Marginal Distributions from Table Totals

What a Marginal Distribution Shows

A two-way table organizes counts for two categorical variables at once. It can also answer a question about just one of those variables: what share of all individuals fall into each of its categories? A marginal distribution answers this by setting aside the other variable and summarizing the categories along one edge of the table.

The word “marginal” refers to the totals shown in the table’s margins. Row totals summarize the counts in each row; column totals summarize the counts in each column. To turn either set of totals into a distribution, divide every category total by the grand total. The resulting proportions describe the full group represented by the table, not just part of it.

Definition: In a two-way table with grand total \(n\), the row marginal distribution gives the proportion of all \(n\) individuals in each row category, and the column marginal distribution gives the proportion of all \(n\) individuals in each column category. Each marginal proportion is its category total divided by \(n\).

Calculating Marginal Proportions

Suppose the table has row categories indexed by \(i\) and column categories indexed by \(j\). Let \(n_{ij}\) be the count in the cell at row \(i\), column \(j\). The row total for row \(i\) is \(n_{i+}\), and the column total for column \(j\) is \(n_{+j}\). The plus sign indicates that counts across the other variable have been added together.

n_{i+}=\sum_j n_{ij} \qquad n_{+j}=\sum_i n_{ij} \qquad n=\sum_i\sum_j n_{ij}

Once the totals are known, divide each by the grand total. The hat notation indicates a proportion calculated from the observed data.

\hat{p}_{i+}=\frac{n_{i+}}{n} \qquad \hat{p}_{+j}=\frac{n_{+j}}{n}

The row marginal distribution consists of all the \(\hat{p}_{i+}\) values; the column marginal distribution consists of all the \(\hat{p}_{+j}\) values. Each distribution describes one variable on its own. For example, a row marginal proportion of \(0.35\) means that 35% of all individuals in the table are in that row category, regardless of which column category they are in.

Worked Example: Marginal Distributions of Commute Mode and Residence

In a hypothetical survey, 120 students report their usual commute mode and whether they live on or off campus. The counts are shown below.

Commute mode	On campus	Off campus	Row total
Walk	25	15	40
Bus	10	30	40
Car	5	35	40
Column total	40	80	120

First, find the row marginal distribution for commute mode. Each row total is divided by the grand total, 120:

\hat{p}_{\text{walk}}=\frac{40}{120}=0.3333,\qquad \hat{p}_{\text{bus}}=\frac{40}{120}=0.3333,\qquad \hat{p}_{\text{car}}=\frac{40}{120}=0.3333

Thus, in this hypothetical group, one-third of all surveyed students usually walk, one-third usually take the bus, and one-third usually travel by car. These are proportions of all 120 students.

Next, find the column marginal distribution for residence. The column totals are 40 and 80:

\hat{p}_{\text{on campus}}=\frac{40}{120}=0.3333,\qquad \hat{p}_{\text{off campus}}=\frac{80}{120}=0.6667

In this group, 33.33% of all students live on campus and 66.67% live off campus. The row and column distributions summarize different variables, but both use the same grand total as their denominator.

Check the row proportions: \(0.3333+0.3333+0.3333\) is approximately \(1\), with the small difference due to rounding. Check the column proportions: \(0.3333+0.6667=1.0000\). The corresponding counts also check: the row totals add to \(40+40+40=120\), and the column totals add to \(40+80=120\).

Why Each Marginal Distribution Sums to One

The proportions in a marginal distribution account for every individual in the table exactly once. Every individual belongs to exactly one row category, so adding all row totals gives the grand total. Likewise, every individual belongs to exactly one column category, so adding all column totals also gives the grand total.

\sum_i \hat{p}_{i+} = \sum_i \frac{n_{i+}}{n} = \frac{\sum_i n_{i+}}{n} = \frac{n}{n} = 1

The same reasoning applies to the column marginal distribution:

\sum_j \hat{p}_{+j} = \sum_j \frac{n_{+j}}{n} = \frac{\sum_j n_{+j}}{n} = \frac{n}{n} = 1

This is a useful check, not merely a formatting convention. If a marginal distribution does not sum to one, possible explanations include a missing category, an incorrect total, or arithmetic or rounding errors. With proportions rounded for display, the sum may be slightly different from one; the unrounded proportions should sum exactly to one.

Worked Example: Building Marginal Distributions from Table Totals

A hypothetical survey of 150 students records whether each prefers morning or afternoon classes and whether the student is in a first-year, second-year, or later-year group. The row totals are 72, 48, and 30. The column totals are 90 for morning preference and 60 for afternoon preference. Find both marginal distributions.

