From Segment Heights to a Pattern
In Constructing a Segmented Bar Chart From a Two-Way Table, you learned to make each group’s bar represent \(100\%\), with segments showing that group’s conditional distribution. Now use the finished chart to ask a new question: do the response-category percentages look different across the groups?
The key is to compare like with like. Look at the same response category in each bar and compare its segment height. If the conditional distributions differ across groups, the chart displays a pattern of association between the group variable and the response variable in the data. If the distributions are very similar, there is little or no visible association in the display.
The word corresponding matters: compare the same response category from bar to bar, not one category in one group with a different category in another. Also consider the entire distribution. One segment may stand out, or several categories may shift together.
A Reliable Way to Compare the Bars
Begin with the percentage scale and locate one response category across all bars. Each bar has the same total height, so the height of that segment represents the percentage of that particular group in the category. State what changes, which groups differ, and by how many percentage points if the chart or accompanying data labels make that comparison clear.
For a bottom segment, its height is the distance from \(0\%\) to its top boundary. For a middle or top segment, its height is the difference between its upper and lower boundaries. The boundary itself is not the segment’s percentage. For example, if a segment extends from \(35\%\) to \(70\%\), its height is \(70\%-35\%=35\%\), not \(70\%\).
A quick scan can help, but do not decide based only on which category is largest in each group. The largest category might be the same across groups even when the distributions differ substantially. Likewise, a category can remain the smallest while its percentage changes. Compare its actual segment height, then check whether the other response categories show a related pattern.
Name the groups and the response categories. The bars represent groups; the segments represent responses.
For each response category of interest, compare its percentage across the bars. Use the labeled scale or segment boundaries.
State which groups have higher or lower percentages in the response categories, using percentage-point differences when the values are available.
Say whether the conditional distributions appear different or similar in the displayed data. Do not claim that the chart proves cause and effect.
Worked Example: A Clear Difference Across Groups
Worked Example: A Clear Difference Across Groups
A fictional survey asks students which type of study space they usually choose: a quiet room, a shared study area, or an outdoor spot. The groups are defined by whether students usually study alone or with others. Compare the conditional distributions and decide whether the chart suggests association.
| Usual study group | Quiet room | Shared area | Outdoor spot | Total |
|---|---|---|---|---|
| Usually alone | 36 | 12 | 12 | 60 |
| Usually with others | 15 | 30 | 15 | 60 |
State. The group variable is usual study group, and the response variable is usual study space. Compare each response category’s conditional percentage across the two groups.
Plan. Use each group’s total of 60 as the denominator. These percentages are the segment heights in the segmented bar chart; compare the same category from one bar to the other.
Do. Among students who usually study alone, the quiet-room percentage is \(36/60\times100\%=60\%\), the shared-area percentage is \(12/60\times100\%=20\%\), and the outdoor-spot percentage is \(12/60\times100\%=20\%\). Among students who usually study with others, the corresponding percentages are \(15/60\times100\%=25\%\), \(30/60\times100\%=50\%\), and \(15/60\times100\%=25\%\). Each group’s percentages add to \(100\%\).
The quiet-room segment is \(60\%-25\%=35\) percentage points taller for students who usually study alone. The shared-area segment is \(50\%-20\%=30\) percentage points taller for students who usually study with others. The outdoor-spot segment is \(25\%-20\%=5\) percentage points taller for that same group.
Conclude. The conditional distributions differ noticeably: students who usually study alone have a larger quiet-room percentage, while students who usually study with others have a larger shared-area percentage. The segmented bar chart shows an association between usual study group and usual study space in these survey data. It does not show that studying with others causes students to choose a shared area.
Worked Example: Read the Segment, Not Just Its Boundary
Worked Example: Read the Segment, Not Just Its Boundary
A fictional survey asks residents in two neighborhoods how they usually travel to a nearby market: walking, cycling, or taking a vehicle. The chart uses walking at the bottom, cycling in the middle, and vehicle at the top. The cycling segment in Neighborhood East extends from \(30\%\) to \(55\%\). In Neighborhood West, it extends from \(20\%\) to \(35\%\). Compare the cycling percentages and interpret the difference.
Do. The East cycling segment is \(55\%-30\%=25\%\) of that group. Its top boundary is at \(55\%\), but the segment itself is only 25 percentage points tall. The West cycling segment is \(35\%-20\%=15\%\) of that group.
