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What do you think is the best way to rank universities participating in a competition if we know for each university the number of recognitions that it's students won. Each university can participate with 3 students, and approximately half of them get a recognition - which can be first (5%), second (15%) or third place (30%).

To be more specific, let's say 5 universities participated in a competition and the first university won 2 first place recognitions and 1 second place recognitions, the second university won 2 third place recognitions, etc... Probably a good way of measuring universities is by the total number of recognitions won, or the number of first place recognitions won, followed by number of second place recognitions won etc.

Now, my question is: How do we rank the quality of education in the universities if we know for each university how many students learn there? I'm asking this because it's probably a good idea to take into account the number of students? One possibility that I was thinking of is to divide the number of achievements with the number of students and use that to compare universities, but the problem here is that each school can participate with only 3 students (probably if the largest university participated with more than 3 students it would have won more recognitions). We can assume that the universities have a lot of students, and that the background of the students enrolled at different universities is the same.

Thank you very much for your time!

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"Quality of education" is a rather vague term, and it's very doubtful that you can quantify it meaningfully using these kinds of result. Presumably the school's team is chosen from the students with the highest probabilities of getting a good result in the competition. The competition results will say nothing at all about what kind of education a typical student gets, just about how well the school attracts top students and how it prepares them for the competition.

Putting that aside, you need a mathematical model for whatever the competition does measure, involving a probability distribution for abilities in the population from which the students are drawn, and how the school can change those abilities. Then you can go about trying to estimate the parameters involving the school's contribution.

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Dividing achievements by university size is probably not a good idea, because a huge school with 3 first places (the best possible score) could then be ranked lower than a tiny school with only one third place (the second-worst score), which is definitely counter-intuitive, because you can't be better than perfect! :)

Perhaps what you want is for large universities to have less of an "excuse" to be less than perfect. Something like this:

rankscore(school) = maxscore - (maxscore-score(school))*size(school)

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I would prefer to measure Quality of Education as follows. Conduct a survey to see what proportion of students actually read their textbook before coming to class.

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