Examining the variables separately

Although our main interest is usually on the relationship between two categorical variables, it can also be of interest to examine the overall distribution of each variable separately. These are called the marginal distributions of the two variables.

The marginal distributions are determined by the row and column totals of a contingency table.

Rank and age in a university

    Rank
Age Full
professor
Associate
professor
Assistant
professor
Instructor Total
  Under 30 002 003 057 06 68
30 to 39 052 170 163 17 402
40 to 49 156 125 061 06 348
50 and over 220 083 039 04 346
Total 430 381 320 33

The yellow highlighted values are the overall frequencies for each age category in the university — i.e. the marginal distribution of age. For example, there were (52+170+163+17) = 402 staff members who were aged 30 to 39.

Similarly, the green highlighted values give the marginal distribution of the ranks of the university staff. The diagram below illustrates the two marginal distributions graphically.

Click the checkbox Stacked to stack the four bars for each age group. The height of each combined bar is the sum of the heights (and therefore the sum of the frequencies) for the four ranks at that age, and therefore describes the marginal distribution of ages.

Uncheck Stacked, select Rank from the pop-up menu, then select Stacked again. This stacks the bars for each rank and therefore shows the marginal distribution of ranks.

In a similar way, the marginal proportions for the variables are obtained by adding the joint proportions across rows and down columns.

This can be expressed more generally as follows. If the joint proportion with row-category x and column-category y is denoted by pxy, then the overall proportion with row-category x is given by

and in a similar way, the marginal proportions for column-category y are

Rank and age in a university

    Rank
Age Full
professor
Associate
professor
Assistant
professor
Instructor Total
  Under 30 2/1164 3/1164 57/1164 6/1164 68/1164
30 to 39 52/1164 170/1164 163/1164 17/1164 402/1164
40 to 49 156/1164 125/1164 61/1164 6/1164 348/1164
50 and over 220/1164 83/1164 39/1164 4/1164 346/1164
Total 430/1164 381/1164 320/1164 33/1164

The highlighted values are the overall proportions for each age (yellow) and rank (green) category in the university — i.e. the marginal distributions of these two variables.