Conclude. The percentage of residents who usually cycle is 10 percentage points higher in Neighborhood East than in Neighborhood West. That comparison describes the displayed conditional percentages; it does not say how many residents cycle in each neighborhood. The chart also suggests the distributions differ in the cycling category, though a complete description should check walking and vehicle segments as well.
This example illustrates why the vertical position of a segment is not always its height. For any segment that begins above zero, subtract its lower boundary from its upper boundary before comparing it with the corresponding segment in another bar.
Worked Example: Similar Shapes Can Suggest Little Association
Worked Example: Similar Shapes Can Suggest Little Association
A fictional school survey asks students in two grade groups which kind of lunchtime activity they usually choose: reading, games, or conversation. The chart displays these percentages, with the categories in the same order in each bar.
| Grade group | Reading | Games | Conversation | Total |
|---|---|---|---|---|
| Grade 9 | 18 | 22 | 20 | 60 |
| Grade 10 | 24 | 28 | 28 | 80 |
Do. For Grade 9, the percentages are \(18/60\times100\%=30\%\) for reading, \(22/60\times100\%\approx36.7\%\) for games, and \(20/60\times100\%\approx33.3\%\) for conversation. They add to \(100.0\%\) after rounding. For Grade 10, they are \(24/80\times100\%=30\%\), \(28/80\times100\%=35\%\), and \(28/80\times100\%=35\%\), totaling \(100\%\).
The reading percentages are equal. The games percentages differ by about \(36.7\%-35.0\%=1.7\) percentage points, and the conversation percentages differ by about \(35.0\%-33.3\%=1.7\) percentage points, using rounded values. The bars should therefore have very similar shapes.
Conclude. There is little visible difference between the conditional distributions of lunchtime activity for these two grade groups. The segmented bar chart suggests little association between grade group and usual lunchtime activity in these data. “Little visible association” is more careful than claiming the variables are exactly independent based only on a visual comparison.
What the Chart Shows—and What It Does Not
A segmented bar chart is useful because it puts conditional percentages on a common \(0\%\)-to-\(100\%\) scale. This makes it easier to judge whether the response distributions differ across groups than it would be to compare raw counts from groups of different sizes. The chart describes the sample or data set displayed; it does not by itself determine whether a difference is statistically significant in a larger population.
Keep the group totals in mind if they are available. A taller segment means a larger percentage of that group, not necessarily a larger number of individuals. For example, \(40\%\) of a group of 20 is 8 individuals, while \(30\%\) of a group of 100 is 30 individuals. The first percentage is larger, but the second group has more individuals in the category.
The chart also cannot establish causation. If the explanatory and response variables are associated in the displayed data, other variables or the way the data were collected could help explain the pattern. In an observational study, a visual difference alone does not show that group membership caused the response difference.
Common Mistakes and AP Exam Tips
- Comparing different categories. Compare the same response category across the bars. A statement about one segment in one group and a different segment in another does not establish how a response percentage changes.
- Calling a boundary the segment height. If a segment runs from \(25\%\) to \(60\%\), its height is \(60\%-25\%=35\%\). Do not report \(60\%\) as the segment’s percentage.
- Comparing counts from bar heights. A segmented bar chart uses within-group percentages, and every bar reaches \(100\%\). It does not display group sizes or category counts unless those are separately labeled.
- Looking only for the largest segment. The largest response category may be the same in every group even when other percentages differ. Compare corresponding heights across the full set of categories.
- Overstating the conclusion. A graph supports a description of association in the displayed data; it does not prove causation or establish a statistically significant result by itself.
- Using vague language. Instead of saying “the bars are different,” name the groups, response category, and direction of the percentage difference. If values are available, report the difference in percentage points.
A strong AP-style interpretation identifies both variables, describes at least one meaningful difference or similarity in the conditional distributions, and makes a careful statement about association in context. For example: “The percentage of students who usually study in a quiet room is higher among those who usually study alone than among those who usually study with others, so the chart suggests an association between usual study group and usual study space in this survey.”
Check Your Understanding
Use corresponding segment heights and describe the conditional distributions in context.
- A segment extends from \(15\%\) to \(48\%\) in one bar. What percentage of that group is represented by the segment?
- In two groups, the same response category is \(42\%\) in Group A and \(27\%\) in Group B. What is the percentage-point difference, and which group has the higher percentage?
- Why can a taller segment in one bar represent fewer individuals than a shorter segment in another bar?
- If all corresponding segment heights are similar across groups, what is a careful statement about association in the displayed data?
- Why does a segmented bar chart showing different conditional distributions not prove that the group variable caused the response differences?