# Pie chart

Pie chart (or pie graph) is a way of displaying data in a circular graph which is divided into sectors. Furthermore, each pie sector represents a certain category.

In order to compare informations, pie chart uses percentages. In other words, the entire circle represents $100 \%$ of a whole, while the sectors represent portions of the whole.

In the following example you will learn how to read and interpret a pie chart.

Example 1:  The following picture shows percentages of types of transport that sample of $200$ people uses most often:

a) How many people use bus most often?

b) How many people don’t use train most often?

c) How many people use bicycle or car most often?

Solution:

Certainly, total frequency is $N = 200$.

a) $20\%$ of $200$ people use bus most often. In other words, the answer is $20 \% \cdot 200 = 0.2 \cdot 200 = 40$ people.

b) $10\%$ of $200$ people use train most often. Therefore, we have

$$(100\% – 10\%) \cdot 200 = 0.9 \cdot 200 = 180$$

In conclusion, $180$ people don’t use train most often.

c) $30\%$ of $200$ people use bicycle most often and $40\%$ of them use car most often.  Therefore, we have

$$(30\% + 40\%)\cdot 200 = 70 \% \cdot 200 = 0.7 \cdot 200 = 140.$$

In conclusion, $140$ people use bicycle or car most often.

Example 2:  The following pie chart shows a survey of the number of pieces of clothes, cosmetics and jewelry that women buy. Furthermore, there were $140$ pieces of jewelry in the survey.

a) What fraction of the articles is cosmetics?

b) Compute the number of items in the survey.

c) How many pieces of clothes were in the survey?

Solution:

a) Fraction of cosmetics is

$$\frac{\alpha_{1}}{360^{\circ}} = \frac{205^{\circ}}{360^{\circ}} = \frac{41}{72}.$$

b) Let $x$ be the number of items. Since there were $140$ pieces of jewelry in the survey, we have

$$\frac{35^{\circ}}{360^{\circ}} \cdot x = 140$$

$$x = 140 \cdot \frac{360^{\circ}}{35^{\circ}}$$

$$x = 1440.$$

Therefore, the number of items is $1440$.

c) There were

$$\frac{120^{\circ}}{360^{\circ}} \cdot 1440 = 480$$

pieces of clothes in a survey.

## Making a pie chart

It is a little bit tricky to draw pie charts by hand. Consequently, there is a variety of computer programs in which the whole procedure is easier.

If we want to draw a pie chart, we need to have a list of categorical variables and numeric variables also. For instance, in the Example 1 types of transport are the categorical variables, while percentages are the numeric variables. Furthermore, it is important that categories don’t overlap.

We must convert the share of each component into a percentage of $360^{\circ}$. In other words, we first need to calculate the angle of the each sector.

In short, the formula for the angle of the sector is:

$$\alpha_{i}=\frac{f_{i}}{N}\cdot 360^ {\circ} = p_{i}\cdot 360^ {\circ},$$

where $f_{i}$ represents the frequency of data, $N$ total frequency and $p_{i}$ relative frequency.

In addition, notice that the angle of the sector is proportional to the frequency of the data.

After that we simply draw a circle and angles for each sector. At the end, we label the sectors.

## Examples

Example 3: Professor Smith teaches $3$ subjects. $640$ students have Subject $1$, $760$ students have Subject $2$ and $230$ students have Subject $3$. Construct a pie chart to represent the number of students who have a certain subject.

Solution:

Obviously, $N = 640 + 760 + 230 = 1630$. Furthermore,

$$\alpha_{1}=\frac{f_{1}}{N}\cdot 360^ {\circ} = \frac{640}{1630}\cdot 360^ {\circ} = 141 ^{\circ} 20′ 58.9”$$

$$\alpha_{2}=\frac{f_{2}}{N}\cdot 360^ {\circ} = \frac{760}{1630}\cdot 360^ {\circ} = 167 ^{\circ} 51′ 9.94”$$

$$\alpha_{3}=\frac{f_{3}}{N}\cdot 360^ {\circ} = \frac{230}{1630}\cdot 360^ {\circ} = 50 ^{\circ} 47′ 51.17”.$$

Now we need to compute the percentages:

$$x \% \cdot 1630 = 640 \rightarrow 1630x = 640 \cdot 100 \rightarrow x \approx 39.3 \%$$

$$x \% \cdot 1630 = 760 \rightarrow 1630x = 760 \cdot 100 \rightarrow x \approx 46.6 \%$$

$$x \% \cdot 1630 = 230 \rightarrow 1630x = 230 \cdot 100 \rightarrow x \approx 14.1 \%$$

Finally, our pie chart is:

Example 4: The following table shows the number of patients in hospital wards.

Draw a pie chart.

Solution:

$$N = 1052 + 2245 + 340 + 552 + 4630 = 8819$$

Cardiology:

$$\alpha_{1}=0.1192 \cdot 360^ {\circ} = 42 ^{\circ} 54′ 43.2”$$

Emergency:

$$\alpha_{2}=0.2545 \cdot 360^ {\circ} = 91 ^{\circ} 37′ 12”$$

Intensive care:

$$\alpha_{3}=0.0385 \cdot 360^ {\circ} = 13 ^{\circ} 51′ 36”$$

Obstetric ward:

$$\alpha_{4}=0.0625 \cdot 360^ {\circ} = 22 ^{\circ} 30′ 0”$$

Surgery:

$$\alpha_{5}=0.525 \cdot 360^ {\circ} = 189 ^{\circ} 0′ 0”$$

Therefore, our pie chart is:

In other words, from the given table you could read that most of the patients are in surgery, but pie chart gives you a clear picture of the whole situation.