The pie charts Pie charts (also known as circle graphs) are one of those visual resources that everyone has seen at some point, even if they don't always know how to explain them well. They often appear in textbooks, economic reports, surveys, and even company presentations to show at a glance how a total is distributed among various categories.
Although they seem very simple, behind a pie chart there is a clear statistical methodologyEach segment of the circle represents a part of the whole, and the size of that segment is not chosen arbitrarily, but rather calculated precisely based on frequencies, percentages, and angles. Understanding how they are constructed and when to use them is key to interpreting the information correctly.
What is a pie chart?
A pie chart is a statistical graph in the shape of a circle which is divided into portions (sectors). Each sector represents a category of the variable we are studying, and its angular amplitude is proportional to the frequency with which that category appears in the data.
In simple terms, it starts with a circle that symbolizes 100% of the data. That circle is divided into pieces; each piece occupies an angle that depends on how many observations belong to that category relative to the total. If a category represents 50% of the data, its sector will occupy half of the circle; if it represents 10%, it will be a much smaller piece.
From a more theoretical point of view, a pie chart is a one-dimensional representationIt collects information on a single variable (for example, preferred sport, favorite fruit, method of transport, etc.), but the graph itself shows at the same time the absolute or relative frequencies and the distribution between categories.
These diagrams are especially common for the qualitative variables (for example, type of sport, brand chosen, favorite color), although they can also be used with quantitative variables when grouped into intervals or classes (for example, age ranges, income ranges or score categories).
Basic characteristics of the pie chart
The fundamental idea behind this type of graph is that the data is represented in a circle divided into sectorsThe key is that the angle of each sector is proportional to the frequency of the corresponding category, so that the area of āāeach piece visually reflects that proportion with respect to the total.
The complete circle has a total amplitude of 360 degreesTherefore, if we add up the angles of all the sectors, the result must be exactly 360Ā°. This ensures that no information is "missing" or "extra" and that the entire distribution is correctly represented in the graph.
Since the area of āāeach sector depends directly on its corresponding central angle, there is a direct proportionality between the amount of data in each category and the angle assigned to it. If one category has twice as many observations as another, its sector will also have twice the angle.
For that reason, very simple formulas based on the following are used to construct the diagram: rule of three, which allow you to go from frequency (absolute, relative or percentage) to the number of degrees that each sector should occupy within the circle.
Elements involved in a pie chart
A well-constructed pie chart combines several statistical elements: absolute frequency, relative frequency, percentage and anglesAlthough it may seem somewhat technical, these concepts are actually used very intuitively when preparing the chart.
La Absolute frecuency It is simply the number of times a particular value or category appears in the sample. For example, if 8 students in a class say their favorite sport is soccer, the absolute frequency of the category "soccer" is 8.
La relative frequency It is the proportion that this category represents of the total data. It is calculated by dividing the absolute frequency by the total number of observations. If there are 20 students and 8 choose soccer, the relative frequency of soccer will be 8/20 = 0,4 (that is, 40%).
El porcentaje It is simply the relative frequency expressed as a percentage. To convert from relative frequency to percentage, multiply by 100. In the previous example, a relative frequency of 0,4 is equivalent to a 40% preference for soccer within the class.
The translation of all this into the language of the circle is done through the degrees of each sectorThe angle of the sector indicates what part of the total 360Ā° corresponds to each category, and is the essential data to be able to draw the diagram using a protractor.
Calculating angles in a pie chart
To go from the data in a table to the concrete drawing of a pie chart, you have to calculate the angular amplitude that corresponds to each category. There are several valid procedures, all based on a direct rule of three.
The most common method uses the relative frequencyIf the relative frequency of a category is fr (for example, 0,25), the sector angle is obtained by multiplying that relative frequency by 360Ā°. Thus, the general formula is: Sector degrees = relative frequency Ć 360Ā°.
You can also start from Absolute frecuencyRelating the number of observations in each category to the total. If N is the total number of data and ni is the absolute frequency of a specific category, we usually set up the proportion: ni / N = sector angle / 360Ā°. Hence the expression sector angle = (ni / N) Ć 360Ā°.
Another possibility is to use the directly percentagesIf a category represents p% of the total, simply calculate p/100 Ć 360Ā° to obtain the angle. For example, if a category represents 25%, the angle will be (25/100) Ć 360Ā° = 90Ā°.
In all cases, if the calculations have been done correctly, adding up the grades of all sectors should yield exactly 360Ā°This is a good way to check if we've made any mistakes in the numbers before finalizing the graph.
Example: sports preferences in a class
Let's imagine a class of students in which we want to study what sport does or prefer Each student. The survey options are: basketball, swimming, soccer, and "does not practice any sport." The responses are collected and a table with the frequencies is created.
Suppose that after asking all the students, a distribution similar to this is obtained: one group plays basketballanother practice swimming, another one plays at CP Football and the rest of the class does not regularly participate in sports. Each of these groups has a specific absolute frequency, which, when added together, gives the total number of students in the class.
To construct the pie chart, the first step is to Calculate the angle corresponding to each sport and the category "does not practice sport". For this, a rule of three is applied using as a reference the 360Ā° of the circumference that represent the total number of students in the class.
The process boils down to establishing a direct proportionality between the number of students in each sport and the total number of students, and between the angle of each sector and 360Ā°. This allows us to determine, for example, how many degrees to allocate to the football sector or the swimming sector based on the number of students in each group.
Although the problem statement mentions a āvector diagramā, what is actually constructed is a classic pie chartwhere each type of sport is represented by a piece of the circle, making it easy to see at a glance which activity is the most practiced and which is the least frequent.
