AP Syllabus focus: 'There are many additional ways to represent frequencies or relative frequencies for categorical data.'
Categorical data can be displayed in several useful forms besides ordinary bar graphs. Choosing the right display helps reveal proportions, dominant categories, and part-to-whole relationships more clearly for a particular audience.
Why use other graphs for categorical data?
A single set of categorical data can often be shown in more than one valid graphical form. Different displays emphasize different ideas, even when they represent the same frequencies or relative frequencies.
Some graph types are especially useful for:
showing how categories make up an entire group
highlighting the largest and smallest categories
presenting information in a compact visual form
communicating results to a general audience quickly
When selecting a graph, the main question is not which display looks best, but which display represents the categories most clearly and accurately.
Common graph types for categorical data
Pie charts
A pie chart shows categorical data as parts of a circle. Because the full circle represents the entire data set, this kind of graph is most useful when the focus is on how each category contributes to the whole.
Pie chart: A circular display in which the whole circle represents all observations and each sector represents a category’s frequency or relative frequency.
Pie charts work best when:
there are only a few categories
categories form one complete set
differences among categories are fairly noticeable
relative frequencies or percentages are the main focus
A pie chart is usually less effective when there are many categories or when several categories have similar proportions. In those cases, the sectors can become difficult to compare. People generally compare lengths more easily than angles or areas, so pie charts are often less precise than other displays for close comparisons.
For a pie chart to be clear, each sector should be labeled with the category name and, when possible, its proportion or percentage. Decorative effects such as tilting the circle or using three-dimensional slices should be avoided because they can distort the visual impression of category size.
Pareto charts
A Pareto chart is a specialized categorical display that orders categories from largest to smallest. This ordering makes the most common categories stand out immediately.
Example of a Pareto chart where categories are sorted from highest to lowest frequency and a cumulative relative-frequency line is drawn on a secondary axis. The ordering makes it easy to identify which few categories account for most observations (a common “prioritize attention” use case). Source
Pareto chart: A bar-type display for categorical data in which categories are arranged from greatest frequency or relative frequency to least.
The defining feature of a Pareto chart is the ranking. Instead of preserving an arbitrary or natural category order, it emphasizes magnitude. This is especially useful when the goal is to identify which categories account for the largest share of the observations.
A Pareto chart is helpful when:
the largest categories are the main interest
the data will be used to prioritize attention
the audience needs to spot dominant categories quickly
However, this graph is less suitable when the categories have a meaningful natural order that should be preserved. For example, if categories already follow a sequence, rearranging them by size may hide that structure.
A clear Pareto chart should:
include labeled categories
use a consistent scale
show either frequencies or relative frequencies, not both without explanation
avoid unnecessary visual effects that distract from the ranking
Segmented bar charts
Another useful display is the segmented bar chart, where one bar represents the entire distribution and is divided into sections for the categories.
Segmented bar chart: A single bar split into pieces so that each segment shows the proportion or frequency of a category within one categorical distribution.
This display emphasizes the idea that all categories together make up one whole. In practice, segmented bar charts are especially readable when they use relative frequencies, since the full bar can naturally represent 100%.
Example of a 100% segmented (stacked) bar chart, where each bar represents a whole (100%) and the segments show category proportions within each bar. This format highlights part-to-whole composition using lengths (percent along a common scale), which often supports more precise comparisons than pie-slice angles. Source
Segmented bar charts are useful because:
they show part-to-whole structure clearly
they use lengths rather than angles, which are often easier to compare
they can present a distribution in a compact space
To be effective, the segments should fill the entire bar, and the graph should include either direct labels or a clear legend. If there are too many categories, however, the bar may become cluttered and some segments may be too small to read comfortably.
Pictographs and icon displays
Some categorical data are shown with repeated symbols or icons instead of plain bars or sectors. These pictographs can attract attention and are common in media, advertisements, and informal reports.
Pictographs should be interpreted carefully. If the symbols change in height, width, or area, the visual impression may exaggerate differences between categories. A trustworthy pictograph should use equal-sized icons and a clear statement of what each icon represents.
