Marginal distribution

A marginal distribution is the full set of proportions for one variable of a two-way table, each one a row total or a column total divided by the grand total. A table of two categorical variables therefore has two marginal distributions, one per variable, and each adds to 1 on its own. One of those proportions on its own is a marginal probability; the marginal distribution is all of them together.

Take 200 students classified by year and by how they travel to school.

The marginal distribution of class is $80/200 = 0.40$ senior and $120/200 = 0.60$ not senior. The marginal distribution of travel is $75/200 = 0.375$ drives and $125/200 = 0.625$ does not. Every denominator is 200.

"The marginal distribution of travel among seniors is $45/80 = 0.5625$ and $35/80 = 0.4375$." Both numbers are right and they add to 1, which is exactly why the mistake survives unnoticed. The denominator is a row total rather than the grand total, so that is a conditional distribution, travel given senior. If your denominator came off a margin, what you computed was conditional.

Margins cannot tell you whether the two variables are related. The 0.375 who drive would read the same whether seniors drove far more than everyone else or exactly as often. Margins also fail to pin down the table: hold them at 80 and 120 down the side and 75 and 125 across the top, and the senior-and-drives cell can be any whole number from 0 to 75 while reproducing them exactly. Cells always determine margins. Margins run back to the cells only in degenerate cases, such as a row total of 0.

Two-way tables are topic 2.1, and the marginal relative frequencies read off them are topic 2.2, Summary Statistics for Two Categorical Variables.