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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.

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My chairman at a place years ago was a statistician, he made it pretty clear that you do not combine distinct populations. When I suggested a bi-modal distribution I made up for class grades, he said any class was four populations, those who work but don't know, those who know but don't work, those who don't know and don't work, and occasionally some who know and also work. – Will Jagy Jul 24 '12 at 19:47

Just give a scholarship to the top 50 in each group. There's not really a point in making assumptions, unless you have students who completed both tests, which you can use to tweak and adjust the distribution.

You also have something to offer - if they desire it you can give them the opportunity to complete a set of standardized tests, which you can use to discriminate against those who had a disadvantageous social background.

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