AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-4)
Unit 1 · Topic 1.4
1.4 Graphical Representations for One Categorical Variable
Bar charts and pie charts turn a table of one categorical variable into a picture. You'll read them, build them and use them to compare two or more groups, and you'll need to spot when a graph is set up in a way that misleads.
Key terms
- bar chart (bar graph)
- pie chart
- frequency
- relative frequency
Bar charts
A bar chart (bar graph) has one bar per category. The height (or length, if the bars run sideways) shows the category's frequency or relative frequency. The categories go along one axis and the counts or proportions along the other.
Bars are separated by gaps because the categories are separate groups with no natural number line between them. You can put the categories in any order. Ordering them from tallest to shortest often makes the chart easier to read.
A bar chart of relative frequencies has the same shape as one of counts; only the scale on the vertical axis changes.
Don't mix up a bar chart with a histogram. A bar chart's categories are labels, so the order is your choice and the bars don't touch. A histogram's bins are intervals on a number line, so the order is fixed and the bars touch (see topic 1.5). Bar charts make it easier than pie charts to compare categories that are close in size, because comparing heights is easier than comparing angles.
Pie charts
A pie chart shows each category as a slice of a circle. Each slice's area, as a fraction of the whole circle, equals that category's relative frequency, so all the slices together make 100%.
Pie charts only work when the categories don't overlap and together cover every unit. If a survey let people pick more than one answer, the percentages can add up to more than 100%, and a pie chart would be wrong. Use a bar chart instead.
Comparing groups
To compare a categorical variable across two or more groups, put the groups side by side using relative frequencies. For example, show how students at two schools get to school with bars for bus, car and walk/bike for each school.
Use proportions, not counts, whenever the groups have different sizes. Otherwise the bigger group will look like it has more of everything.
Reading graphs critically
Check the vertical axis before you judge. A bar chart whose axis starts at 40% instead of 0% can make a small difference look huge, because bar height no longer matches the actual value. Pictographs and 3-D pie charts can distort area in a similar way.
When you use a graph to justify a claim, refer to specific values you read from it: "About 50% of School B's students arrive by car, compared with about 35% at School A."
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Compare two groups with a graph
School A (200 students surveyed): bus 90, car 70, walk/bike 40. School B (50 students surveyed): bus 15, car 25, walk/bike 10. Describe how you would graph these to compare the schools, and write one comparison.
Show the solutionHide the solution
- Step 1: Different sample sizes mean you should graph relative frequencies.
- Step 2: School A: bus 0.45, car 0.35, walk/bike 0.20. School B: bus 0.30, car 0.50, walk/bike 0.20.
- Step 3: Make a side-by-side bar chart: for each way of getting to school, draw one bar for School A and one for School B, with relative frequency (0 to 0.5) on the vertical axis starting at 0.
- Step 4: Compare: Riding in a car is more common at School B (50%) than at School A (35%), while bus riding is more common at School A (45% vs. 30%). Walking or biking is the same at both (20%).
Answer: Use a side-by-side bar chart of relative frequencies. Car travel is more common at School B (50% vs. 35%), bus travel is more common at School A (45% vs. 30%), and walk/bike is 20% at both.
- Example 2Calculator allowed
Trap: a truncated axis
A bar chart shows that 52% of voters favor Plan X and 48% favor Plan Y. The vertical axis runs from 45% to 55%, so the Plan X bar looks more than twice as tall as the Plan Y bar. What should a reader conclude?
Show the solutionHide the solution
- Step 1: Bar heights measured from 45% are 7 points for X and 3 points for Y, which is why X looks more than twice as tall.
- Step 2: The actual values are 52% and 48%, a difference of only 4 percentage points.
- Step 3: With the axis starting at 0, the bars would look almost the same height.
Answer: Plan X is slightly more popular (52% vs. 48%). The chart exaggerates the gap because the axis doesn't start at 0.
Common mistakes
- Comparing groups of different sizes with counts instead of relative frequencies.
- Using a pie chart when the categories overlap or the percentages don't add to 100%.
- Leaving gaps out of a bar chart, or drawing it like a histogram. Bar charts are for categories; histograms are for numbers.
- Judging a difference by bar heights without checking where the axis starts.
On the exam
- Multiple-choice questions may show bar charts for two groups and ask which statement is true. Read off the proportions and compare them; don't rely on visual impressions alone.
- If you're asked to construct a graph on free response, label both axes, include a scale and use relative frequencies when comparing groups.
Connected topics
Videos
Check yourself
3 questions on 1.4 Graphical Representations for One Categorical Variable. Pick an answer to see if you got it, and why.
School A has 1,200 students and School B has 400 students. Each school asks every student which of four clubs they'd most like to join. To compare the two schools' preferences, which display is most appropriate?
A survey asked 300 teens, "Which of these streaming services do you use? Select all that apply." The percents for the five services add to 182%. Which statement is correct?
| Lunch choice | Number of students |
|---|---|
| Pizza | 70 |
| Tacos | 48 |
| Salad | 34 |
| Sandwich | 28 |
| Pasta | 20 |
Invented data: a survey of 200 students about their favorite cafeteria lunch
Which graph would be appropriate for displaying these data?
0 of 3 answered