Math in Society, 2nd Edition: Tools for decision making

A contingency table summarizes all the possible combinations for two categorical variables. Each value in the table represents the number of times a particular combination of outcomes occurs. For example, suppose we randomly select 250 households from the greater Portland area and ask whether they have a cat and whether they have a dog. In this case, “have a cat” and “have a dog” are the two variables, and each variable has two categories: Yes and No. To create the contingency table, we make columns for the categories of one variable, and rows for the categories of the other variable. We also add a row and column for the subtotals of each category. Each cell of the resulting table contains the number of outcomes having the characteristics of the intersecting row and column categories. For our dog and cat example, the table would look like this:

Suppose that of the 250 households surveyed, 180 said they have a cat, 95 said they have a dog, and 52 said they have both a cat and a dog. We can use this information to fill in the cells of the table.

The first cell we can fill in is the grand total, which is the total number of subjects in the study. In this case, there are 250 households participating in the survey. The next two cells we can fill in are the total number of households that have a cat, 180, and the total number of households that have a dog, 95. The final cell we can fill in from the given information is the intersection of the having a dog column and a having a cat row, which is 52 households.

Since each row and column must sum to their totals, we can use subtraction to find the missing numbers as shown below.

Now that we have our contingency table completed, notice that the numbers in the central four cells add to the grand total as shown in the table on the left. The total row and the total column also add to the grand total as shown in the right table.

If the subtractions we just did seem familiar, they should! This is very similar to what we did for reporting data with a Venn diagram. The Venn diagram for this data is shown below. We also subtracted the intersection from the total of the cat and dog owners to find numbers in the crescent regions.

Notice that the numbers in the four regions of the Venn diagram are the same as the four cells in the center of the contingency table and add to the grand total.

Now we can use the contingency table or the Venn diagram to determine the percentage of households that meet certain conditions. For instance, what percent of those surveyed own a cat and do not own a dog? In the Venn diagram, this is 128 households in the cat only region.

In the contingency table we see the 128 households at the intersection of the row of households who own a cat and the column of households who do not own a dog. As a percentage, the total number of households surveyed, is \(\frac{128}{250}=0.512\) or 51.2% that have a cat and no dog.

How about the percentage of households surveyed that have a cat or a dog? We know from Venn diagrams that the inclusive or includes the number of households who own a cat only, a dog only, and both a cat and a dog, or \(128+52+43=223\) households. As a percentage of the total surveyed, we get \(\frac{223}{250}=0.892\) or 89.2% of households in the sample have a dog or a cat (or both).

We can get the same answer from the contingency table. by adding the cells for households who have a cat and not a dog, a dog and not a cat, and the households that have both a cat and a dog. This also gives us 223 households.

There is another way to calculate an or statements from a contingency table. We could add the row and column totals for having a cat and having a dog, but then we have counted the 52 households in the intersection twice. We can subtract that number to get \(180+95-52=223\) households with a dog or a cat, which we know is 89.2% of those surveyed.

Another question we can answer using a contingency table is what percentage of dog owning households also own a cat? In this case the group that we are interested in isn’t every household surveyed (the grand total), but just those households that own a dog.

We call this a conditional statement because we are only considering the households with a certain condition. If we focus on the column representing the households that own a dog, we see that there is a total of 95 households with a dog, and that 52 of those 95 households also have a cat. Therefore, \(\frac{52}{95} \approx 0.547\) or approximately 54.7% of the households with a dog also have a cat. Another way to phrase this conditional statement is, “What percent of households have a cat given they have a dog.” You will see the word given quite a bit in this chapter and that makes the denominator change. It is also possible to find this conditional percentage using the Venn diagram by taking the number in the intersection and dividing it by the total in the whole dog circle.

When there are only two categories for each variable, like yes/no questions, Venn diagrams and contingency tables provide basically the same information and can be used interchangeably. A Venn diagram works well for yes/no variables since a subject is either inside the circle (has the characteristic) or outside the circle (does not have the characteristic). If we have more than two possibilities for any of the variables, though, we cannot use a Venn diagram. We can use a contingency table, though. Here is an example where one variable has four categories and the other has three categories.

If our sample is representative of the population, then we can also interpret a percentage we calculate from a contingency table as a probability, or the likelihood that something will happen. Since a contingency table is constructed from data collected through sampling or an experiment, we call it an empirical or experimental probability. This is different from a theoretical probability which we will look at in the next section.

Suppose that 60% of students in our class have a summer birthday (June, July, or August). Now suppose everyone’s name and birth month are written on slips of paper and thrown into a bag. If we pull a slip of paper out of the bag at random, what is the probability that the selected student has a summer birthday? If you think there should be a 60% chance, you are right! The relative frequency of the characteristic of interest will be equal to its empirical probability. To write this as a probability statement, it would look like

Probability is a function named P, and the function is applied to what follows in the parentheses. Let’s look at another example where we write probability statements and find empirical probabilities.

We have mentioned conditional probabilities, which we find by isolating our attention to the given row or column. Here is another example of finding conditional probabilities.

In this section we have learned about empirical probability. In the next section we will discuss another kind of probability that you may be familiar with – theoretical probability.