48 | 24 | 8 | 80 |
| No | 40 | 40 | 20 | 100 |
| Total | 88 | 64 | 28 | 180 |
Because the outcome has three categories, compare all three percentages within each app-use group. Among the 80 visitors who use the app, the percentages are:
Among the 100 visitors who do not use the app, the percentages are:
The distributions differ in each category. For example, 60% of app users were satisfied, compared with 40% of nonusers, a difference of 20 percentage points. The dissatisfied percentages are 10% and 20%, respectively. The neutral percentages are 30% and 40%. Within each group the three percentages add to 100%.
A complete response could be: “Among these surveyed library visitors, app use and rating of the reservation system appear to be associated because the conditional distributions differ. Sixty percent of app users were satisfied, compared with 40% of nonusers; the dissatisfied percentages were 10% and 20%, respectively. Thus, satisfaction was more common and dissatisfaction less common among app users in this survey.”
The response gives more than one piece of evidence because the outcome has multiple categories. It compares the distribution within each app-use group, rather than comparing counts alone. The statement is still descriptive: it does not claim that using the app caused visitors to be more satisfied.
Common Mistakes and AP Exam Tips
- Writing a claim without evidence. “There is an association” is not a supported response by itself. Cite conditional percentages and identify the groups they describe.
- Using counts instead of within-group percentages. Groups may have different totals, so a larger count does not necessarily mean a larger share. Divide by the appropriate group total before comparing.
- Giving percentages with no labels. “The values are 60% and 30%” leaves the reader unsure which group is which. Name both groups and the outcome category.
- Confusing percentage points with percent change. When comparing two percentages for an association conclusion, subtraction gives a percentage-point difference. A change from 30% to 60% is 30 percentage points, not a 30% increase.
- Claiming causation. “The app made visitors satisfied” goes beyond a descriptive table. A careful response says satisfaction was more common among app users in the surveyed group.
- Generalizing beyond the data. If the table summarizes a survey of particular respondents, make the conclusion about those respondents unless the data collection supports a broader claim.
- Ignoring other outcome categories. For a multi-category outcome, compare the conditional distributions. One selected percentage can help, but describing the other categories can make the pattern clearer.
For full-credit communication, write a sentence that connects the variables, reports specific conditional percentages with the correct group labels, and explains what the comparison shows. If you give a difference, state it in percentage points and keep the direction clear. A conclusion should describe the observed association, not supply a cause that the table did not establish.
Check Your Understanding
For each prompt, use the relevant conditional percentages and write a conclusion in context.
- A table shows that 42 of 70 students who walk to school bring a reusable bottle, while 30 of 75 students who use another way to get to school bring one. Calculate and compare the two percentages. What do they suggest about association in these students?
- Why would comparing the two counts of students who bring a bottle be less informative than comparing the percentages within each travel group?
- Two groups have conditional percentages of 48% and 51% in a specified outcome category. What is their difference in percentage points? Write one cautious sentence describing the comparison.
- A table shows that 70% of people using a community garden reported satisfaction, compared with 45% of people not using it. Write a conclusion supported by those percentages, without claiming that garden use caused satisfaction.
- What information should accompany a percentage in a written conclusion so that the reader knows which conditional group it describes?