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A contingency table, also called a cross-tabulation or two-way frequency table, summarizes the relationship between two categorical variables. One variable's outcomes are arranged in rows, while the other variable's outcomes are arranged in columns. This structure displays observed frequencies and helps systematically organize and analyze data from two variables.

In a COVID-19 and vaccination contingency table, rows indicate infection status (infected or not infected), while columns show vaccination status (unvaccinated, partially vaccinated, or completely vaccinated). The table displays how many individuals fall into each combination. For example, of 236 total observations, 150 were infected and 86 were not infected across all vaccination categories.

After performing a test of independence, the chi-square statistic determines whether observed frequencies in a contingency table match expected frequencies. This comparison helps establish whether two categorical variables are independent or dependent on each other, revealing patterns in the data.

Contingency tables display observed frequencies of two variables, enabling quick calculation of conditional probabilities. By organizing data into rows and columns, they show how one variable may depend on another, allowing students to determine the probability of events occurring dependent or independent of other variables.

Observed frequencies are the actual counts displayed in a contingency table showing how many cases fall into each category combination. Expected frequencies are theoretical values calculated under the assumption of independence. A test of independence compares these two to determine whether variables are truly dependent or independent.

Contingency tables help display and find probabilities of events, which are often visualized using histograms or bar graphs. These visual representations make it easier to interpret relationships between categorical variables and communicate findings about data patterns and dependencies to audiences.

Contingency tables systematically organize, analyze, and quantify data by displaying how two categorical variables may be dependent or contingent upon each other. They provide a structured method for portraying frequency distributions that facilitates calculating probabilities and understanding relationships between variables in research.