Marginal Distribution — Definition, Formula & Examples
Marginal distribution is the distribution of a single variable by itself, found by looking at the row totals or column totals in a two-way table. It tells you the overall breakdown of one variable, ignoring the other variable entirely.
Given a joint frequency table of two categorical variables, the marginal distribution of one variable is the set of frequencies (or relative frequencies) obtained by summing across all categories of the other variable. These sums appear in the margins of the table.
Key Formula
Where:
- = A specific category of variable A
- = Grand total of all observations
How It Works
To find a marginal distribution, add across a row or down a column of a two-way table. The resulting totals, divided by the grand total, give marginal relative frequencies. For example, if your two-way table crosses gender with sport preference, the column totals give the marginal distribution of sport preference, while the row totals give the marginal distribution of gender. Marginal distributions are typically expressed as percentages or proportions of the grand total.
Worked Example
Problem: A survey of 200 students records whether they prefer dogs or cats and whether they are in grade 10 or grade 11. The two-way table shows: Grade 10 prefers dogs = 60, cats = 30; Grade 11 prefers dogs = 70, cats = 40. Find the marginal distribution of pet preference.
Sum each pet column: Add down the Dogs column and the Cats column to get the marginal totals.
Convert to relative frequencies: Divide each total by the grand total of 200.
Answer: The marginal distribution of pet preference is 65% dogs and 35% cats.
Visualization
Why It Matters
On the AP Statistics exam, you are regularly asked to compare marginal and conditional distributions when analyzing two-way tables. Understanding marginal distribution is also essential in the real world when researchers need to report overall rates — such as the total percentage of patients who recovered — before breaking results down by subgroup.
Common Mistakes
Mistake: Confusing marginal distribution with conditional distribution by dividing by a row or column total instead of the grand total.
Correction: Marginal distribution always uses the grand total as the denominator. Conditional distribution uses a specific row or column total as the denominator because it restricts attention to one subgroup.