# Contingency table

What is a Contingency table? Contingency tables are used in descriptive statistics to get an overview of two, mostly categorical variables. In the crosstab you can read how often the combination of the values of two characteristics occurs.

The frequency is given either in absolute or relative frequency. Therefore, with a cross-tabulation in statistics, one gets an insight into how two variables are related.

## Crosstabs in statistics

If two categorical variables are present, a crosstab is obtained by entering the values of the variables in the table. For the first variable, the values are plotted from left to right, for the second variable from top to bottom. The individual cells are then filled with either the absolute or the relative frequency.

### Crosstabs and market research

Crosstabs are very often used in market research because they can be used to compare customers or products very well. For example, one of the following questions can be answered:

• Which insurance is preferred by which age group?
• Are the car brands different in the city and in the country?
• Which apple variety sells best in which season?

## Interpret crosstabs

How do you interpret a crosstab? A crosstab shows the frequencies of two variables.

In each cell of a crosstab, the frequencies of the characteristic combinations are plotted; in the example above, female and without a degree occur exactly 6 times.

### Rows and columns for the crosstab

In this way, the values of one variable are plotted in the rows and the values of the other variable are plotted in the columns. Usually, the independent variable is plotted in the columns and the dependent variable in the rows.

## Absolute and relative frequencies for crosstabs

When creating a crosstab, either the absolute or the relative frequencies can be output:

### Absolute frequencies

Absolute frequencies are those values that indicate how often the respective combination of two characteristic values occurs.

### Relative frequencies

Relative frequencies, on the other hand, indicate how often the respective combination of expressions occurs in relation to all cases and is therefore usually expressed as a percentage.

## Example contingency table

The creation of crosstabs will now be examined in more detail using an example. In the example it is assumed that on a rainy day a student counts how many people "with" and how many "without" umbrellas come to the statistics lecture. In addition, the student makes a note of the sex of the students.

That's how it works with DATAtab: Open the Online Statistics Software, copy the table below to the Calculator and select the variable Gender and with umbrella in the Descriptive Statistics area. DATAtab will automatically create a Contingency table for you.

Gender With umbrella
female yes
male yes
female yes
female yes
male yes
male no
female no
male no
female no
female no
male no
female yes
male yes
female yes
male yes
male yes
male no
female no
male no
female no
female no
female no

The result can now easily be displayed in a contingency tables. The cross-classified table now contains the absolute frequencies of the respective characteristic combinations. This is calculated as follows:

With umbrella
yes no Total
Gender female 5 7 12
male 5 5 10
Total 10 12 22

## Testing a crosstab for significance

A crosstab can be used to examine whether there is a relationship between the two variables. However, since a crosstab is a descriptive statistic, a statement can only be made about the sample. If a statement is to be made about the population, the chi-square test is required.

Cite DATAtab: DATAtab Team (2023). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net