D. Statistical Analysis · One-Variable Data AnalysisGraphing Categorical Data: Bar Graphs and Circle Graphs
D1.3About 65 minutes
Learning goals- I can choose an appropriate graph for one-variable data based on the type of data.
- I can draw an accurate bar graph and an accurate circle graph, calculating each sector angle.
- I can compare two groups of different sizes with a relative-frequency (percent) bar graph.
- I can explain when a circle graph should not be used.
Words to know
bar graph
A graph that uses separated bars of equal width to show the frequency (or relative frequency) of each category. The bars do not touch, because the categories are separate.
circle graph
A graph (also called a pie chart) that shows each category as a sector of a circle. The sector angle is the category's relative frequency × 360°.
sector angle
The angle at the centre of a circle graph for one category.
double (side-by-side) bar graph
A bar graph with two bars for each category, one for each group being compared.
Pareto chart
A bar graph of nominal data with the bars sorted from tallest to shortest, often with the categories that matter most first.
The right picture for the right data
A school council wants to run a winter safety campaign and needs to know how students get to school. They survey 240 students. The answers are categories: school bus, car, walk, transit, bike or other. You cannot draw a histogram of categories, and you cannot find their mean. So which graph should they use?
The curriculum asks you to generate graphs "based on the type of data provided". This lesson covers the graphs for categorical data. The next lesson covers graphs for numerical data.
| Type of data | Good graphs | Examples |
|---|
| Categorical, nominal | bar graph, circle graph, Pareto chart | favourite sport, mode of transport |
| Categorical, ordinal | bar graph with categories kept in order | survey ratings, T-shirt sizes |
| Numerical, discrete with few values | bar graph or dot plot | people in a home, goals in a game |
| Numerical, continuous (or many values) | histogram, frequency polygon, stem-and-leaf plot, box plot | heights, times, masses |
Bar graphs
- Put the categories on one axis and the frequency (or percent) on the other.
- All bars have the same width and there are gaps between them. The gaps say "these are separate categories". (A histogram has no gaps, because its classes form a continuous number line.)
- The frequency axis must start at 0 and have an even scale, so the lengths of the bars are proportional to the frequencies.
- For nominal data you may order the bars any way you like. Sorting them from tallest to shortest (a Pareto chart) makes the most common categories easy to see.
- For ordinal data, keep the natural order (strongly disagree → strongly agree). Sorting by height would scramble the meaning.
Circle graphs
A circle graph shows how a whole is split into parts. Each sector's angle is proportional to its frequency:
sector angle = (f ÷ n) × 360°
Use a circle graph only when:
- the categories are parts of one whole (every individual is in exactly one category), and
- there are only a few categories (about 6 or fewer; thin slivers are hard to read).
Do not use a circle graph when people could choose more than one answer ("Which apps do you use?"), because the percents add to more than 100% and there is no whole to divide.
Worked exampleExample 1: Calculating sector angles
Of 240 students surveyed in January: school bus 84, car 66, walk 54, transit 30, bike or other 6. Find the sector angles for a circle graph.
- Check the total: 84 + 66 + 54 + 30 + 6 = 240 ✓.
- School bus: (84 ÷ 240) × 360° = 0.35 × 360° = 126°.
- Car: (66 ÷ 240) × 360° = 0.275 × 360° = 99°.
- Walk: (54 ÷ 240) × 360° = 0.225 × 360° = 81°.
- Transit: (30 ÷ 240) × 360° = 0.125 × 360° = 45°.
- Bike or other: (6 ÷ 240) × 360° = 0.025 × 360° = 9°.
- Check: 126 + 99 + 81 + 45 + 9 = 360° ✓.
Sector angles 126°, 99°, 81°, 45°, 9°. Draw each with a protractor, starting from a radius at the top and going clockwise.
