Introduction to Comparative Representations

In your Foundation studies, you learned that a pie chart shows how a total is split into different categories using angles. However, what if you want to compare two different sets of data where the total number of people or items is different? That is where comparative pie charts and 2D/3D representations come in!

In this chapter, we move beyond just looking at the "slices" and start looking at the total size of the diagrams. This is a Higher tier topic that focuses on the relationship between the physical size of a shape and the frequency it represents.

Comparative Pie Charts

When we compare two pie charts, we don't just look at the angles inside. We use the area of the whole circle to represent the total frequency.

The Golden Rule: The area of a pie chart is directly proportional to the total frequency it represents.
\( \text{Area} \propto \text{Total Frequency} \)

Because the area of a circle is \( \pi r^2 \), if we want to draw a second pie chart for a larger group of data, we can't just double the radius to double the frequency—that would actually make the area four times larger! Instead, we use a specific formula to find the correct radius.

Calculating the New Radius

If you are given the radius of one pie chart (\( r_1 \)) and its total frequency (\( F_1 \)), and you need to find the radius of a second pie chart (\( r_2 \)) for a different total frequency (\( F_2 \)), use this formula from your syllabus:

\( r_2 = \sqrt{\frac{F_2 \times r_1^2}{F_1}} \)

Step-by-Step Example:

Imagine Pie Chart A represents 100 students and has a radius of 3cm. You need to draw Pie Chart B to represent 200 students.

1. Identify your values: \( F_1 = 100 \), \( r_1 = 3 \), \( F_2 = 200 \).
2. Square the first radius: \( 3^2 = 9 \).
3. Multiply by the new frequency: \( 200 \times 9 = 1800 \).
4. Divide by the old frequency: \( 1800 \div 100 = 18 \).
5. Take the square root: \( \sqrt{18} \approx 4.24 \text{ cm} \).

So, the radius of the second pie chart should be 4.24 cm, not 6 cm!

Quick Tip: If the frequency doubles, the radius increases by a factor of \( \sqrt{2} \), not by 2!

Comparative 2D and 3D Representations

Statisticians sometimes use other shapes like squares, pictures (pictograms), or even 3D objects like cubes to compare data. The same logic applies: the size of the shape must match the frequency.

2D Representations (Area)

In 2D comparisons (like squares or rectangles), the Area represents the frequency.
If a square with side length 2cm represents 10 people, and you want to represent 40 people (4 times the data), the area must be 4 times larger.
Since \( \text{Area} = \text{length}^2 \), you would only need to double the side length to 4cm (\( 4^2 = 16 \), which is four times the original area of 4).

3D Representations (Volume)

In 3D comparisons (like cubes or spheres), the Volume represents the frequency.
\( \text{Volume} \propto \text{Frequency} \)

Don't worry if this seems tricky! The examiners usually want you to spot when these diagrams are misleading. If a company doubles the height and width of a 3D product image to show that sales have "doubled," they are actually misleading the audience because the volume has increased by \( 2 \times 2 \times 2 = 8 \) times!

Summary of Dimensional Growth:

  • If you double the 1D length, the 2D Area becomes \( 2^2 = 4 \) times larger.
  • If you double the 1D length, the 3D Volume becomes \( 2^3 = 8 \) times larger.

Misleading Construction Errors

A big part of the Higher tier syllabus is critiquing diagrams. You need to recognize construction errors that cause "graphical misrepresentation."

What to look for:
  • Distorted Sizing: The most common error is making the radius or side length proportional to the frequency instead of the area or volume.
  • Incorrect Scales: Check if the radius of a comparative pie chart has been calculated correctly using the square root formula.
  • 3D Effects: 3D pie charts or bars can often make the slices at the "front" look much larger than they actually are.

Did you know? Using 3D images to represent 1D data (like simple counts) is almost always considered misleading because the human eye perceives the volume, making the difference between data points look much larger than it really is!

Quick Review: Key Takeaways

1. Area-Frequency Relationship: In comparative pie charts, the Area of the circle represents the total frequency.

2. The Radius Formula: To find a new radius, use \( r_{new} = \sqrt{\frac{\text{new total} \times \text{old radius}^2}{\text{old total}}} \).

3. 2D vs 3D: 2D shapes use Area for frequency; 3D shapes use Volume for frequency. Always check if a diagram is "exaggerating" a difference by scaling dimensions incorrectly.

4. Common Trap: If an exam question asks why a diagram is misleading and shows two 3D cubes where one is twice as tall as the other to represent "double the data," your answer should be: "The volume of the larger cube is 8 times greater, which misrepresents the data as being 8 times larger rather than 2 times larger."

Note: For more on how to choose the best way to display data, see the chapter on "Choosing representations and spotting misleading graphs".