A categorical variable sorts individuals into groups, so the first summary is a count for each group. Counts alone can mislead when groups differ in size, so we also turn them into relative frequencies: the fraction or percent of the whole. That is the same skill as finding a percent of a number, run backward.
Key idea
$$\text{relative frequency} = \dfrac{\text{count in the category}}{\text{total count}}.$$ A bar graph shows one bar per category (counts or percents). A pie chart shows each category as a slice whose angle is that fraction of $360^\circ$. The relative frequencies of all categories add up to 1 (or 100%).
Why it works
A pie chart works because a circle is one whole. If a category is 30% of the data, its slice must be 30% of the circle: $0.30 \times 360^\circ = 108^\circ$. A bar graph compares categories by height, so its bars should start at 0 and be separated by gaps: the categories are separate groups, not a number scale.
| Way to school | Count | Relative frequency | Pie slice |
|---|---|---|---|
| Bus | 28 | $28/80 = 0.35$ | $126^\circ$ |
| Car | 20 | $0.25$ | $90^\circ$ |
| Walk | 24 | $0.30$ | $108^\circ$ |
| Bike | 8 | $0.10$ | $36^\circ$ |
| Total | 80 | 1.00 | $360^\circ$ |
Worked example: Percents from counts
Of 80 students, 28 ride the bus. What percent is that, and how large is the bus slice of a pie chart?
- Relative frequency: $\dfrac{28}{80} = 0.35$.
- As a percent: $0.35 \times 100 = 35\%$.
- Pie slice: $0.35 \times 360^\circ = 126^\circ$.
Answer: 35%, a slice of $126^\circ$.
Worked example: Working backward from a slice
In a pie chart of a survey of 300 people, the slice for "Winter" is $72^\circ$. How many people chose winter?
- The slice is $\dfrac{72}{360} = 0.2$ of the circle, so 20% chose winter.
- Count: $0.2 \times 300 = 60$ people.
Answer: 60 people.
Common mistake
Using a histogram for categorical data, or a pie chart for numbers that are not parts of one whole. Histograms need a number line (heights, times); categories get bar graphs. A pie chart only makes sense when every individual lands in exactly one slice, so the slices add to 100%. Check: do the percents add to 100?
To compare two groups of different sizes (say, 80 students at one school and 500 at another), compare relative frequencies, not counts.