# What are some typical questions we can ask about the result of two university exams with a third final one yet to come?

I'm trying to understand the performance of students on an on-going university course. I built a histogram of grades of two exams of the same course. Here's the first exam.

It seems to me that there's a relevant portion of the students that are not doing well at all and the rest seems to show something that seems expected (a cluster around the average).

Here's the distribution of the second exam.

It seems that the students improved considerably. What questions can I ask (that I can easily answer from the grades) that would allow me to get some safe conclusions of the performance of the course?

I've computed the averages and the standard deviation in both exams. The first exam has an average of 4.51 and standard deviation of 3.08. The second has an average of 6.33 and standard deviation of 3.20.

One more question, if I may. I'd like to forecast how many students are going to pass. As a simple method (which I need because my skills are null here), I'm thinking about just making the second exam's grades as the third exam grades and see what happens. In other words, suppose the performance on the third is the same as on the second.

I can also suppose the third will be like the first --- and then take the average of the two suppositions as one third forecast. (Thank you!)

• I assume the maximum score is 10 and not 10.5? Assuming it's 10, you may want to change your bin width in the histograms to 1 instead of 1.5.
– mkt
Jul 1, 2023 at 17:22
• Other than the summary statistics, I don't think that there's much we can suggest based on these plots. There's not a lot of information in two histograms. It could be informative to plot them against each other to see if the students who performed poorly in the second also performed poorly in the first (a reasonable assumption). It's also perhaps reasonable to assume that improvement will continue, meaning that the third test ought to be better than the second. And perhaps you could focus more effort on guiding the 20-odd students whose performance is weak.
– mkt
Jul 1, 2023 at 17:33

The data show clearly that the second exam worked out in a very different manner from the first one (maybe because the experience of the first exam has set up many students better for the second one). Although you could think that this should make us expect that the third exam may work out even better, there may be reasons against it, namely (a) the professor may have thought that the first exam was too difficult after seeing results and may have made the second exam easier, but after seeing results of the second exam they may well think that this was too easy, and the third exam may be more difficult again. (b) It all also depends on the rules for passing and actually on the general degree rules. For example some students who have done well up to now may not need that good a mark in the third exam anymore to pass the course, and may therefore not prepare well. It also depends on whether it is only important to pass or whether doing as well as possible is also important for the students in the end (even then some students may know they only need a 5 for securing the final result they need). Then some may care much about their mark, some others only about passing. (c) Material covered in the exams may be different with some easier material covered in the second exam, and about the third we don't know.

So there are lots of considerations that would need to be taken into account when making predictions, and the data alone can hardly say anything. It would be different if there were data from previous years of the same course and/or the same professor, so that one could say something about general tendencies regarding how the third exam relates to the first and the second.

Also, if you were happy making lots of assessments a priori, i.e., before seeing the data, regarding all kinds of things that may happen and how likely they would be, you could do a Bayesian analysis based on a prior distribution chosen by you depending on what you believe, and then update it using the data. Obviously you have seen the data now, so you can't do that anymore, but if you were forced to make probabilistic predictions in this situation, probably going this way trying to set up a "prior" as independently as possible from the actual data (i.e., pretending to yourself that you haven't seen them) would not be the worst way to go. To some extent invalid but maybe better than available alternatives.

Better accept that the given information in the data alone tells you only about those actual exams but doesn't give you much for prediction or generalisation.

If I'd need to predict the number of passing students, the first thing needed are of course the passing conditions. Then I could go through the list of students and see how many marks each single one needs from the third exam for passing. Just predicting from their performance up to that point what for every student is the probability to get the missing marks is probably somewhat easier but still requires a lot of assumptions (for example how the level of difficulty of the third exam will be relative to the first two). But seriously, going through the list of students and just guessing probabilities for each student seems much better to me than using the histograms only without looking at data student-wise.

Next thing I'd do is try to find past data for this course (and maybe courses with similar regulations for passing) that give me some information about the third exam that I currently don't have.