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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 '14 at 2:59
  • "Of course, the mean is such a crude measure that it is possible to return bizarre distributions (e.g. every mark is either 79 or 39) that meet the requirement, without passing too many people who shouldn’t, but most people find this equally unpalatable." (source) – Lyndon White Oct 11 '16 at 1:35
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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.

  • 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. – Collin Apr 30 '14 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 '14 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 '14 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 '14 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 '14 at 3:32
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https://divisbyzero.com/2008/12/22/how-to-curve-an-exam-and-assign-grades/ gives a nice rundown on a slew of different methods that a feasible, as well as a nice analysis of the pros/cons of each.

EDIT

The suggested forms of curving include:

  1. Returning graded exams, having students fix their errors, and giving them a percentage of the points they missed back. Has the advantage of forcing students to understand why they were marked down and hopefully learn something, but doubles the amount of grading necessary and gives a larger curve to students who performed poorly.
  2. Flat scale: $f(x)=x+b$, basically giving everyone $b$ points. Nice and simple, but can be problematic if there's a "curve-breaker" that would end up over 100%.
  3. Rescale percentages so that the highest grade $H$ becomes 100%: $f(x) = 100 *x / H$. Similar pros and cons to the previous, except that higher scores get a larger curve.
  4. Linear rescaling: $f(x) = ax + b$. Combination of the previous two, particularly in regards to being able to pick two raw scores $x_0$ and $x_1$ and choosing what curved scores they correspond to.
  5. Drop a question: $f(x) = 100*x / (100-p)$ where a problematic question is worth $p$ points. Must be careful to ensure that it was truly an unfair question, and has the con of disenfranchising students that may have spent unproductive time on it.
  6. Root functions (and generalized root functions): $f(x) = \sqrt{100 x}=10 \sqrt{x}$ or $f(x) = 100^{1-a} x^a$ where $a$ is between 0 and 1. Gives a nice boost to lower scores (Calc I optimization problem), but is hard to explain to students and perhaps needlessly complicated.
  7. Classic "Bell Curve": Mean becomes a C, other letter grades by standard deviations from the Mean. Creates a cutthroat environment of students competing against each other and automatically fails half the students. Really only feasible for large quantities of scores (like standardized tests) and is computationally more complex.
  8. Extra Credit Problems: assign a separate problem that students can use to earn points back on the exam. Can be problematic in that the stronger students will generally be the ones that actually do the problem correctly, effectively giving the higher scores a larger curve than the lower scores.
  9. Grading by Gravity: throw the exams down a hallway and assign grades by how far they go (facetious)
  10. Tenured & just waiting until retirement: Everyone gets an A (or F) (also facetious)

From there, the article gives a breakdown of some excel code for converting percentages to letter grades, as well as giving examples of several implementations of curves.

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    While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review – gman Oct 10 '16 at 11:38
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    Thank you @gman for the feedback. I'm new to contributing on stackexchange and appreciate your comment. I have edited my post to give a synopsis of the various curving methods discussed, as well as some of the (dis)advantages of each---is this more in line with the guidelines? – erfink Oct 10 '16 at 20:49
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    Yes, this is a great answer now! Thanks for contributing, and welcome to Academia.SE! – ff524 Oct 10 '16 at 20:57
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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.


(source: freakonomics.com)

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