Drawing and Interpreting Two-Way Frequency Tables

Categorical data is data belonging to one or more categories that have a set amount of possible outcomes or values. This is opposed to numerical data, for which all the possible values can't be listed. However, numerical data can be grouped into intervals, making it categorical data. One example of categorical data is whether a person is rocking a mullet. The possible values are then "yes" and "no."

A frequency table is a type of table that is used to present categorical data from some kind of measurement. It lists the possible values or outcomes of the category, and how many times each value or outcome has been measured. The results of asking a group of ten people if they prefer cats, dogs, quokkas, or turtles can be presented as follows.

Preference	Frequency
Cats	$4$
Dogs	$4$
Quokkas	$1$
Turtles	$1$

When categorical data belongs to two categories, such as if people are asked whether they own a car and whether they have a driver's license, it can be presented in a two-way frequency table. One of the categories is represented by the rows of the table, and the other by the columns. The above survey, with $100$ participants, could result in the following answers.

		Driver's license
		Yes	No
Car	Yes	$43$	$4$
	No	$24$	$29$

The two categories are then "car" and "driver's license," both with the possible answers "yes" and "no." The entries in the table are called joint frequencies. Often, two-way frequency tables include the total of the rows and columns. These totals are called marginal frequencies. The sum of the "total" row and "total" column are each equal to the sum of all joint frequencies, $100$ in this case.

		Driver's license
		Yes	No	Total
Car	Yes	$43$	$4$	$47$
	No	$24$	$29$	$53$
	Total	$67$	$33$	$100$

From the table, it can, for instance, be read that $43$ out of the $100$ people both own a car and have a driver's license, and that $33$ of the $100$ do not have a driver's license.

A joint relative frequency is the ratio of a joint frequency and the total number of values or observations. Similarly, a marginal relative frequency is the ratio of a marginal frequency and the total. For the example above, the joint and marginal relative frequencies are found by dividing the frequencies by $53,$ the number of participants.

		Hat preference
		Top hat	Beret	Total
Gender	Male	$\frac{6}{53}\approx 0.11$	$\frac{12}{53}\approx 0.23$	$\frac{18}{53}\approx 0.34$
	Female	$\frac{15}{53}\approx 0.28$	$\frac{20}{53}\approx 0.38$	$\frac{35}{53}\approx 0.66$
	Total	$\frac{21}{53}\approx 0.40$	$\frac{32}{53}\approx 0.60$	$1$

Notice that, ignoring the error margin introduced by rounding, a marginal relative frequency can be found by adding a row or column of joint marginal frequencies. This table shows, for instance, that female's with a preference for berets make up about $38\%$ of the participants.

A conditional relative frequency is the ratio of a joint frequency and either of its corresponding two marginal frequencies. Alternatively, it can be calculated using relative joint and marginal frequencies. As an example, the following data will be used.

		Driver's license
		Yes	No	Total
Car	Yes	$43$	$4$	$47$
	No	$24$	$29$	$53$
	Total	$67$	$33$	$100$

Using the column totals, the left column of joint frequencies should be divided by $67,$ and the right column by $33.$ Since the column totals are used, the sum of the conditional relative frequencies will be $1.$

		Driver's license
		Yes	No
Car	Yes	$\frac{43}{67}\approx 0.64$	$\frac{4}{33}\approx 0.12$
	No	$\frac{24}{67}\approx 0.36$	$\frac{29}{33}\approx 0.88$

This table shows that, for instance, out of all the participants with a driver's license, about $64\%$ of them own a car, and out of those without a driver's license, $88\%$ do not have a car.