Contingency Table

A contingency table (or cross-tabulation) is an effective way to show the joined distribution of two variables, that is to say, the distribution of one variable within the different categories of another. Data in the table are organized in rows and columns. Each row corresponds to one category of the first variable (usually considered as the dependent variable), while each column represents one category of the second variable (usually considered as an independent variable). The intersection of a row and a column is called a “cell.” Each cell contains the cases that have a certain combination of attributes corresponding to that row and column (see Table 1). Inside each cell a variety of information can be displayed, including (a) the total count of cases in that cell, (b) the row percentage represented by the cell, (c) the column percentage represented by the cell, and (d) the proportion of the total sample of cases represented by that cell.

Generally, a contingency table also contains the sums of the values of each row and column. These sums are called the “marginals” of the table. The sum of column or row marginals corresponds to the sample size or grand total (in the lower right-hand cell of the table).

The product of the number of the rows by the number of the columns is called the “order” of the table (Table 1, for example, is a 2 × 2 table), while the number of the variables shown in the table represents its dimension.

A bivariate contingency table represents the first device the researcher can use in the exploration of the relationship between two variables (including ones that are nominal or ordinal). In order to establish whether the variables are associated or not, however, the researcher has to abandon the raw frequencies in favor of the percentages, because only these allow a proper comparison. One can calculate three types of percentages: (1) row, (2) column, and (3) total percentages. However, not all these percentages are generally reported in the contingency table, as that would be more information than needed in most instances; although they are shown in each cell in Table 1 below the cell count. Which percentages the researcher takes into account depends on the specific research question. However, if the researcher aims at exploring the influence of the variable shown in the columns (considered as independent) on the variable shown in the rows (considered as dependent), she or he should report the column percentages. Therefore, keeping fixed the first category of the dependent variable (in the rows), the researcher will analyze how the values change along the categories of the independent variable (in the columns). If one considers the column percentages in the Table 1 (i.e. the 2nd percentage below the count in each cell) for example, keeping fixed the category “low educated,” one can see that females in this sample are significantly more likely to be “less educated” than are males. Of note, if the percentages in a cell are based on too small a number of cases, the results will not be reliable.

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