What a Pie Chart Shows
In Choosing Between Bar Chart, Pie Chart, and Histogram, you learned that a pie chart can display a categorical distribution. In Why Histograms and Bar Charts Are Different, you saw why graph choice depends on what the variable’s values represent. Here we focus on how to read and make a pie chart: each slice represents a category’s proportion of the whole circle.
A circle has \(360\) degrees. A pie-chart slice’s central angle—the angle formed at the center of the circle by the slice’s two radii—is proportional to the category’s relative frequency. A category with twice the proportion of another category gets twice the angle. The entire circle represents all the individuals or items in the group being described.
If a category makes up \(25\%\) of the group, its slice angle is \(25\%\) of \(360^\circ\), or \(90^\circ\). To read a chart in the other direction, divide a slice’s angle by \(360^\circ\). For example, a \(90^\circ\) slice represents \(90/360=0.25\), or \(25\%\) of the group.
A pie chart emphasizes how the categories divide one whole. It does not show the individual cases or values. A reader can compare a slice with the full circle, but comparing two similar-sized slices precisely can be difficult. That limitation matters when choosing whether a pie chart is the most useful display.
When a Pie Chart Is Appropriate
A pie chart is most appropriate for one categorical variable measured on one group when its categories divide that group into distinct parts. Each individual or item should contribute to exactly one category, and the listed categories should account for everyone or everything represented. In other words, categories should be mutually exclusive and collectively exhaustive.
For example, if a survey asks each student to name one primary way they get to school, a student’s answer belongs to one commute category. A pie chart can show how the students in that survey are divided among the categories. By contrast, if students can select every commute method they use during a week, one student could appear in multiple categories. Those counts are not separate pieces of a single whole, so a pie chart would misrepresent them as if they were.
The chart should make clear what the whole represents. Include a descriptive title or an explanatory label, such as “Primary commute method among 45 surveyed students.” Label slices with category names and, when practical, percentages or counts. Percentages help readers judge each category’s share; counts tell how many individuals are represented. Showing both can be especially helpful, provided the values are calculated from the same total.
Making a Pie Chart From Counts
To construct a pie chart, first find the total count and check that every case is included exactly once. Then find each category’s relative frequency by dividing its count by the total. Multiply each relative frequency by \(360^\circ\) to get the slice angle. The angles must add to \(360^\circ\); this is a useful check on the arithmetic.
When drawing by hand, use a protractor to mark the angles from the center of the circle. Start from a radius you choose, mark one category’s angle, and continue around the circle for the remaining categories. You can use any starting point and category order, as long as the sectors match the calculated angles and every category is labeled. If rounding makes the angles add to slightly more or less than \(360^\circ\), use unrounded values for the calculations and adjust a final angle by the small rounding difference.
The slices represent proportions, not counts directly. The count determines the slice size only in relation to the total count. If a chart uses a full circle for 20 responses, each response accounts for \(360^\circ/20=18^\circ\). If the total is 100, each response accounts for \(3.6^\circ\). The same count can therefore represent a different share when the total changes.
Worked Example: Reading a Pie Chart’s Angles
Worked Example: Reading a Pie Chart’s Angles
A fictional community garden survey asks 45 volunteers to choose one favorite task. A pie chart shows that the “watering” slice has an angle of \(144^\circ\). What proportion and percentage of the surveyed volunteers chose watering, and how many volunteers is that?
The angle represents the watering category’s share of the full \(360^\circ\) circle. Divide the angle by \(360^\circ\):
Thus, \(40\%\) of the surveyed volunteers chose watering. To find the count, multiply that proportion by the total:
So, 18 of the 45 surveyed volunteers chose watering. Check by calculating the angle from the count: \((18/45)\times 360^\circ=144^\circ\), matching the chart. In context, the watering slice represents \(40\%\), or 18, of the volunteers in this survey.
Worked Example: Constructing a Pie Chart
Worked Example: Constructing a Pie Chart
A fictional school club asks 80 members to choose one preferred type of club event. The results are shown below. Find the percentage and angle for each slice, then describe how to draw and check the chart.
| Preferred event | Count |
|---|---|
| Outdoor activity | 30 |
| Game night | 20 |
| Workshop | 18 |
| Community project | 12 |
| Total | 80 |
The counts add to \(30+20+18+12=80\), so each of the 80 members is represented once. For each category, divide its count by 80 to find its proportion. Multiply by \(100\%\) for the percentage and by \(360^\circ\) for the angle.
| Preferred event | Calculation for angle | Percentage | Angle |
|---|---|---|---|
| Outdoor activity | \((30/80)\times 360^\circ\) | 37.5% | 135° |
| Game night | \((20/80)\times 360^\circ\) | 25% | 90° |
| Workshop | \((18/80)\times 360^\circ\) | 22.5% | 81° |
| Community project | \((12/80)\times 360^\circ\) | 15% | 54° |
Check the percentages: \(37.5\%+25\%+22.5\%+15\%=100\%\). Check the angles: \(135^\circ+90^\circ+81^\circ+54^\circ=360^\circ\). These checks confirm that the slices account for the complete circle. Draw the sectors with the listed angles and label each with its event name and percentage. A title such as “Preferred club event among 80 members” identifies the group represented.
