Bar Charts and Pie Charts
Master the two most common categorical data displays. Learn when to use each, how to draw them correctly, and how to calculate exact pie chart sector angles.
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When would you use a pie chart instead of a bar chart? Can you think of an example where one works better than the other?
Bar charts and pie charts both display categorical data, but they do different jobs. Bar charts compare category sizes; pie charts show parts of a whole.
Data: Pets owned by 20 students — Dog 8, Cat 6, Fish 4, Other 2. The bar chart shows each frequency as a bar height. The pie chart shows each as a slice proportional to 360°.
Know
- The rules for drawing a correct bar chart (scale, gaps, labels, title)
- The formula for pie chart sector angles: angle = (f ÷ n) × 360°
- When to choose a bar chart vs a pie chart
Understand
- Why bars in a bar chart must be equal width with gaps between them
- Why pie chart angles must sum to exactly 360°
- Why pie charts become unreadable with more than ~6 categories
Can Do
- Draw a correctly labelled bar chart from a frequency table
- Calculate all sector angles for a pie chart and verify the sum
- Identify errors in a given chart
Wrong bar chart: Bars of different widths. A wider bar looks bigger even with the same height, creating a misleading impression.
Right: All bars must be equal width with equal gaps between them. Only height encodes the frequency.
Wrong pie chart: Sector angles that sum to 359° or 361°. Even 1° off means the chart doesn’t represent the data faithfully.
Right: Calculate all angles using the formula, then check: $\text{sum} = 360\degree$. If you’re 1° off due to rounding, adjust the largest sector.
A bar chart has these required features:
- Title — descriptive, tells us what the chart shows
- Axis labels — what each axis measures, including units
- Even scale — y-axis starts at 0, equal intervals (e.g. 0, 2, 4, 6, 8, 10)
- Equal bar widths — all bars the same width
- Gaps between bars — distinguishes bar chart (categorical) from histogram (continuous)
- All bars drawn accurately — height corresponds exactly to frequency
Each sector’s angle is calculated using:
$$\text{Sector angle} = \frac{f}{n} \times 360\degree$$
where $f$ = frequency of the category and $n$ = total number of data values.
Example: Pets: Dog 8, Cat 6, Fish 4, Other 2. Total $n = 20$.
| Pet | Freq | Calculation | Angle |
|---|---|---|---|
| Dog | 8 | $\frac{8}{20} \times 360$ | 144° |
| Cat | 6 | $\frac{6}{20} \times 360$ | 108° |
| Fish | 4 | $\frac{4}{20} \times 360$ | 72° |
| Other | 2 | $\frac{2}{20} \times 360$ | 36° |
| Total | 20 | 360° ✓ |
The choice of chart depends on what you want to communicate:
Never use a pie chart when: categories don’t add up to a whole (e.g. people can vote for multiple options), or when you need to compare exact values (bar chart is more precise).
Watch Me Solve It · Calculate pie chart angles
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1Confirm total $n$$30 + 48 + 24 + 12 + 6 = 120$ ✓Always verify the total before calculating angles.
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2Apply: Angle = (f ÷ 120) × 360° for each categoryWalk: $\frac{30}{120} \times 360 = 90\degree$ Bus: $\frac{48}{120} \times 360 = 144\degree$ Car: $\frac{24}{120} \times 360 = 72\degree$ Cycle: $\frac{12}{120} \times 360 = 36\degree$ Train: $\frac{6}{120} \times 360 = 18\degree$
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3Check: angles sum to 360°$90 + 144 + 72 + 36 + 18 = 360\degree$ ✓This confirms no arithmetic errors. Now draw the sectors using a protractor.
Brain Trainer · 4 problems
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1 In a survey of 40 students, 10 prefer Drama. What is the pie chart angle for Drama?
$\frac{10}{40} \times 360 = \frac{1}{4} \times 360 = 90\degree$90° -
2 A pie chart sector is 72°. The total sample was 150. How many people does that sector represent?
Proportion = 72 ÷ 360 = 0.2. Number = 0.2 × 150 = 30.30 people -
3 Name THREE features that must appear on a correctly drawn bar chart.
Any three from: descriptive title; labelled axes (with units); even scale starting at 0; equal bar widths; gaps between bars; bars drawn to correct height.Title, axis labels, even scale -
4 A researcher wants to show what share of the national energy supply comes from solar, coal, gas, and hydro. Should they use a bar chart or pie chart? Why?