The circle graph for the same survey. The school bus sector (126°) is just over a third of the circle. The 9° sector is a thin sliver: with many categories like that, a bar graph would be clearer.Comparing two groups: use percents
A second school, Lakeview, surveyed only 80 students: school bus 12, car 30, walk 18, transit 15, bike or other 5. If you put Lakeview's counts beside Hilltop's counts (Hilltop is the school with 240 students), every Lakeview bar would look small simply because fewer students were asked. To compare groups of different sizes, graph relative frequencies (percents) in a double bar graph.
Worked exampleExample 2: A relative-frequency comparison
Convert both schools' results to percents and describe the main difference.
- Hilltop (n = 240): bus 84 ÷ 240 = 35%, car 27.5%, walk 22.5%, transit 12.5%, bike/other 2.5%.
- Lakeview (n = 80): bus 12 ÷ 80 = 15%, car 30 ÷ 80 = 37.5%, walk 22.5%, transit 18.75%, bike/other 6.25%.
- Compare: walking is the same at both (22.5%). Hilltop students rely much more on the school bus (35% vs 15%), while Lakeview students are more often driven (37.5% vs 27.5%) or take transit (18.75% vs 12.5%).
In counts, Hilltop has more students in every category. In percents, the real difference appears: Hilltop is a bus school, Lakeview is a car-and-transit school.
A double bar graph of percents, not counts, so the two schools are compared fairly even though one surveyed 3 times as many students.Just for funWhy did the circle graph refuse dessert?
It was already 360 degrees of pie.
Technology
In Google Sheets or Excel, select the category column and the frequency column, then Insert → Chart, and choose Column (bar) or Pie. Always check that the software did not start the value axis above 0, did not add a 3-D effect (3-D tilts make front slices look bigger) and that the title and axis labels say what was measured, from whom, and when.
Hands-on activitySurvey your household's pantry
You need: your kitchen cupboards (ask first), paper, a pencil, a protractor, and a compass or a round lid to trace.
- Choose one shelf or cupboard and classify every item into categories you decide in advance, for example: cans, boxes, bags, jars or bottles, other. Every item must fit exactly one category.
- Make a frequency table with relative frequencies.
- Draw a bar graph (frequencies, scale from 0) and a circle graph (calculate each sector angle and check that they add to 360°).
- Write two sentences: which graph shows the largest category most clearly, and which shows the part of the whole most clearly?
What to look for: the sector angles must add to 360° (within a degree or two if you rounded), the bars must be equal widths with gaps, and both graphs need titles and labels. A typical answer: the bar graph makes it easier to compare categories of similar size; the circle graph shows at a glance that one category is, say, about half of the shelf.
Your turn
1For each variable, name a suitable graph: (a) favourite winter sport of 50 students, (b) reaction times of 200 drivers, (c) responses from "strongly disagree" to "strongly agree".
(a) bar graph or circle graph (nominal categories); (b) histogram (or box plot), since it is continuous; (c) bar graph with categories in their natural order (ordinal).
2In a survey of 60 people, 21 prefer tea. What sector angle represents tea?
(21 ÷ 60) × 360° = 0.35 × 360° = 126°.
3A sector in a circle graph has angle 54°. What percent of the data does it represent?
4A circle graph of a 400-person survey has a 72° sector for "no opinion". How many people had no opinion?
72 ÷ 360 = 0.2, and 0.2 × 400 = 80 people.
5A survey asks "Which streaming services do you use? (Choose all that apply)". Why is a circle graph a poor choice?
People can choose several answers, so the percents add to more than 100%. There is no single whole to divide into sectors. Use a bar graph of percents instead.
6Why do bars in a bar graph have gaps, but bars in a histogram do not?
Bar graph categories are separate groups. Histogram classes are touching intervals on a continuous number line, so the bars touch.
More practice: warm-up
1Convert 0.25 of a circle into degrees.
2Which kind of data is "province of birth"?
3Sector angles of 150°, 90° and 60° are drawn. What angle remains for the last category?
360 − 150 − 90 − 60 = 60°.
4Should the frequency axis of a bar graph start at 0?
