The contingency table, cross-tabulation, or two-way frequency table, statistically summarizes a relationship between two categorical variables.

Here, the outcome of one variable is arranged in rows, while that of the other variable is arranged in columns.

Consider two variables: COVID-19 cases and vaccination status. The contingency table for this data will contain two rows and three columns. The row variable identifies whether individuals are infected or not. The column variable identifies the individual's vaccination status.

Of the 236 total observations, 150 individuals are infected, and 86 are not. Furthermore, 129 individuals are unvaccinated, 27 are partially vaccinated, and 80 are completely vaccinated.

The contingency table displays the observed frequencies of two variables. After performing a test of independence, the chi-square statistic is used to determine whether these values mirror the expected frequency.

Contingency tables help to display and find the probability of any event occurring dependent or independent of other variables, which are visualized using histograms or bar graphs. Thus, they help systematically organize, analyze and quantify data.