Let’s say a professor has seen that the average score for the midterm exam was a grade and a half higher than it was in previous years. Because of this he gained a reasonable suspicion of cheating (he has been teaching the course for several years in the same way and this is the first time he has...
@BenCrowell To be honest I am not quite sure, I think the professor submits the students names to the university academic affairs and he/she might get expelled because it is against the University's honesty policy. I am pretty sure they have to sign a statement when he/she get enrolled that all test are taken with academic integrity(no cheating).
There is a possibility that the class has a higher number of smart students in it. I've seen one math department have one incredibly good calculus 1 class out of 8 total one semester. Much better final exam scores than the rest (nearly half got A's, when 20% overall got A's). That class had a much higher proportion of (good) engineering students (these usually have better math skills than the typical student), because their other classes largely excluded the other calculus course times as an option, so nobody seriously suspected cheating.
The question stands by itself of course, but the case that prompted this has important additional points: the instructor was tipped off to cheating, and had actual chains of emails documenting cheating (students bragging about it) sent to him. It's a dirty, muddy case though, if only as he said he'd write his own questions, to then rely on book-publisher provided questions that students (illegally) accessed too.
@gnometorule: That way to hold exams (relying on widely used test banks and prohibiting their usage or relying on people not creating their own test banks; having people take the same exam at different times), sounds like it was a disaster waiting to happen anyway.
Two data points: (1) My class test grades are usually bimodal in distribution. (2) A year ago I had two sections of a basic algebra class, exact same schedules, lectures, tests, days, etc.: on the final exam one section had a 40% passing rate, while the other had 80%. Scores between different sections of the same course vary widely by nature.
@DanielR.Collins: I have come to believe that a bimodal distribution is expected and good. The first time I aw one, I asked a professor with much more experience than I. Tapping the high end of the chart he said, "It means that these students get it and," tapping the low end, "...these students don't." The ones who work and study get it, and the ones who don't, don't.
If I saw average midterm test scores go up in a class I taught, my first thought would be "maybe I've finally figured out a good way to teach this stuff".
One very good student, around whom others flock to ask questions/study together can very well explain such an outcome. A few excellent students among average ones can make the difference between most students failing/barely scraping by to almost everybody getting good grades.
@vonbrand: In a relatively small class, for sure. In a class of over 500 students -- which is the real life example from 2010 from which the OP has borrowed the histograms -- this seems less likely. (If so, that student should be getting paid as a TA!) In fact, because the question is about the merits of a statistical argument, I think it's a bit irresponsible of the OP not to mention the sample size. In classes of the size I normally teach -- 30 or fewer students -- I would essentially never allow statistics to influence policy. 500 students is another thing entirely.
THe second diagramm is not fine grained enough. It could have been bimodal as well. If basing any reasoning on this diagramm the prof will probably be asked to take stats 101 again....
Discussion on question by James Smith…
Discussion on question by James Smith…