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It's a very basic high school statistical question, but I'm struggling to solve it.

Suppose I have a school with $287$ students and each one made a test with $50$ questions (multiple choice questions with $5$ items each, they have to choose one item in each question). These questions are divided into the following subjects:

Subject A - $8$ questions

Subject B - $6$ questions

Subject C - $10$ questions

Subject D - $6$ questions

Subject E - $6$ questions

Subject F - $14$ questions

We say a student fail the test when he doesn't solve any question.

Then we have the following result:

Subject A - $3$ students failed

Subject B - $16$ students failed

Subject C - $1$ students failed

Subject D - $1$ students failed

Subject E - $8$ students failed

Subject F - $0$ students failed

So how can I compare the performance of the students? In another words, which subject were they best and which one were they worse?

Remark: Each student has only two options: successful or failure, so in this case the overall score in each subject is not important.

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user42912
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    wouldn't it just be subject B, since that is the greatest percentage failure, and all subjects had the same number of students taking it? – Saketh Malyala Jun 22 '17 at 19:28
  • @SakethMalyala the problem is the chance of failure is bigger when you have more questions. – user42912 Jun 22 '17 at 19:33
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    @SakethMalyala's Comment is about the best one can do: Best in F (no failures among 287 students) and worst in B (16 failures) Without knowing the definition of 'Fail' in terms of questions right and wrong, I don't see how any formal statistical analysis is possible. // Depending on the quality of the questions, perhaps there is more info about Subject F (14 questions) and least about Subjects D and E (6 each). – BruceET Jun 22 '17 at 19:36
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    why so? shouldn't each problem in each section be of the same average difficulty – Saketh Malyala Jun 22 '17 at 19:36
  • @SakethMalyala sorry, I made a mistake, the chance of failure is bigger when you have less questions. – user42912 Jun 22 '17 at 19:38
  • If all 287 students took every test, and if we have no data beyond the number who failed, then the best you can do is say that they did better in the ones in which fewer students failed. But of course it would be better to look at the scores of those who did not fail. – Michael Hardy Jun 22 '17 at 19:39
  • @SakethMalyala The failure is a student who doesn't solve any question at all. I'm going to edit this question. – user42912 Jun 22 '17 at 19:39
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    this is still not nearly enough information to make a meaningful analysis. for example, subject B: we could have had 16 students get 0 right, and 271 get all 6, whereas in subject F, all students got 1 out of 14. and that completely fullfills the question's premises. – Saketh Malyala Jun 22 '17 at 19:44
  • @SakethMalyala let's suppose for each student we have only two options: successful or failure. The score is not important in this case. – user42912 Jun 22 '17 at 19:49
  • How many multiple choice options in each @user42912 – Ahmed S. Attaalla Jun 22 '17 at 19:49
  • @AhmedS.Attaalla 5 options – user42912 Jun 22 '17 at 19:50

1 Answers1

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You define failure as scoring a zero on the test. Then of course, the students are more likely to score a zero if there are fewer questions. Let us say that a test has $n$ questions, and $5$ options per question. Then the probability of getting all the answers wrong is $(\frac{4}{5})^n$. So the probability of "not failing" is $1-(\frac{4}{5})^n$. So with $n$ questions and $287$ students, we expect that $(\frac{4}{5})^n(287)$ should fail.

If the students together just put it to chance we expect $(\frac{4}{5})^n(287)$ to fail, but if they do something to sway the probability of getting an answer wrong down, then we should expect less to fail.

Now I make a suggestion, I will define "better", you need not agree with it but I think it is logical: the subject which is "better" is the subject for which the signed percent error from the expected number of failures is minimum. When it is maximum that is when it is worse.

$$(\frac{\text{observed}-(\frac{4}{5})^n(287)}{(\frac{4}{5})^n(287)})(100)$$

According to this definition of better/worse if you put the subjects in order of better to worse this is what you get:

$$F,D,C,A,E,B$$

Ahmed S. Attaalla
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  • Thank you very much for your answer. I used your idea to do something a little bit different. I took the division: $(4/5)^n(287)/\text{number of people failed}$, higher this number, best they did in the subject. What do you think? – user42912 Jun 23 '17 at 03:00
  • I agreed with your definition of "better", I want only to know which index is the best, the one I wrote in the last comment or yours. – user42912 Jun 23 '17 at 03:07
  • I think it is the same thing. Because $f(x)=x-100$ is increasing, my definition can be stated as the better subject is when $\frac{\text{observed}}{(4/5)^n(287)}$ is minimum. Because $f(x)=\frac{1}{x}$ is decreasing for $x>0$ that is when $\frac{(4/5)^n(287)}{\text{observed}}$ is maximum. But your quantity is of course better numerically for computation. – Ahmed S. Attaalla Jun 23 '17 at 03:22
  • I'm having problems only with the subject F. I got something divided by 0. So how can I get rid of this problem? maybe your index is better when we have subjects with zero failure. – user42912 Jun 23 '17 at 03:25
  • Oh I see $x=0$ in $\frac{1}{x}$ when $\text{observed}$ is zero. In that case, you can fix this, if you just use $\text{observed}{(4/5)^n (287)}$ instead. The lower the value, the better the subject. – Ahmed S. Attaalla Jun 23 '17 at 03:30
  • I just realized a down side with how I defined better: when $\text{failirues}=0$ the quantities do not distinguish between tests with different number of questions. – Ahmed S. Attaalla Jun 23 '17 at 03:38
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    I think it's not a problem, in this case we can put the zero failure subjects aside and compare them looking the number of questions, less questions, better they did. – user42912 Jun 23 '17 at 03:42