While on a train, suddenly, our journey was halted. A murder had been committed on the train, and a detective was brought to investigate. During the investigation, the detective first considered the boarding class of the culprit and had to take the census of all the passengers on the first class and the economic class. Afterwards, we were grouped based on handwriting into left-handed, right-handed, and ambidextrous.
This detective had just considered two categorical variables, boarding class, and handwriting, but was he able to solve the crime? Here, you will be learning about the correlation, graph, tests, and more regarding two categorical variables. You can also be a detective!
Graphs of Two Categorical Variables
Earlier in the crime story, it was mentioned that the detective had approached the case by considering two categorical variables. What is a categorical variable?
A categorical variable, also known as a qualitative variable, is a variable whose properties are described, rather than measured.
If the properties of a variable can be measured or counted, they are known as quantitative variables. You will not focus on these variables in this article.
Definitions are always better understood with examples!
You get thirsty while on the train, so you go and get a can of soda. To be more specific, you get a \(12\) oz lime-flavored soda, which comes in a green can and has \(40\) calories.
In this example, the categorical variables are those that you can describe, such as the flavor and the color of the can. The amount of liquid in the can and the calorie count are both measurable, so they are quantitative variables.
And what does the detective mean when talking about two categorical variables?
When talking about two categorical variables, you are talking about the combinations you can get from looking at two separate categorical variables.
Let's go back to the investigation. The detective considered two categories: boarding class and handwriting. So, there are six possible combinations that are produced using these two categories:
- First-class right-handed
- Economic class right-handed
- First-class left-handed
- Economic class left-handed
- First-class ambidextrous
- Economic class ambidextrous
A two-way table, or contingency table, is a table that organizes the observations according to two categorical variables. Each cell in a contingency table represents a combination of two factors, and the frequency of the subjects that fall within those categories is written in that cell.
The detective used a contingency table to classify the passengers of the train based on-boarding class and handwriting.
| Boarding class |
| First class | Economic class |
| Handwriting | Right | \[30\] | \[35\] |
| Left | \[13\] | \[11\] |
| Ambidextrous | \[4\] | \[7\] |
For example, by looking at the table, you can tell that \(30\) of the first-class passengers are right-handed. You can find the rest of the frequencies of the other combinations of factors by looking at the respective cell.
The frequencies of a contingency table show how many subjects fall within each combination of the two categorical variables.
Typically, contingency tables also include an extra row at the bottom and an extra column to the right to count totals.
| Boarding class |
| First class | Economic class | Total |
| Handwriting | Right | \[30\] | \[35\] | \[65\] |
| Left | \[13\] | \[11\] | \[24\] |
| Ambidextrous | \[4\] | \[7\] | \[11\] |
| Total | \[47\] | \[53\] | \[100\] |
For example, there are \(65\) right-handed passengers, and there are \(53\) passengers in the economic class. By looking in the bottom-right corner, you find that there is a total of \(100\) passengers.
Relative Frequency
Sometimes, rather than the actual numbers, you just need to know which fraction of the subjects fall within each category. This fraction, or ratio, is known as relative frequency.
The relative frequency is the ratio of the frequency of an observation divided by the total of observations.
Perhaps, you wish to know what fraction of the total suspects consists of left-handed first class passengers, then, the relative frequency of left-handed first class passengers to the total passengers is:
\[\frac{13}{100} \]
or written as a percentage:
\[ \frac{13}{100} \cdot 100 \% = 13\%\]
You might also find the marginal relative frequency and conditional relative frequency, which are two kinds of relative frequencies.
Marginal Frequency and Marginal Relative Frequency
Contingency tables typically write totals at the extreme right column and the bottom row. These totals are known as marginal frequencies.
The marginal frequency is the number of subjects that fall within each individual category. The marginal distribution consists of all the marginal frequencies of the table.
In the train scenario, the marginal distribution will tell you the frequency of first-class, economic class, right-handed, left-handed, and ambidextrous passengers.
The marginal distribution receives its name from the fact that the totals are shown on the margins of the table.
The marginal frequencies of a contingency table show how many subjects fall within each individual categorical variable.
If you know how to find marginal frequencies and relative frequencies, then you also know about marginal relative frequencies! Whenever you are using a marginal frequency to find a ratio, then you are finding a marginal relative frequency.