The row totals describe the three year groups. Their sum is \(72+48+30=150\), which agrees with the stated grand total. Divide each by 150:

\frac{72}{150}=0.4800,\qquad \frac{48}{150}=0.3200,\qquad \frac{30}{150}=0.2000

The row marginal distribution is therefore 0.4800 first-year, 0.3200 second-year, and 0.2000 later-year. In context, 48% of all 150 students are first-year students, 32% are second-year students, and 20% are in later years.

The column totals describe class-time preference. They also sum to the grand total: \(90+60=150\). Divide each by 150:

\frac{90}{150}=0.6000,\qquad \frac{60}{150}=0.4000

The column marginal distribution is 0.6000 morning preference and 0.4000 afternoon preference. Thus, 60% of all surveyed students prefer morning classes and 40% prefer afternoon classes. The checks are \(0.4800+0.3200+0.2000=1.0000\) for the row distribution and \(0.6000+0.4000=1.0000\) for the column distribution.

Marginal Distributions Versus the Full Table

A marginal distribution deliberately summarizes one variable at a time. It preserves the overall proportion in each category, but it does not show how the categories of the two variables occur together. In the commute example, the marginal proportion who travel by bus is \(40/120\). That total combines the bus counts in both residence categories.

This summary is appropriate when the question concerns the overall distribution of one variable. If the question concerns how a category of one variable is distributed across categories of the other, the marginal distribution alone is not enough; information from the cells is needed. Keep the distinction clear: a marginal proportion uses a category total over the grand total, while a cell count is the number observed in one particular pair of categories.

Worked Example: Finding and Checking Marginal Proportions

In a hypothetical group of 100 library users, a table records whether a user borrowed a printed book or an e-book and whether the user visited on a weekday or weekend.

Format borrowed	Weekday	Weekend	Row total
Printed book	18	12	30
E-book	22	8	30
No book	10	30	40
Column total	50	50	100

For the row marginal distribution, divide each format total by 100:

\hat{p}_{\text{printed}}=\frac{30}{100}=0.3000,\qquad \hat{p}_{\text{e-book}}=\frac{30}{100}=0.3000,\qquad \hat{p}_{\text{no book}}=\frac{40}{100}=0.4000

Among all 100 users, 30% borrowed a printed book, 30% borrowed an e-book, and 40% borrowed no book. For the column marginal distribution, divide each visit-day total by 100:

\hat{p}_{\text{weekday}}=\frac{50}{100}=0.5000,\qquad \hat{p}_{\text{weekend}}=\frac{50}{100}=0.5000

Half of all users in this hypothetical group visited on a weekday and half on a weekend. The row proportions sum to \(0.3000+0.3000+0.4000=1.0000\), and the column proportions sum to \(0.5000+0.5000=1.0000\). The calculations use the table’s margins and grand total; they do not tell us how borrowing format and visit day are related.

Common Mistakes

Using the wrong denominator. Dividing a row total by that row total, or by a column total, does not give a marginal proportion for the full group. For a marginal distribution, divide every category total by the grand total.
Mixing up row and column totals. Row totals summarize the row variable; column totals summarize the column variable. Read the table headings before naming the distribution.
Reporting counts as proportions. “40 students” is a count. “40 out of 120, or 0.3333” is a proportion and can be interpreted as 33.33% of the full group.
Ignoring the sum check. If the proportions are far from summing to one, recheck the totals, denominators, and category coverage before interpreting them.
Interpreting a marginal proportion as a relationship. A marginal distribution describes one variable overall. By itself, it does not show how that variable’s categories vary across the other variable.

Key takeaway: To find a marginal distribution, divide each row total or each column total by the grand total. Interpret each result as a proportion of the full group, and check that each complete set of marginal proportions sums to one.

Check Your Understanding

Use the definitions and checks in this tutorial to answer the following questions.

A two-way table has a grand total of 80 and a row total of 28 for one category. What is that category’s row marginal proportion, and how would you interpret it?
Why do the row totals add to the grand total? Give the corresponding reason for the column totals.
A table has column totals 45 and 55 out of a grand total of 100. State the column marginal distribution and verify its sum.
A learner divides each row total by the total for the first column to calculate a row marginal distribution. Explain the error and give the correct denominator.
What information about the relationship between two variables is not conveyed by their marginal distributions alone?