How Marginal Totals Determine Expected Counts in a Contingency Table

Expected counts describe what a table would look like under a model of independence while retaining its row and column totals. They are model-based reference counts, not new observations.

Construct a table from its margins

Suppose a two-row, three-column table contains 100 observations. Row totals are 40 and 60; column totals are 30, 50, and 20. Under independence, each row has the same expected column proportions: 30%, 50%, and 20%.

For example, the first cell is $40(30)/100=12$. The expected counts reproduce every margin. Dividing 100 equally among six cells would ignore the differing category frequencies.

Compare observed and expected counts

Given actual observed counts $O_{ij}$, Pearson's statistic is

$$\chi^2=\sum_{i,j}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}.$$

For a standard independence test in an $r\times c$ table, the reference degrees of freedom are $(r-1)(c-1)$. This example has two degrees of freedom. The margins alone do not supply the observed cell counts, so they do not determine a test statistic or p-value.

Check the sampling conditions

Use counts of observations in mutually exclusive categories and ensure the observations are independent. A common introductory adequacy rule requires every expected count to be at least five; all six expected counts here satisfy it. Sparse tables may require an exact or simulation-based method rather than an unexamined chi-square approximation.