An Urn Model

The example analysed in the preceeding subsections has been based on continuous distributions to avoid problems with the definition of the median of discontinuous distributions.¹⁸ But after the median paradox has been established by an example without difficulties of this kind, the reader is invited to rediscover the principles explicated in Subsection 2.2 in the following urn model.

Suppose that there are two urns i and j containing red and green lottery tickets revealing a number after unfolding (the colours representing experimental conditions, and the process of drawing a ticket, unfolding it and reading the number standing for the performance of an experiment resulting in a value of the dependent variable). Assume furthermore that the numbers on the tickets are given by the following scheme:

	Red tickets	Green tickets
Urn i:	11 12 13 23 30	11 13 15 17 19
Urn j:	21 23 25 27 29	10 17 27 28 29

In both urns, the median of the numbers on the red tickets is smaller than on the green tickets (13 resp. 15 for urn i, and 25 resp. 27 for urn j). But if all twenty tickets are collected in one urn, then the median of the numbers on the red tickets is greater (23, vs. 17 for the green tickets). Of course, the numbers could also been interpreted as measurement values in four groups, the subgroups (defined by the values i and j of a dichotomous variable) being pooled to obtain the medians for all tickets with identical colour.

For a probability interpretation, consider the experiment of selecting an urn and a colour by two tosses of a coin and drawing a ticket of the selected colour from the selected urn. Then f_u(c,x)(with $u \in \{i,j\}$, $c \in \{red, green\}$, and $x \in R$) would be the conditional probability of drawing a ticket with a number up to x, conditioned upon the selection of urn uand colour c. Similarly, $y_\pi(c,x)$would be the conditional probability of getting a number up to x, given colour c.¹⁹ Obviously, $y_\pi(c,x)$is identical with the conditional probability of drawing a ticket with a number up to x from the pooled urn, conditioned upon colour c. If we accept for these probability distributions a definition of the median making it identical with the medians of the numbers on the respective 5 or 10 tickets, then we again have a median paradox in the sense of Subsections 2.1 and 2.2.