Summary Statistics for Two Categorical Variables

One table of counts yields three tables of proportions: joint (each cell over the grand total, all cells summing to 1), row conditional (each cell over its row total, each row summing to 1), and column conditional (each cell over its column total). For 500 employees with 200 trained and 300 untrained, the cell 150 gives $\frac{150}{500} = 0.30$ jointly, $\frac{150}{200} = 0.75$ conditional on training, and $\frac{150}{300} = 0.50$ conditional on passing. Association is then summarized either as a difference in conditional proportions ($0.75 - 0.50 = 0.25$, i.e. 25 percentage points) or as a ratio ($\frac{0.75}{0.50} = 1.5$ times the rate), and both are reported with the groups and the context named.

The two failures are the denominator and the comparison. Reporting the share of passers who were trained when asked for the share of trained employees who passed reverses the condition and answers a different question. And comparing raw counts, or comparing one group's conditional proportion against the overall marginal proportion, produces a difference that measures group sizes rather than rates. When the underlying proportions are small, a ratio of 3 and a difference of 0.4 percentage points describe the same two numbers, so the summary chosen is itself a claim about what matters.