Two Categorical Variables

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.

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:

\[ \frac{13}{100} \cdot 100 \% = 13\%\]

In the train scenario, the marginal distribution will tell you the frequency of first-class, economic class, right-handed, left-handed, and ambidextrous passengers.

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:

\[ \frac{53}{100} \cdot 100 \% = 53\%\]

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.

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:

\[ \frac{11}{53} \cdot 100 \% = 20.75 \%\]

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:

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.

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.

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.

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.

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

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.

1. Terrible
2. Bad
3. Regular
4. Good
5. Excellent

If you feel like the service was almost excellent, this method will also allow you to give intermediates, like \(4.8\).

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.

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.

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

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.

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.

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:

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