Mathematics and Statistics · Ch 13 — Bivariate Frequency Distribution and Chi-Square Statistic

Marginal and Conditional Frequency Distributions

A two-way table actually carries three kinds of distribution inside it. The cell frequencies describe the two variables jointly; but by looking only at the totals, or only at a single row or column, two further, one-variable distributions can be read straight off the same table.

Note

Marginal Frequency Distribution

The marginal distribution of $x$ is the set of row totals — the frequency distribution of $x$ on its own, obtained by adding across each row (i.e. ignoring $y$ entirely). The marginal distribution of $y$ is the set of column totals — the distribution of $y$ on its own. They are called marginal because they are written in the margins (the total row and total column) of the table.

Because a marginal distribution is just an ordinary single-variable frequency distribution, every earlier tool applies to it unchanged. In particular the mean of $x$ from its marginal distribution is

\bar{x} = \dfrac{\sum f_x \cdot x}{N},

where $f_x$ is each $x$ -value's row total, and $\bar{y}$ is found the same way from the column totals.

Note

Conditional Frequency Distribution

A conditional distribution fixes one variable and looks at how the other is distributed within that fixed slice. The conditional distribution of $y$ given a particular value of $x$ is simply the frequencies in that one row; the conditional distribution of $x$ given a particular value of $y$ is the frequencies in that one column. Dividing each such frequency by that row's (or column's) total converts it to relative (proportional) form. …

Definition 1Marginal Distribution

The single-variable frequency distribution of one variable obtained from a two-way table's totals: the row totals give the marginal distribution of $x$ , the column totals give t …

Definition 2Conditional Distribution

The distribution of one variable within a fixed value of the other: the frequencies in a single row give the distribution of $y$ for that fixed $x$ ; the frequencies in a single column give the dist …

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