Imagine you were to determine the marginal frequency of economic class passengers relative to the total suspects from the table. Knowing that the marginal frequency of economic class passengers is \(53\) and the total frequency is \(100\), the marginal relative frequency of economic class passengers is:
\[\frac{53}{100}\]
or written as a percentage:
\[ \frac{53}{100} \cdot 100 \% = 53\%\]
You can also apply this reasoning to find more frequencies. Try finding the marginal frequency of left-handed people, or the marginal relative frequency of first-class passengers.
Conditional Frequency and Conditional Relative Frequency
By using the same table, if you choose to focus on a particular row, then you will be working with a particular handwriting. Likewise, if you decide to focus on a particular column, then you are dealing with a specific boarding class.
In this case, you are placing a condition on the values that you are reading from the table.
The conditional frequency is the number of subjects that fall within a category, considering that the other category has already been specified.
The conditional frequency makes more sense when talking about conditional relative frequency.
A conditional relative frequency is the ratio of a conditional frequency divided by the marginal frequency of the specified category.
Typically, the word “given” is used to emphasize that you are dealing with a conditional frequency.
As usual, this idea is better understood with an example.
Using the information in the table, determine the conditional relative frequency that a suspect is left-handed given that it is on the economic class.
Solution:
Here is the table again, so you do not have to scroll back up.
| Boarding class |
| First class | Economic class | Total |
| Handwriting | Right | \[30\] | \[35\] | \[65\] |
| Left | \[13\] | \[11\] | \[24\] |
| Ambidextrous | \[4\] | \[7\] | \[11\] |
| Total | \[47\] | \[53\] | \[100\] |
Table 1. Categories of people and hand dexterity.
You are asked to find a certain conditional relative frequency given that the passenger is from the economic class. This means that you will focus on the column that contains the frequencies of the economic class passengers.
Since you are asked to find the conditional relative frequency that a subject is left-handed, you now look at the row with the left-handed passengers. There are \(11\) economic class, left-handed passengers.
Finally, divide this frequency by the marginal frequency of economic class passengers. The number at the bottom of the economic class column tells you that there are \(53\) passengers in the economic class, so the conditional relative frequency that a suspect is left-handed, given that is in the economic class, is:
\[ \frac{11}{53}\]
which you can write as a percentage with the help of a calculator, that is:
\[ \frac{11}{53} \cdot 100 \% = 20.75 \%\]
The use of relative frequency in analyzing a contingency table enables you to re-express the table in percentages of the desired condition, which may require just one category or a combination of both. When such is achieved, a graph may be drawn in the form of a pie chart or bar chart.
Graphing two categorical variables
As mentioned earlier, the two-way table is essential for visualizing two categorical variables. The following example is a quick illustration of the graphing of two categorical variables.
Considering the table which has repeatedly been used in this text, graph a pie chart considering all data given.
Solution:
In this case, all two categories must be represented in one pie chart. This means that the contingency table has to be re-expressed in percentage to easily plot the graph. Below is the table from the data given:
| Boarding class |
| First class | Economic class | Total |
| Handwriting | Right | \[30\] | \[35\] | \[65\] |
| Left | \[13\] | \[11\] | \[24\] |
| Ambidextrous | \[4\] | \[7\] | \[11\] |
| Total | \[47\] | \[53\] | \[100\] |
Table 2. Categories of people and hand dexterity.
Each cell has to be represented in percentage as a fraction of the total, which is \(100\). For example, the relative frequency of left-handed first-class passengers would be calculated as:
\[\frac{13}{100}\cdot 100\%=13\%\]
By repeating this process with all the frequencies, you obtain the following table.
| Boarding class |
| First class | Economic class | Total |
| Handwriting | Right | \[30 \%\] | \[35 \% \] | \[65 \%\] |
| Left | \[13 \%\] | \[11 \%\] | \[24 \%\] |
| Ambidextrous | \[4 \%\] | \[7 \%\] | \[11 \%\] |
| Total | \[47 \%\] | \[53 \%\] | \[100 \%\] |
Table 3. Categories of people and hand dexterity.
Figure 1. A pie chart of the boarding class and handwriting of the train passengers
The pie chart above illustrates the graphical representation of the data collated by the detective and is an example of a two categorical variables graph.
