Comparative pie charts Revision Notes for Edexcel GCSE Statistics

Comparative pie charts

What are comparative pie charts?

Comparative pie charts are a powerful tool used when you need to compare data from groups of different sizes. Unlike regular pie charts that show proportions within a single group, comparative pie charts allow you to make meaningful comparisons between two populations that may have very different total numbers.

infoNote

The key principle behind comparative pie charts is that the physical size of each pie chart represents the total number of people or items in that particular sample. This means that if one group has twice as many people as another group, its pie chart will appear noticeably larger.

Understanding chart sizes and sample sizes

When you look at comparative pie charts, the first thing to notice is that they're usually different sizes. This size difference isn't random - it's carefully calculated to represent the actual difference in sample sizes between the two groups being compared.

infoNote

Practical Example: If you're comparing the favourite TV programmes of 100 primary school children with those of 400 secondary school students, the pie chart for the secondary students would be four times larger in area than the primary school chart.

The mathematical relationship

The relationship between chart sizes and sample sizes follows a specific mathematical formula:

chatImportant

Key Formula for Comparative Pie Charts:

$\frac{\text{area of large pie chart}}{\text{area of small pie chart}} = \frac{R^2}{r^2} = \frac{N}{n}$

Where:

$R$ = radius of the large circle
$r$ = radius of the small circle
$N$ = total number in the large sample
$n$ = total number in the small sample

This can be rearranged to: $\frac{r^2}{R^2} = \frac{n}{N}$

This formula is essential for solving comparative pie chart problems, as it allows you to work out missing sample sizes or chart dimensions.

Calculating actual numbers from angles

Once you understand the relationship between chart sizes, you can calculate the actual number of people in each category using the angle formula:

$\text{Number} = \frac{\text{angle}}{360} \times \text{total}$

This works the same way for both charts, but remember that the "total" will be different for each chart based on their respective sample sizes.

Step-by-step worked example

lightbulbExample

Worked Example: Holiday Destinations

Problem: Two pie charts show holiday destinations for samples in 2016 and 2017. The 2016 chart has radius 3cm and represents 200 people. The 2017 chart has radius 4.5cm. France shows 72° in both charts. How many more people went to France in 2017 than in 2016?

Step 1: Find the total number of people in 2017 Using the formula: $\frac{R^2}{r^2} = \frac{N}{n}$

$\frac{4.5^2}{3^2} = \frac{N}{200}$
$\frac{20.25}{9} = \frac{N}{200}$
$N = \frac{20.25}{9} \times 200 = 450$ people in 2017

Step 2: Calculate people who went to France in 2016

Number = $\frac{72}{360} \times 200 = 40$ people

Step 3: Calculate people who went to France in 2017

Number = $\frac{72}{360} \times 450 = 90$ people

Step 4: Find the difference

$90 - 40 = 50$ more people went to France in 2017

Important points to remember

When working with comparative pie charts, always remember that even though the angle for a particular category might be the same in both charts, the actual number of people it represents will be different because the total sample sizes are different.

bookmarkSummary

Key Points to Remember:

This is a common source of confusion in exams, so always:

First work out the total sample size for each chart
Then use the angles to calculate actual numbers
Finally, make your comparison using the actual numbers, not just the angles

Common exam traps

chatImportant

Watch Out for These Common Mistakes:

Don't assume same angles mean same numbers - A 90° sector in a chart of 100 people represents 25 people, but a 90° sector in a chart of 400 people represents 100 people.
Check your radius calculations carefully - Small errors in squaring the radii will lead to wrong sample sizes.
Remember to find the difference - Questions often ask how many more/fewer, not just the individual totals.

Remember!

bookmarkSummary

Essential Points for Success:

Comparative pie charts show data from groups of different sizes
The size of each pie chart represents the total sample size
Use the formula $\frac{r^2}{R^2} = \frac{n}{N}$ to relate chart sizes to sample sizes
Same angles in different sized charts represent different actual numbers
Always calculate total sample sizes first, then work out individual category numbers

Comparative pie charts (Edexcel GCSE Statistics): Revision Notes