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.
Reading pie charts
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.
Answer these questions:
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.