It is also common to draw charts using conditional relative frequencies.
The detective decides to focus his attention on the first-class passengers. Draw a pie chart of the conditional relative frequencies of the passengers given that they are first-class passengers.
Solution:
Since you are asked to draw a pie chart for the passengers that meet the condition of being in first-class, you should focus on such a column from the table.
| First class passengers |
| Right-handed | \[30\] |
| Left-handed | \[13\] |
| Ambidextrous | \[4\] |
| Total | \[47\] |
Table 4. Categories of people and hand dexterity.
As usual, to draw a pie chart you need to find the relative frequencies, which in this case will be conditional relative frequencies. For right-handed first-class passengers, this is:
\[ \frac{30}{47} \cdot 100 \% = 63.8\%\]
and find the rest of the conditional relative frequencies in a similarly, obtaining the next table.
| First class passengers |
| Right-handed | \[63.8 \%\] |
| Left-handed | \[ 27.7 \% \] |
| Ambidextrous | \[ 8.5 \% \] |
| Total | \[ 100\%\] |
Table 5. Categories of people and hand dexterity.
Here is the resulting pie chart.
Figure 2. A pie chart of the handwriting of the first-class passengers
Keep in mind that you can also use other types of graphs to study two categorical variables, such as bar graphs or stacked bar charts.
Two Categorical Variables and Correlation
There are some tests to calculate the correlation between two categorical variables, like the tetrachoric correlation, the polychoric correlation, and Cramer's V. However, these methods are not addressed in AP, so will remain out of the scope of this article.
Whenever you are dealing with correlation at an AP level, you are talking about the correlation between quantitative variables. For more information about this, please take a look at our article about Two Quantitative Variables.
The Two Categorical Variables Test
To check the association between variables in a contingency table, the chi-square (or \(X^ 2\)) test is used. Two questions are asked, which form the null hypothesis and the alternate hypothesis. The null hypothesis is denoted as \(H_0\), and represents “no association exists between both variables”, which implies that both variables are indeed independent. Meanwhile, the alternate hypothesis denoted by \(H_a\) represents “association exists between both variables”, and implies that both variables are dependent.
For more information about the chi-square test and how to perform it, please reach out to our Chi-Square Tests article.
Regression With Two Categorical Variables
Besides studying the relation between the data you have collected, statistics can also be used for predicting outcomes. Given a significant enough data set, you can start making predictions based on the data you previously gathered. This is the main idea behind regression analysis.
Regression analysis is a collection of techniques used in statistics to find a mathematical model that can describe the relationship between two (or more) variables.
Regression analysis is usually done on quantitative variables because you are working with the numerical values of such variables. However, in some scenarios, it is possible to assign a numerical value to a categorical variable, so the techniques used in regression become available. The following example might sound familiar.
For administrative purposes, restaurants often rely on surveys to evaluate a customer's satisfaction. This satisfaction can be seen as a categorical variable, which will typically be described using words like:
- Terrible
- Bad
- Regular
- Good
- Excellent
However, you cannot do operations with these words. One way of dealing with this situation is to assign numerical values to each of the possible categories, so the following phrase might now sound familiar to you:
“On a scale from \(1\) to \(5\), where \(1\) is terrible and \(5\) is excellent, how would you rate the service?”
This way, you can assign a numerical value to each of the possible words that you would have used.
- Terrible
- Bad
- Regular
- Good
- Excellent
If you feel like the service was almost excellent, this method will also allow you to give intermediates, like \(4.8\).
Once you have assigned numerical values to categorical variables, you can now start doing regression with it. You just have to make sure that the numerical values assigned are reasonable enough. For more information about regression analysis please take a look at our Linear Regression article.
Examples of Two Categorical Variables
You should try as many examples as possible to develop competency on tasks involving two categorical variables.
The detective now decides to focus on investigating right-handed people. Draw a pie chart of the conditional relative frequencies of the passengers given that they are right-handed.
Solution:
Since you are asked to draw a pie chart for the passengers that meet the condition of being right-handed, you should focus on the corresponding row of the table.
| First class | Economic class | Total |
| Right-handed people | \[30\] | \[35\] | \[65\] |
Table 6. Categories of people and hand dexterity for right-handed people.
