Attributes, Contingency Tables and Independence

Not every characteristic can be measured as a number. Sex (male / female), literacy (literate / illiterate), or whether a customer liked a product (yes / no) are qualitative characteristics, called attributes. Each attribute sorts the units into a few mutually exclusive categories. When two attributes are recorded for the same set of units, the counts are laid out in a two-way table of frequencies called a contingency table — structurally the same as a bivariate frequency table, but with categories of attributes labelling the rows and columns instead of numerical classes.

The central question about two attributes is whether they are associated (related) or independent (unrelated). For example: is preferring a product associated with the customer's sex, or does the split of likes-to-dislikes look the same for men as for women? The tool for answering this begins by asking what the table would look like if the two attributes were perfectly independent.

The logic is proportional: if an attribute's categories are unrelated to the other's, then each row should split across the columns in the same proportion as the overall column totals do. So the proportion $C_j / N$ of the whole sample that falls in column $j$ should, under independence, also be the proportion of row $i$ that falls in column $j$ — giving $E_{ij} = R_i \times (C_j / N)$, which is the formula above. …