Conditional distributions of X given Y and Y given X

The conditional proportions for X given Y can be quite different from the corresponding conditional proportions for Y given X.

You must be careful to distinguish between these.

Rank and age

The clustered bar chart below is identical to that on the previous page.

Select Propn within Age from the pop-up menu with bars still clustered by Age. This shows a conventional bar chart of the ranks separately for each age group.

Now select Rank from the menu to cluster the same bars by rank. This is a valid display but takes a little more thought to understand than the previous displays in which each cluster of bars was a separate bar chart. In this display, the bar chart giving the conditional distribution of ages for assistant professors is split between all of the clusters of bars.

This diagram clearly shows how the proportion of full professors increases steadily with age, and the proportion of assistant professors decreases steadily with age.

With the bars still clustered by Rank, consider the difference between the bar charts that are found with the options Propn within Age and Propn with Rank. For example, notice that:

  • 84% of those aged under 30 were assistant professors
  • 18% of assistant professors were aged under 30

A more extreme example of the difference between the conditional proportions for X given Y and of Y given X, is that under 5% of women are pregnant at any time, but 100% of pregnant people are women!