Relative frequencies are always required for drawing pie charts, so find them using the usual method. This will result in the following table.
| First class | Economic class | Total |
| Right-handed people | \[46.2 \% \] | \[ 53.8 \%\] | \[ 100 \%\] |
Table 7. Categories of people and hand dexterity for right handed people.
Using this table, you can draw the pie chart of these conditional relative frequencies.
Figure 3. A pie chart of the boarding class of the right-handed passengers
Remember the crime scene? While the detective carried out his investigation, he confirmed that the crime had been carried out by an ambidextrous person on the first class, who also had flu. By placing an additional condition, the detective narrowed down the search! However, the only ambidextrous fellow with the flu on the train was... ME. Did I commit murder ? Wait up after the following example to confirm this.
A survey was carried out to determine the relationship between a population's sociability and the size of their family. Here, an individual was asked whether they considered themselves sociable or not and whether they came from a family size of four and below, or from a family size of above four. The results are shown below.
| Categories | Sociable | Not sociable |
| Family size of four and below | \[40\] | \[50\] |
| Family size of above four | \[90\] | \[20\] |
Table 8. Categories of people and their sociability.
a. Find the relative frequency of individuals that come from a family size above four, relative to the total individuals sampled.
b. Determine the conditional relative frequency that an individual is from a family of four and below, given that is not sociable.
c. Make a graph of the information given.
Solution:
Since you are not provided such information, you should begin by finding the marginal frequencies and adding them to the contingency table. Here, you only have to add the values of the rows and the columns.
| Categories | Sociable | Not sociable | Total |
| Family size of four and below | \[40\] | \[50\] | \[90\] |
| Family size of above four | \[90\] | \[20\] | \[110\] |
| Total | \[130\] | \[70\] | \[200\] |
Table 9. Categories of people and their sociability..
Now, you can answer the questions.
a. This is the marginal frequency of individuals that come from families that are above four in size, divided by the total individuals sampled. This is:
\[\frac{110}{200}=\frac{11}{20}\]
or written as a percentage
\[ \frac{11}{20} \cdot 100 \% = 55 \%\]
b. You are asked to find the conditional relative frequency that an individual is from a family of four and below, given that is not sociable. The condition placed here is that the individual is not sociable, so you will focus on the corresponding column.
Next, go to the corresponding row to find that \(50\) individuals under this category also come from a family size of four and below. To find the conditional relative frequency, you divide this number by the total of not sociable individuals, so:
\[\frac{50}{70}=\frac{5}{7}\]
or written as a percentage
\[ \frac{5}{7} \cdot 100 \% = 71.4\%\]
c. A bar graph may be drawn to give a visual interpretation of the data given. However, it would be beneficial to have a separate table of the data in percentages, as shown below:
| Categories | Sociable | Not sociable | Total |
| Family size of four and below | \[ 20\%\] | \[25 \%\] | \[45 \%\] |
| Family size of above four | \[ 45 \% \] | \[ 10 \% \] | \[ 55 \% \] |
| Total | \[ 65 \% \] | \[ 35 \% \] | \[ 100 \% \] |
Table 10. Categories of people and their sociability..
The bar graph below is a representation of the data given:
Figure 4. Sociability and family size bar chart
After the detective revealed that I was the culprit, I woke up from my weary dream. Nonetheless, all you have learned here are based on statistical principles and would prove very useful when you attempt more tasks. See how statistics can be useful even when solving crimes?
Two Categorical Variables - Key takeaways
- Two categorical variables are data representations arranged by considering two factors or groups, which are otherwise termed categories.
- When studying two categorical variables, they are typically arranged in contingency tables, which are also known as two-way tables.
- Each value in a contingency table represents the frequency of the individuals that fall under each combination of the two categorical variables.
- Contingency tables typically also include totals in their margins. These totals are known as marginal frequencies.
- The marginal frequencies of a contingency table show how many subjects fall within each individual categorical variable.
- Relative frequency is the fraction of an event out of the total frequency in a statistical experiment.
- A conditional relative frequency can be obtained by dividing one of the frequencies of the table by the marginal frequency of the category that is being used as the condition.
- The data from studying two categorical variables can be arranged using the typical charts that are used for categorical data, like bar charts and pie charts.
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