Meaning of Marginal Distributions in a Contingency Table

A contingency table summarizes the conditional frequencies of two attributes and shows how these two attributes are dependent on each other with the information on a partition of universe generated by these attributes. Mainly, in statistical analysis, this table is used to check statistical independence of two attributes. For the analysis, marginal sum, called marginal distribution plays an important role. This paper focuses on marginal distributions from the viewpoint of linear algebra and discusses the decomposition of a contingency matrix into expected matrix and residual matrix. When the rank of a matrix is full, say, r, the difference between a original matrix and the expected matrix will become r -1 at most. The sum of rows or columns will become zero, which means that the information of one rank correponds to information on the frequency of a contingency matrix. Furthermore, the elements of residual matrix can be represented by linear combination of 2 times 2 subdeterminants, whose total number is equal to the degree of freedom of a contingency table.