Comparative Pie Charts and 2D and 3D Representations

Comparing a share with a total

Pictorial representations use the sizes of shapes to communicate quantities. To interpret them correctly, first identify what their size represents: a length, an area or a volume. Then read the labels and key to establish the categories, units and totals being compared.

A frequency is the number of observations in a category. A proportion is that category’s fraction of the total. These answer different questions: ‘How many?’ and ‘What share?’ A category can become a larger share of a smaller data set while its actual frequency falls.

Reading comparative pie charts

An ordinary pie chart divides a circle into sectors to show proportions. The whole circle represents the whole data set, so a sector with angle $\theta$ represents the proportion $\theta/360$.

Two ordinary pie charts can have equal-sized circles even when their totals are different. Their sector angles still allow comparison of proportions, but equal sector angles do not establish equal frequencies.

Comparative pie charts also communicate the totals: the area of each whole circle is proportional to the total frequency it represents. A circle with twice the area represents twice the total. Within each circle, sector angles continue to show proportions.

Suppose School A surveys 200 pupils and School B surveys 800 pupils about travelling to school. Half of School A’s pupils walk, compared with a quarter of School B’s pupils. School A therefore has the larger proportion walking: 50% rather than 25%. However, the frequencies are $200\times\frac12=100$ pupils and $800\times\frac14=200$ pupils. School B has twice as many pupils walking.

On correctly scaled comparative pie charts, School B’s whole circle has four times School A’s area. Its walking sector has a smaller angle but a larger area. This is why angles compare proportions, whereas areas compare frequencies.

Using radii to find totals

A circle’s area is $A=\pi r^2$, where $r$ is its radius. The area depends on the square of the radius, not the radius alone. Doubling a radius makes the area four times as large.

For two comparative pie charts, let $F_1$ and $F_2$ be their total frequencies, and and their radii. Since their areas are proportional to their totals,

$$\frac{F_2}{F_1}=\frac{\pi r_2^2}{\pi r_1^2}=\left(\frac{r_2}{r_1}\right)^2.$$

For example, suppose one chart represents 500 people and has radius 2 cm. A second chart on the same scale has radius 6 cm. Its radius is three times as large, so its area and total frequency are nine times as large. It represents $500\times9=4500$ people.

Once the total is known, find a category’s frequency using its sector angle:

$$\text{category frequency}=\frac{\theta}{360}\times\text{total frequency}.$$

If instead a category’s frequency and angle are known, reverse the calculation to find the total:

$$\text{total frequency}=\text{category frequency}\times\frac{360}{\theta}.$$

A useful order is therefore to establish each chart’s total, calculate the relevant category frequencies, and then make the comparison requested.

Comparing two-dimensional representations

Other two-dimensional displays may use the areas of circles, squares or similar pictures to represent quantities. When area encodes the data, compare areas rather than widths or heights alone.

For similar shapes, every corresponding length changes by the same linear scale factor, $k$. Their areas change by $k^2$. This happens because both dimensions contributing to area are scaled: for a rectangle, the new area is , which is times the original area.

For instance, suppose squares with side lengths 2 cm and 6 cm represent quantities through their areas. They have a length ratio of $1:3$ but an area ratio of $1:9$. On a common scale, the larger square represents nine times the quantity. If the smaller square represents 40 items, the larger represents $40\times9=360$ items.

You can also work backwards: if one similar picture represents four times the quantity, its area must be four times as large, so its corresponding lengths are $\sqrt4=2$ times as large.

This rule depends on similarity. If a rectangle becomes twice as wide but stays the same height, its area doubles rather than quadruples. Check which dimensions have changed before drawing a conclusion.

Comparing three-dimensional representations

A three-dimensional display may use the volumes of solids to represent quantities. For similar solids, multiplying every corresponding length by $k$ multiplies volume by $k^3$. For a cuboid, length, width and height each contribute a factor of $k$, giving $k\times k\times k=k^3$.

For example, suppose two cubes represent quantities through their volumes. Their edge lengths are 2 cm and 4 cm, so the linear scale factor is 2. Their volumes are $2^3=8\text{ cm}^3$ and $4^3=64\text{ cm}^3$. The larger cube represents eight times the quantity, not twice the quantity. If the smaller represents 50 items, the larger represents items.

The reverse relationship is also useful: a volume ratio of $1:27$ corresponds to a length ratio of $1:3$ for similar solids, because $\sqrt[3]{27}=3$.

When reading a drawing of a solid, use its labelled dimensions and the key rather than judging its apparent size on the page. Perspective can make some edges look shorter. Check that volume really represents the quantity and that the solids are similar before cubing a corresponding-length ratio. You interpret these representations; you are not expected to draw them.

Judging whether a comparison is fair

A pictorial display is useful when its key clearly explains what size represents and all groups use a consistent scale. Enlarging a picture’s width and height by a factor of 2 to show ‘twice as much’ actually gives it four times the area. Enlarging all three dimensions of a solid by the same factor gives it eight times the volume. Such choices exaggerate a difference if area or volume is meant to represent the quantity.

A strong comparison states the category and groups, distinguishes frequency from proportion, and supports the conclusion with a value or ratio. For example: ‘School A has the higher proportion walking, but School B has the higher number walking: 200 pupils compared with 100.’