The outdoor-activity slice is the largest, representing \(37.5\%\) of the members. The community-project slice is the smallest, representing \(15\%\). The chart shows each category’s share of the 80 responses; it does not imply that the same percentages would necessarily occur among all students or in another club.
When a Pie Chart Can Mislead
A pie chart can be misleading when its slices do not truly divide one whole. This happens if cases can belong to more than one category, if some cases are left out, or if the category counts come from different groups or totals. In those situations, the slice angles may add to more or less than a full circle, or may falsely suggest that the categories are parts of a single distribution.
Even when a pie chart is valid, it may not be the clearest choice. A chart with many categories can become crowded with small slices and hard-to-read labels. If several categories have similar proportions, judging which slice is slightly larger is difficult. A bar chart is often better for comparing category sizes because readers can compare bar heights against a common scale. As in Ordering Bars for Clarity, categories without a meaningful sequence can be arranged consistently, such as from greatest to least, to help readers compare them.
Use caution with three-dimensional pie charts and exploded slices that pull sectors apart. Perspective can make slices at the front look larger than slices at the back, even when their angles are the same. Separating a slice can also draw attention to it beyond what its proportion alone warrants. A simple two-dimensional circle with clear labels is easier to interpret.
Rounding percentages for display can make their sum appear to be \(99\%\) or \(101\%\), even if the underlying proportions total \(100\%\). That small difference may be due to rounding rather than a missing category. Keep calculations based on the original counts or unrounded proportions, and do not use rounded percentages as though they were exact counts.
Worked Example: A Pie Chart for Overlapping Responses
Worked Example: A Pie Chart for Overlapping Responses
A fictional recreation survey asks 30 residents which activities they did last weekend and allows them to check every activity that applies. There are 18 checks for walking, 14 for gardening, and 10 for cycling. A student proposes making a pie chart with one slice for each activity. Is that appropriate?
The counts total \(18+14+10=42\), which is greater than the 30 residents. That can happen because one resident may have checked more than one activity. The categories overlap, so the 42 checks are not 42 distinct residents divided into non-overlapping parts of the group.
The problem is also visible in the proposed slice angles. If each count is divided by 30 and multiplied by \(360^\circ\), the angles would be:
Those angles total \(216^\circ+168^\circ+120^\circ=504^\circ\), not \(360^\circ\). Equivalently, the percentages based on 30 residents would be \(60\%+46.7\%+33.3\%=140\%\), rounded to one decimal place. Neither check produces one whole because the same resident can be counted in multiple activities.
A pie chart would therefore be inappropriate for these responses. A bar chart of the activity counts or percentages of residents who selected each activity can display the results, but its percentages may add to more than \(100\%\). The chart and its labels should make clear that residents could select multiple activities.
Common Mistakes and What a Strong Answer Says
- Using the count as the slice angle. A count is not measured in degrees. Divide it by the total, then multiply by \(360^\circ\).
- Forgetting the whole. A slice’s proportion uses the total number of cases represented in the chart. State what that group is, and use the same total for each category.
- Assuming any categorical counts make a pie chart. Check that every case belongs to exactly one category and that the categories include all cases. Multiple-response counts usually overlap.
- Making angles that do not fill the circle. Check that the angles total \(360^\circ\), allowing for a small rounding adjustment when needed. Also check that category counts sum to the stated total.
- Choosing a pie chart for precise comparisons. If the slices are close in size or there are many categories, a bar chart may make comparisons clearer.
- Leaving the chart’s meaning unstated. Include a title or label naming the group, and identify what the counts or percentages describe.
A full-credit explanation gives both the calculation and its meaning. For example: “The gardening slice would have angle \((14/30)\times360^\circ=168^\circ\), but these responses overlap because residents could select multiple activities. The counts do not form parts of one whole, so a pie chart is not appropriate.” This connects the arithmetic to the structure of the data.
Check Your Understanding
Answer each question using the relationship between a slice angle and the proportion it represents.
- A category occupies \(72^\circ\) of a pie chart. What proportion and percentage of the group does it represent?
- In a group of 50 people, 15 choose one particular category. What angle should its pie-chart slice have?
- A survey asks respondents to select all the fruits they eat, so one respondent may select several. Explain why a pie chart of the fruit counts may be inappropriate.
- Why might a bar chart be more useful than a pie chart when several categories have very similar proportions?
- A pie chart displays favorite music categories for 40 students. What information should its title or labels provide to make the whole clear?