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I'm working on a small application which would hypothetically allow instructors to perform a variety of grade transformation techniques on a given distribution. (I don't intend for the application to actually be used)

What are some of the more commonly used transformation techniques assuming a distribution is ~normally distributed? Online resources on the topic are somewhat sparse.

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I normally apply an affine transformation, translating the mean to where i want it and scaling so that one standard deviation equals one grade change. –  Jim Conant May 1 at 2:59

2 Answers 2

up vote 8 down vote accepted

It's not perfect, but I often use a piecewise linear transformation. Specifically, I fix some cutoffs (typically the cutoff between a B+ and an A-, between C+ and B-, between D+ and C-, and between F and D) and then scale all the A's linearly with their range, the B's linearly in their range, and so on.

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I'd give you an upvote if I could. So for example, maybe: the average was a 55. You decide that anyone who scored between 50 - 59 should receive a C. So then anyone who scored in that range would have 20 points added to their score. This would put the average at 75 which is in the middle of C range. –  user2079802 Apr 30 at 22:10
    
That's an example, but the general system is a little more general. For instance, if the average was a 55 but the standard deviation was higher, I might decide that anyone who scored between 45 and 64 should receive a C, so if 45<=x<=64, their new score is 70+(x-45)/2. –  Henry Apr 30 at 23:01
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so then anyone who scored in that range would have 20 points added to their score — Why are you adding points at all? Shouldn't the conclusion be "So anyone who scored in that range gets a C"? –  JeffE May 1 at 0:20
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But if it's not a final grade, what's the point of the curve? –  JeffE May 1 at 11:11
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@Dennis: The advantage over not curving is that if the median for the exam ends up being a 65, I may not want half the class to get a D or worse. My experience has been that the top 25-30% of the class (roughly the A's) tends to be somewhat clustered, while the next 30% (roughly the B's) tends to be a little more spread out, and the next 30% tends to be much more spread out. This scheme curves those to A's, B's, and C's respectively, and is reasonably simple among schemes that do so. –  Henry Aug 22 at 3:32

One I have not used although I've heard of being used before is to rank the final numerical score, and then use that to assign the final grade. Typically these ranks are then used to bin the results into corresponding letter grades of arbitrary proportions. E.g. the top 5% get and A, (5 - 10%] get a B etc. (I have no idea how (un)common this is.)

If you wanted the end result to be as close to normal as possible (frequently not possible if you have ties - or pretty much meaningless if you have small class sizes), you could convert the ranks to quantiles and then take the inverse CDF of your specified normal distribution (mean and variance) you desired. I don't know of anyone who goes quite that far in curving grades though.

This is actually how all civil service exams (exams that state agencies use to hire individuals here in the US) are curved that I know of. After the minimum score cut off, people are ranked into specific bins, and then cohorts of interviews are arranged for the people in the first bin (and if they don't work out they go further down the list).

I would speculate the most common form of curving is simply bumping grades above a particular cut-off. See the Freakonomics blog for one example.

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