Pie chart is more appropriate here because the researcher is showing parts of a whole (the 4 sources together make 100% of supply). The pie chart emphasises proportions clearly with only 4 categories, which is within the readable limit.Pie chart: parts of a whole, 4 categories
Bar Chart Rules
- Title, axis labels with units
- Even scale starting at 0
- Equal bar widths, equal gaps
- Height = frequency
Pie Chart Formula
- Angle $= \dfrac{f}{n} \times 360\degree$
- Check: all angles sum to 360°
- Max ~6 categories for readability
Reverse: angle to count
- Proportion $= \dfrac{\text{angle}}{360}$
- Count $= \text{proportion} \times n$
Choosing the Chart
- Bar chart: comparing categories, many categories, change over time
- Pie chart: parts of a whole, few categories (max 6)
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. A class survey found: Red 5, Blue 8, Green 4, Yellow 3. Total = 20. Calculate the pie chart angle for each colour. Show your working and verify angles sum to 360°.
Q7. A bar chart shows 4 bars: Apples 12, Bananas 8, Grapes 15, Oranges 5. (a) What is the total frequency? (b) What is the relative frequency of Grapes? (c) Which is the modal category?
Q8. A pie chart sector has an angle of 72°. The total sample size was 150. (a) What fraction of the sample does this sector represent? (b) How many people are in this category? (c) What percentage does this sector represent?
Quick Check
1. B — $(15 \div 60) \times 360 = 90\degree$.
2. C — Equal widths with gaps between bars.
3. A — $(25 \div 100) \times 360 = 90\degree$.
4. D — Bar chart for 10 categories; pie chart becomes unreadable.
5. B — Y-axis starting at 3 makes differences appear much larger than they are.
Model Answers
Q6 (3 marks): Red: $(5 \div 20) \times 360 = 90\degree$ Blue: $(8 \div 20) \times 360 = 144\degree$ Green: $(4 \div 20) \times 360 = 72\degree$ Yellow: $(3 \div 20) \times 360 = 54\degree$ [1 mark each for correct calculation]. Check: $90 + 144 + 72 + 54 = 360\degree$ ✓ [included in Q6 marking].
Q7 (3 marks): (a) Total = $12 + 8 + 15 + 5 = 40$ [1]. (b) Relative frequency of Grapes = $15 \div 40 = 0.375 = 37.5\%$ [1]. (c) Modal category: Grapes (highest frequency = 15) [1].
Q8 (3 marks): (a) Fraction $= 72 \div 360 = \tfrac{1}{5} = 0.2$ [1]. (b) Count $= 0.2 \times 150 = 30$ people [1]. (c) Percentage $= 0.2 \times 100 = 20\%$ [1].
The School Travel Pie Chart
A survey of 120 students finds: Walk 30, Bus 48, Car 24, Cycle 12, Train 6. (a) Calculate all five pie chart angles and draw the pie chart in your workbook. (b) Which two categories together make up exactly 50% of the data? Show your working.
Reveal solution
Angles: Walk: $(30 \div 120) \times 360 = 90\degree$ Bus: $(48 \div 120) \times 360 = 144\degree$ Car: $(24 \div 120) \times 360 = 72\degree$ Cycle: $(12 \div 120) \times 360 = 36\degree$ Train: $(6 \div 120) \times 360 = 18\degree$. Check: $90 + 144 + 72 + 36 + 18 = 360\degree$ ✓.
(b) 50% of 120 = 60 students. Walk (30) + Car (24) = 54 — no. Walk (30) + Cycle (12) = 42 — no. Bus (48) + Cycle (12) = 60 — Yes! Bus and Cycle together = exactly 60 students = 50%.
Bar chart
Equal widths, gaps between bars, y-axis from 0
Pie chart angle
$(f \div n) \times 360\degree$ — check sum = 360
Reverse calculation
Count = (angle ÷ 360) × n
When to use bar chart
Many categories, comparing frequencies, change over time
When to use pie chart
Parts of a whole, max 5–6 categories
Common error
Y-axis not starting at 0 → misleading chart
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