# Ranking students from 2 separate exams in single scale.

Is there a way to rank 2 student groups who face 2 separate exams in a single scale using z-score, given that there are enough student in each group to consider each score distribution a normal distribution?

For instance say 2 student groups answered to 2 separate maths paper. Each of these groups has at lease 2000 students in it. We want to give 100 scholarships to those students, so we need to rank them accordingly and give this scholarship to the top 100 of them. And now they know their marks and we need a way to rank them in a single scale.

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I have never seen an exam in which raw scores were even close to normally distributed. But you can transform so they are close to normal with identical mean. These are then the "official scores." Then take the top $100$. Or just if $a$ students took first exam, and $b$ the second, use top $\frac{100a}{a+b}$ from first group, top $\frac{100b}{a+b}$ from second. But just because we have done some math doesn't mean the procedure is mathematically justified. – André Nicolas Jul 24 '12 at 17:37

It depends on your prior beliefs about these students. If they're all from the same population and just happened to take two different tests, you can compare their $z$ scores from the separate distributions. On the other hand, if there's reason to believe that the two groups differ statistically, then there's no way of knowing how to compare the tests. Simplifying assumptions might be to ignore the differences between the groups and use the separate $z$ scores anyway, or to ignore the differences in difficulty between the two tests and compare their test results directly, but you have no systematic way of knowing how good either of these approximations is.