No, the resit exam is not more difficult

It’s January. Exam time. And for the unfortunate ones, next month is resit-exam time. Each year I get the same complaints after the resit exam, so I’ve decided to write a blog post about it. From now on, complaining students will receive no more than the hyperlink to this blog.

Each year, I end my course with an exam. For those that have to miss the exam due to force majeure, a resit opportunity exists. Fortunately, it’s a relatively small number of students that have to miss the exam due to illness, bereavement, etc. The resit opportunity is also open to students that did participate in the first exam, but failed.

Most of the time, the pass rate of the resit exam is considerably lower than the pass rate on the first exam. Students – and especially those that failed the resit exam – see this as proof of some evil plan of mine: I made the resit much more difficult than the first opportunity. Why else would the pass rate be lower? Statistics don’t lie!

True, they don’t. But people – including students that fail a statistics course two times in a row – are prone to misinterpreting statistics. And they are misinterpreting the numbers here.

The grade a student receives at an exam depends on three aspects: (1) his/her proficiency (usually a combination of motivation and intelligence); (2) the difficulty level of the exam; (3) coincidence (being (un)lucky with guessing multiple choice questions, ‘not having a good day’, etc.).

Let’s make a thought experiment. Suppose that all students would participate in both the first opportunity as the resit exam. That way, you end up with two grades for everyone. Let’s keep things easy and only look at whether a student passes or fails the exam opportunity.

Let’s make up some numbers now (homework exercise: re-do this example with different numbers and observe that the conclusion still holds true). Suppose we’ve got 100 students  (because 100 is easy to compute with). Here they are:

Opportunity 2
Pass Fail
Opportunity 1 Pass 48 10
Fail 12 30

As you can see, the pass rates of Opportunity 1 (48% + 10% = 58%) and Opportunity 2 (48% + 12% = 60%) are comparable; the performance on the resit is actually even slightly better.  Nearly half the students are able to pass twice, nearly a quarter of students passes once, and 30 students won’t pass this year.

This, however, is not data that you would observe in reality. The students that pass the first opportunity need not take the second one. (Even more so: if you pass the first and earn the course credits, you will lose these again by failing in the resit). Thus, we only observe the resit-results for those that failed the first opportunity:

Opportunity 2
Pass Fail
Opportunity 1 Fail 12 30

Now the pass rate suddenly is 12/42 = 28.6%, much lower than the 58% pass rate of the first opportunity. Thus, the fact that the resit exam has a much lower pass rate than the first opportunity exam does not imply that the resit exam is more difficult. If you still believe it does, you don’t receive a passing grade on your statistics exam.

Thanks to Nynke for proofreading.