Proportional Venn diagrams

A proportional Venn diagram is drawn from the marginal probabilities of one variable and the conditional probabilities for the other variable,

Rewriting the definition of conditional probabilities,

The area of any rectangle in the diagram therefore equals the joint probability of the categories it represents.

An alternative proportional Venn diagram can be drawn from the marginal probabilities of Y and the conditional probabilites of X given Y. The area for the rectangle corresponding to any (x, y) is its joint probability, pxy.

Example

The table below is based on the world population in 2002, categorised by region and by age group. It shows the joint probabilities for a randomly chosen person being in each age/region category.

Joint probabilities
  Age
  0-19 20-64 65+
Africa and Near East 0.085 0.073 0.006
Asia 0.215 0.315 0.035
America, Europe and Oceanea 0.084 0.158 0.030

The two proportional Venn diagrams are shown below.

Note that the areas are the same in both diagrams — they are simply rearranged.