Comparative Pie Charts
Revise Comparative Pie Charts for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. Comparative pie charts scale the area (radius) of two pie charts to compare data sets of different total sizes.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Comparative pie charts allow two pie charts with different totals to be compared fairly. This is a Higher-only topic.
The area of each chart is made proportional to the total frequency it represents. Because area depends on the square of the radius, the radii must be calculated using \(\frac{r_1^2}{r_2^2} = \frac{n_1}{n_2}\), where \(n\) is the total frequency.
Rearranged, this gives \(r_2 = r_1 \sqrt{\frac{n_2}{n_1}}\). The most frequent error is scaling the radius directly in proportion to the frequency, which makes the area wrong by a factor equal to the ratio itself, exaggerating the difference substantially.
Revision notes
Why area, not radius
The visual impression of a pie chart comes from its area, not its radius.
So the area must be proportional to the total frequency. Since area depends on the square of the radius, the radius must be scaled by the square root of the frequency ratio.
The formula
\(\frac{r_1^2}{r_2^2} = \frac{n_1}{n_2}\), rearranging to \(r_2 = r_1 \sqrt{\frac{n_2}{n_1}}\).
Substitute the known radius and both totals, then take the square root. Always show the square root step, as it usually carries a mark.
The common error
Scaling the radius directly in proportion to the frequency.
If one total is four times the other, the radius should be doubled, not quadrupled. Quadrupling it makes the area sixteen times larger instead of four times, greatly exaggerating the difference.
Key points
- Comparative pie charts have areas proportional to totals.
- Area depends on the square of the radius.
- \(\frac{r_1^2}{r_2^2} = \frac{n_1}{n_2}\).
- \(r_2 = r_1 \sqrt{\frac{n_2}{n_1}}\).
- Take the square root of the frequency ratio.
- Scaling the radius directly is the common error.
Worked examples
Example 1
A pie chart of radius 3 cm represents 50 people. Work out the radius for a chart representing 200 people. [3 marks]
Working
Example 2
Explain why the radius is not scaled in direct proportion to the total frequency. [2 marks]
Working
Example 3
A chart of radius 4 cm represents 100 items. Work out the radius representing 25 items. [3 marks]
Working
Common mistakes
Scaling the radius in direct proportion.
The square root of the frequency ratio must be used.
Inverting the ratio inside the square root.
It is the new total over the old total when finding the new radius.
Forgetting the square root.
It is the step that converts an area ratio into a radius ratio.
Not showing the formula.
It usually carries a method mark.
Exam tips
- Write the formula before substituting.
- Show the square root as its own step.
- Check the larger total gives the larger radius.
- Remember this is a Higher-only topic.
Key terms
- Comparative pie charts
- Pie charts with areas proportional to their totals.
- Radius
- The distance from centre to edge, scaled by a square root.
- Area
- What must be proportional to the total frequency.
- Square root
- The operation converting an area ratio to a radius ratio.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.