Although pictographs can make a display more engaging, they are usually less precise than standard statistical graphs. In AP Statistics, clarity and accuracy are more important than decoration.
Choosing an appropriate display
The best graph depends on what the display needs to communicate.
Use a:
pie chart when the part-to-whole relationship is the main idea
Pareto chart when ranking categories from most common to least common is most important
segmented bar chart when proportions should be shown as parts of one complete distribution
pictograph only when the scale is extremely clear and precision is not the primary goal
When evaluating or creating a graph, check:
whether category labels are readable
whether the display shows frequencies or relative frequencies clearly
whether the categories cover the full set without overlap
whether the visual design makes comparisons easy rather than difficult
Presentation issues to watch for
Even when the graph type is appropriate, poor design can make a categorical display misleading.
Common problems include:
too many categories in one display
missing labels or unclear legends
categories that are not mutually exclusive
distorted shapes or three-dimensional effects
ordering that confuses rather than clarifies
tiny categories that are impossible to distinguish visually
A strong categorical graph should make the data easier to understand, not harder to decode.
FAQ
The starting position of the first slice is usually a design choice, not a statistical one.
Different software programs may:
begin at the top, side, or another position
move clockwise or counterclockwise
sort categories automatically or keep the original order
These choices do not change the data values, but they can affect how easy the chart is to read.
For clarity, the graph should make labels easy to match to slices and should avoid unnecessary rotation tricks that make comparisons harder.
Yes. This usually happens because of rounding.
For example, if several category percentages are rounded to the nearest whole percent, the total may end up slightly below or above 100%.
This is generally acceptable if:
the rounding is small
the graph is labeled clearly
the underlying proportions were calculated correctly
If exact totals matter, use one more decimal place or show counts alongside percentages.
Very small categories can be hard to display clearly, especially in pie charts and segmented bar charts.
One option is to combine several tiny categories into an Other category. If this is done:
the combination should be stated clearly
the grouped categories should make substantive sense
the graph should not hide an important category just because it is small
If the small categories matter individually, a different display with better labeling may be more appropriate.
A donut chart is basically a pie chart with the center removed.
It can still display categorical proportions, but it usually does not improve statistical clarity. In fact, removing the center may make sector sizes a little harder to judge.
For classroom and exam settings, a standard pie chart or another clearer display is usually preferred. If a donut chart is used, it should still follow the same rules as a pie chart:
categories must represent one whole
labels must be clear
visual distortion should be avoided
Software often applies a default rule when it makes a categorical graph.
A program may:
keep the categories in the order entered
sort alphabetically
sort from largest to smallest
use an internal category order
This matters because the order can change what the viewer notices first. For instance, sorting by size turns the display into something like a Pareto arrangement, while alphabetical order does not emphasize frequency.
Before accepting the graph, always check whether the software’s ordering matches the purpose of the display.
Practice Questions
A class survey records the favorite type of pet for each student. The teacher wants a graph other than a standard bar graph that shows how each category makes up the whole class. Identify an appropriate graph and give one reason it is appropriate.
1 mark: Identifies a valid graph, such as a pie chart or a segmented bar chart.
1 mark: Gives a valid reason tied to part-to-whole display, such as showing each category as a proportion of the full class.
A student council surveys 200 students about their preferred after-school activity. The results are:
Sports: 80
Music: 50
Clubs: 40
Jobs: 20
Other: 10
The council wants a display other than a standard bar graph that makes it easy to see which category is most common. Identify the most appropriate graph, describe how it should be constructed, and explain one advantage and one limitation of that graph for these data.
1 mark: Identifies a Pareto chart.
1 mark: States that categories should be arranged from greatest frequency to least frequency.
1 mark: States that the bars should represent the category frequencies or relative frequencies clearly and with labels.
1 mark: Explains one valid advantage, such as quickly showing the most common categories or making ranking easy to see.
1 mark: Explains one valid limitation, such as not emphasizing part-to-whole as directly as a pie chart or not preserving a natural category order.
