Tired and underprepared

It's common for university exams nowadays to offer a lot of choice about which questions to answer. I'm not sure what the aim of this is - I can't construe it in any other way than throwing a lifeline to the underprepared student - but as I have on more than one occasion been one of those underprepared students, I'm not complaining.

It does, however, introduce an interesting statistic which I like to think summarises the size of the class relative to the complexity of the course: the number of students who answered each possible set of exam questions. Let this be called x_i, where i denotes the particular set of exam questions chosen. For example, in a standard first-year accounting course of 300 students, you might offer them a choice of any four questions from a set of five. There are five possible exams the students could choose (i=1, 2, 3, 4, 5), and if they choose questions randomly, you'd expect ³⁰⁰/₅=60 to sit each exam; that is, E(x)=60. This means you could assign one marker to each of the five possible exams, each would mark 60 scripts, and it would all be over relatively quickly. (I know they don't really mark exams that way, but that's not the point.)

50 or 60 is a reasonable value for E(x) for lower-level courses. In an upper-level undergrad course, with maybe 20 students and the same five possible exams, E(x) gets a lot lower; it's likely to be less than 10. Given that the real marking situation will probably a different lecturer marking each question, we now have such a small sample that it's possible some questions will have been chosen a lot more often than others. That means the marking load is a lot less spread, and, probably, marking becomes a more specialised task - as you'd expect for an advanced course.

But what happens at Honours level? Then things start getting seriously crazy. Here are the values of E(x) for each of the exams I just sat.

Course	Students	Choice of	From	E(x)
Growth	11	2	9	0.31
Macro	14	3	4, one of which has a subchoice	2
Micro	5	2 and 2	3 and 3	0.56
Metrics	10	4	4	10

The figure for Metrics is high because there was no choice in that exam. In all the others, the figure is so low, there's a real chance that some questions weren't answered by anyone at all - in two of the exams, there were more possible exams than students. Is that a win or a lose for the lecturer? Students with a great deal of specialised knowledge are probably a good thing, but what about the effort that went into composing the unanswered questions?