Are professors allowed to accuse someone of cheating based on a general increase in test scores? 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 received such high scores). He looked at the score distributions and found it to be bimodal:



He went on to assume that, since this distribution wasn’t like his normal distribution he had received on his test scores in previous years, one third of all his students cheated on the mid-term.



Now here comes the question, is it ethical and reasonable to assume that one third of the people in his class were cheating? Maybe the distributions were different and bimodial because some students studied harder (causing peak at better scores), while others didn’t (peak at lower scores). Are professors allowed to make everyone retake the test because of this data (even the ones who took it honestly). This punishment seems like he is punishing the supposed two thirds of the class that took it honestly.

The professor goes on to claim that those students who had significantly higher scores than in their previous test must have cheated! Are professors allowed to that? What if they got a bad score on the previous test because they didn’t study and they got a good score on the mid-term because they did study? Also what if a student has a good score on the previous test and the mid-term because he cheated on both? Or what if the student was honest and received a good score on both test because he is a hard working student? So my question, Is it normal practice and ethical for professors to accuse someone of cheating just by looking at the student’s test scores?

PS: This didn't happen to me. I just heard about it from a friend.
First, as you already noted, the better scores and the bimodal distribution may have other reasons. For example:


One of three TAs for the course did an outstanding job.
The professor reused questions from previous exams (perhaps without noticing) and one third of the students informed themselves about previous tests.


Most importantly, the following thing is odd: It’s highly implausible that the rate of students who decide to cheat on their own increases by that amount from one year to the next. Thus, they would have to decide for cheating together or due to some common motivation. This in turn only makes sense if the cheating is a collaborative endeavour or if somebody is helping the cheaters.

Now, while such scenarios are not unthinkable, they are usually as likely as possible explanations that do not involve cheating (your mileage may vary depending on the circumstances). Moreover, even such a scenario happened, just repeating the exam is unlikely to solve it. Therefore the professor should first find out what actually happened.

So, the statistics may point at something being unusual, but without further investigations you cannot say that it’s cheating. Moreover, even if it is cheating, statistics can only tell you that it happened, put cannot point to any individual involved in it.

To address your individual questions:



  Are professors allowed to accuse someone of cheating based on a general increase in test scores?
  
  
  
  The professor goes on to claim that those students who had significantly higher scores than in their previous test must have cheated! Are professors allowed to that?


Technically, yes. Almost everybody can accuse everybody else of everything as long as we are not entering the domain of libel laws. Will it be successful or a good idea? Probably not, at least in any reasonable university or jurisdiction. And that’s not even considering that he has not one but many students against him.

  is it ethical and reasonable to assume that one third of the people in his class were cheating?


Assuming something cannot be ethical or unethical; acting upon an assumption on the other hand may be, but that depends on the action.

  Is it normal practice and ethical for professors to accuse someone of cheating just by looking at the student’s test scores?


Normal practice – I never heard of it. Ethical – certainly not. There are several possible reasons for a good individual test result, e.g., hard work and plain luck. The professor should be aware of this and thus the accusation is unfounded and hence unethical.