Contingency Table

A contingency table, sometimes called a two-way frequency table, is a tabular mechanism with at least two rows and two columns used in statistics to present categorical data in terms of frequency counts. More precisely, an contingency table shows the observed frequency of two variables, the observed frequencies of which are arranged into rows and columns. The intersection of a row and a column of a contingency table is called a cell.

For example, the above contingency table has two rows and five columns (not counting header rows/columns) and shows the results of a random sample of adults classified by two variables, namely gender and favorite way to eat ice cream (Larson and Farber 2014). One benefit of having data presented in a contingency table is that it allows one to more easily perform basic probability calculations, a feat made easier still by augmenting a summary row and column to the table.

The above table is an extended version of the first table obtained by adding a summary row and column. These summaries allow easier computation of several different probability-related quantities. For example, the estimated probability that a person in the sampled population prefers ice cream in a cup is , while the estimated probability that a random participant is female is . Computing conditional probabilities is also made easier using contingency tables. For example, the probability that a person prefers ice cream sandwiches given that the person is male is , while the conditional probability that a person is male given that ice cream sandwiches are preferred is .

Other common statistical analyses can be performed on data given in contingency table form. Let be the total of row , the total of column , and the sample size. The expected frequency of the cell at their intersection under independence is

(1)

Computing says that the value one would expect at cell --i.e., the expected number of men who prefer to eat ice cream from a cup--is approximately

(2)

so the observed count 592 is greater than the expected count under independence. Knowing also gives without repeated application of (1):

(3)

One of the major benefits of computing expected frequencies is the ability to test whether the two variables being examined--in this case, gender and favorite way to eat ice cream--are independent. This is done by computing, for each cell , the expected frequency , comparing it to the observed frequency , and then performing a chi-squared test. For an table, the Pearson statistic is

(4)

Under the null hypothesis of independence and the usual large-sample conditions, has approximately a chi-squared distribution with degrees of freedom. Its upper-tail probability is the P-value for the chi-squared test. Individual cell frequencies can also be divided by the sample size to estimate cell probabilities, with uncertainty summarized by an appropriate confidence interval under a binomial distribution or multinomial distribution model.

Another common test associated with contingency tables is the homogeneity of proportions test, a form of chi-squared test used to determine whether several proportions are equal when samples are taken from different populations (Larson and Farber 2014). The usual large-sample rule of thumb for these chi-squared approximations calls for randomly selected observations and expected frequencies of at least 5 in each cell. These tests play important roles throughout various branches of statistics.