Unit 1 · Topic 1.4 · about 20 minutes
Graphical Representations for One Categorical Variable
Draw and read bar charts and pie charts, and use relative frequency graphs to compare groups and back up a claim.
Predict first
You draw a bar chart of the homework survey using counts. Then you draw it again using relative frequencies. How will the two charts compare?
Bar charts
A bar chart, or bar graph, shows one categorical variable with one bar for each category. The height of each bar is that category's frequency or relative frequency. Bars can also run sideways, and then their length carries the number.
The categories are labels, not numbers, so the bars stand apart with space between them, and their order is up to you. Sorting from tallest to shortest makes the ranking easy to read. If the categories have a natural order, such as survey answers from never to always, keep that order instead.
Where 200 students usually do their homework, as counts
The same survey as relative frequencies
Same bars, different axis. Dividing every count by 200 scales every bar by the same amount.
Pie charts
A pie chart shows the same information as slices of a circle. Each slice is a category, and the slice's share of the circle's area equals that category's relative frequency. In the homework survey the bedroom slice is 47% of the pie. To draw it by hand, give it 47% of 360 degrees, about 169 degrees. All the slices together fill the whole circle, 100% of it.
That last fact is the catch. A pie chart only works when the categories split one group into parts, with every unit in exactly one category, so that the relative frequencies add to 1. Survey questions that say check all that apply break this rule, and so do percents that come from different groups.
Sort it
Could each data set be shown as a pie chart? Tap a card, then tap its bin.
A pie chart works
Use a bar chart instead
Comparing groups
Tables, bar charts and pie charts can all compare two or more groups on the same categorical variable. The rule from 1.3 Tabular Representation and Summary Statistics for One Categorical Variable matters even more in a picture: when the groups are different sizes, graph relative frequencies. In a bar chart of counts, every bar of the bigger group is inflated by its size.
A school district surveyed a random sample of 250 freshmen and a random sample of 500 seniors about how they usually get to school. Here are both groups as relative frequencies. Both charts use the same vertical scale, so the bars can be compared directly.
Freshmen (250 sampled)
Seniors (500 sampled)
Freshmen are too young to drive alone, and it shows: 40% of the sampled seniors drive themselves, while the freshmen rely on buses and family cars.
Worked exampleWho is more likely to walk or bike?
A senior on the student council says, "Seniors are more likely than freshmen to walk or bike to school. In the survey, 70 seniors walk or bike and only 60 freshmen do." Do the data support the claim?
Notice the group sizes. The district sampled 500 seniors and 250 freshmen, twice as many seniors. Comparing 70 with 60 ignores that.
Compute relative frequencies. Seniors: . Freshmen: .
Compare. In the samples, 24% of freshmen walk or bike, compared with 14% of seniors. The charts show the same thing: the walk-or-bike bar is taller for freshmen.
Conclude in context. The data do not support the claim. A sampled freshman was more likely than a sampled senior to walk or bike, even though the senior sample contained more walkers and bikers.
No. As proportions, 0.24 of the freshmen walk or bike, compared with 0.14 of the seniors.
Check your understanding
Use the chart for the 500 sampled seniors above. About how many of the seniors in the sample take the bus to school?
In a pie chart of the favorite sport of 160 students, the soccer slice takes up one quarter of the circle. Which must be true?
A survey asked 400 students, "Which of these do you use to listen to music? Check all that apply." The results were phone 91%, laptop 38%, smart speaker 22% and car stereo 47%. Which display is appropriate?
Use the two charts for the freshman and senior samples above. Which claim do the charts support?
A youth soccer league wants to compare the jersey sizes ordered by its 120 under-10 players and its 300 under-14 players. Which display makes a fair comparison of the two groups?
Course alignment, for teachers
AP Statistics topic 1.4, Unit 1: Exploring One-Variable Data and Collecting Data.
- Skill 3.A: Construct tabular and graphical representations of data and distributions.
- Skill 4.A: Describe and compare tabular and graphical representations of data, as well as summary statistics.
- Skill 4.B: Justify a claim based on statistical calculations and results.