Yes, so the bar lengths are proportional to the frequencies.
5What is a bar graph with bars sorted from tallest to shortest called?
More practice: on your own
1A food bank sorted 500 donated items: canned goods 210, pasta and rice 130, cereal 75, baby food 35, other 50. Find each sector angle.
Canned: 210/500 × 360° = 151.2°; pasta and rice: 130/500 × 360° = 93.6°; cereal: 54°; baby food: 25.2°; other: 36°. Check: 151.2 + 93.6 + 54 + 25.2 + 36 = 360 ✓.
2A school club has 18 Grade 9s, 12 Grade 10s, 6 Grade 11s and 4 Grade 12s. Another club has 30 Grade 9s, 30 Grade 10s, 25 Grade 11s and 15 Grade 12s. Which club has the greater proportion of Grade 12s?
Club 1: 4 of 40 = 10%. Club 2: 15 of 100 = 15%. Club 2 has a greater proportion, even though the first club looks small in every count.
3A newspaper shows a circle graph whose sectors are labelled 45%, 30%, 20% and 15%. What is wrong?
The parts add to 110%. A circle graph's parts must add to 100%. Either the respondents could pick more than one answer (so a circle graph is the wrong choice) or there is an error.
4Explain why sorting the bars of a satisfaction survey (very unhappy, unhappy, neutral, happy, very happy) from tallest to shortest would be a mistake.
The data is ordinal. Its order carries meaning (from unhappy to happy). Sorting by height would break that order and hide patterns such as "most answers lean to the happy side".
5A circle graph of 75 responses has a sector of 100.8°. How many responses does it represent?
100.8 ÷ 360 = 0.28, and 0.28 × 75 = 21 responses.
Challenge
1In a circle graph, category A's sector is twice category B's, and category C's sector is 30° more than category B's. These are the only three categories. Find all three angles.
Let B = x. Then A = 2x and C = x + 30. So 2x + x + x + 30 = 360, 4x = 330, x = 82.5. A = 165°, B = 82.5°, C = 112.5° (check: 165 + 82.5 + 112.5 = 360 ✓).
2A survey of 1 200 Ontario residents had 5 answer categories, and the smallest category was 1.5% of responses. A student wants a circle graph. Find the smallest sector's angle and explain whether a circle graph is a good idea.
0.015 × 360° = 5.4°, a very thin sliver that is hard to draw, label or compare. With 5 categories a circle graph is acceptable, but a bar graph (or Pareto chart) would show the small category and the comparisons between similar categories more clearly.
3Hilltop (n = 240) has 35% bus riders and Lakeview (n = 80) has 15%. A student says: "Overall, 25% of students at the two schools ride the bus, the average of 35% and 15%." Is the student right? Find the correct overall percent.
No. The schools are different sizes. Bus riders: 84 + 12 = 96 out of 240 + 80 = 320 students, so the overall percent is 96 ÷ 320 = 30%. A simple average of two percents only works when the groups are the same size; otherwise use a weighted average.
Quiz
1. Which graph is the best choice for the heights of 300 Grade 12 students?
2. A category has 18 of 72 responses. What is its sector angle?
3. When is a circle graph not appropriate?
4. Two classes of different sizes answered the same survey question. Which graph compares them fairly?
5. Which statement about bar graphs is true?
6. A sector of a circle graph is 63°. What percent of the data is in that category?
Big idea (D1.3): the type of data decides the graph. Categorical data uses bar graphs and circle graphs; numerical data uses histograms, dot plots, stem-and-leaf plots and box plots (next lessons).
How to help: Ask your teen to find a circle graph or bar graph in the news or on a website and check it: do the parts add to 100%? Does the axis start at 0? Is it comparing counts from groups of different sizes?
Common difficulties:
- Forgetting to multiply by 360° (writing a 25% category as 25°).
- Comparing counts instead of percents for groups of different sizes.
- Using a circle graph for "choose all that apply" questions.
- Sorting ordinal categories by height.