Detailed example: graph of favorite sports
Let's now look at a more complete example with specific numerical dataA classmate of Marta's conducts a survey among his classmates to find out what their favorite sport is, and collects the answers in a table showing the absolute frequencies, relative frequencies, percentages and the grades of each sector.
The table is as follows: for Dance There are 5 students (absolute frequency 5), for Soccer There are 8 students, for Tennis There are 2, for Basketball There are 3 and for Athletics There are 2. The sum of all absolute frequencies is 20 students in total.
If we calculate the relative frequencies, we find that Dance has a relative frequency of 0,25 (25%), Soccer one of 0,4 (40%), Tennis one of 0,1 (10%), Basketball one of 0,15 (15%), and Athletics another of 0,1 (10%). Adding all the relative frequencies together gives us 1, which is equivalent to 100% of the students.
To find the degrees of each sector, the formula for the degrees as a function of relative frequencyFor example, for Dance, the calculation is 0,25 Ć 360Ā° = 90Ā°. For Soccer, it's 0,4 Ć 360Ā° = 144Ā°. For Tennis, it's 0,1 Ć 360Ā° = 36Ā°. For Basketball, it's 0,15 Ć 360Ā° = 54Ā°. For Athletics, again, it's 0,1 Ć 360Ā° = 36Ā°.
If we add the amplitudes of all the sectors: 90Ā° + 144Ā° + 36Ā° + 54Ā° + 36Ā°, the result is 360Ā°This confirms that we have correctly distributed the full circle among the five sports categories of the survey.
Another practical example: favorite fruits in a class
In another class, a survey is conducted with the question āWhat is your favorite fruit?āBased on the students' responses, a data table is created showing the number of students who prefer each fruit, and then this information is represented in a pie chart to visualize the proportions.
The four fruits considered in the example are: mango, strawberry, grape and appleThe total number of students surveyed is 30. Of these, 12 choose mango, 6 prefer strawberry, 7 opt for grape and 5 choose apple as their favorite fruit.
To construct the pie chart, one starts from the idea that the measure of the surface area of āāeach sector It is directly proportional to the central angle that defines it. Therefore, a direct proportionality relationship between the number of students and the degrees of the circle is again used.
In the case of the mango, the proportion is set up as follows: 30 students correspond to the 360Ā° of the full circle, while 12 students correspond to the angle of the sector representing that fruit. Therefore, the angle of the mango is calculated using the expression 360Ā° Ć 12 / 30, resulting in an angle of 144Ā°.
Repeating the same procedure for the rest of the fruits yields the following angles: for strawberry, the angle is 360Ā° Ć 6 / 30 = 72Ā°; for grape, 360Ā° Ć 7 / 30 = 84Ā°; for apple360Ā° Ć 5 / 30 = 60Ā°. By adding the four angles, 144Ā° + 72Ā° + 84Ā° + 60Ā°, the total 360Ā° of the circumference are recovered.
This example clearly shows how, starting only from the absolute frequencies of each category (the number of students who choose each fruit), it is possible to construct a complete pie chart that visually reflects the distribution of preferences within the class.
How to build a pie chart step by step
Constructing a pie chart involves combining Numerical calculation and graphical representationAlthough in practice computer programs are often used to automatically create the drawing, it is important to know the manual process to fully understand what the graphic is showing.
The first step always involves creating a frequency table Based on the collected data, a table is created. It lists the categories of the variable (for example, each sport or each fruit), the absolute frequencies, the relative frequencies, and, if desired, the percentages. This table is the foundation of the entire process.
Next, for each category, the sector angleThis can be done using any of three methods: relative frequency, absolute frequency, or percentage. The key is to maintain the same criteria for all categories and verify that the sum of angles reaches 360Ā°.
Once the degrees corresponding to each sector are known, a circle is drawn with a compass and the center of the circleUsing a protractor, the different sectors are drawn from an initial radius with their corresponding amplitude, placing one after the other until the circle is completed.
Finally, you can add Differentiated colors for each sector and a legend indicating which category each slice of the circle represents. In this way, the pie chart becomes a very intuitive visual tool that helps to quickly interpret the proportions.
When is it appropriate to use a pie chart?
Pie charts can be used with all kinds of variables, but they are especially frequently used in the case of... qualitative variables, in which the aim is to show how the population is distributed among different categories without inherent order.
They are very useful when you want to highlight the part-whole relationshipThat is, when we need to see at a glance what fraction of the total each category represents. For example, what percentage of customers choose each product, how the budget is divided among different departments, or what proportion of votes each party receives in an election.
They are also suitable when the number of categories is not excessive. few categories (for example, between 3 and 6) the chart is clearly interpreted. However, when there are too many categories or the differences between them are very small, the pie chart loses readability and it may be preferable to use other types of charts, such as bar charts.
In both education and fields like economics and finance, pie charts are used to make data more accessible to people who may not be familiar with statistical language. Their main strength is that they allow quick visual comparisons between the different parts of the whole.
In the field of financial education, for example, it is very common to see pie charts that show the allocation of expenses of a family or a company: how much is allocated to housing, transportation, food, leisure, savings, etc. With a simple glance, it's clear which category is absorbing the most resources.
In short, the pie chart is a very powerful graphical resource for showing how a total is divided among several parts using a circle divided into proportionate portionsBehind each sector are simple but precise calculations based on frequencies, percentages, and angles, ensuring that the representation is faithful to the original data and that the reader can interpret the information quickly and without needing extensive statistical knowledge.
