# How to Scale or Curve the Grades for an Examination?

I am the teaching assistant for a course in economics, and one of my responsibilities is marking examinations.

I had just finished marking the midterm, and the grades were very poor. The details are (approximately):

Mean: 23%
Standard Deviation: 5%
Min: 4%
Max: 68%
Number of Students: 84

Upon consultation of the professor, I was told to scale the grades. The method he had given me was to add a fixed amount to each student's grade such that the mean would be 72%.

To me, this seems to be an inadequate way to correct the distribution of grades and would be unfair to the students who did score much higher than the class mean.

My question is, what is the best way to rescale the grades for an examination such that the it is fair to all the students in the class?

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This is subjective; there is no "best" way. Also: talk to the professor you're grading for about your concerns. – Zev Chonoles Feb 26 '13 at 21:30
I wouldn't rescale the grades at all. I'd just throw out the exam entirely (but separately track the few students who did well). – JeffE Feb 26 '13 at 22:48

First, let's acknowledge that this was a horrible test. You have almost no power to discriminate between the different abilities of the students with a mean of 23% and a standard deviation of 5%.

Then you have to decide what to do. Personally, I'd decrease the weight of this test in acknowledgement that it was a poor test (or allow students the option of letting something else count for more). But if you just want to solve it numerically, there are the following constraints:

1. You can't have a top score of more than 100%
2. The mean is required to be 72%
3. You want all differences to still mean something

If you add 49% to the grades, you get a mean of 72% but your top score is now 117%. You could cap it at 90-95% or so, but the top students would be indistinguishable. So you'd have to scale back by squashing the grades also, e.g. by multiplying the difference from 23% by (28/45) to get it back in range. This will make the students distinguishable but not meaningfully so.

Really the best option is to change the grading scheme so that you have a higher mean and more distinction between different scores (to the extent that this is possible). For example, you can give a number of points for attempting a problem at all, a number of points for writing down anything which has a bearing on the right answer, etc..

You can just directly transform your old scores into new scores with the mean and deviation you like, however. Let `icdfG(p)` be the value at which the probability of getting a lower percentile score `p` from a Normal distribution with mean 0 and standard deviation is 1. Then you can just calculate percentiles and map

``````new score = 72 + icdfG(percentile)*10
``````

or something to get a mean of 72 and a standard deviation of 10. (Your top student in a class of 84 would have a percentile of 83.5/84 if you did it this way.) Your top student would have a score of 97 under this scheme (and the second to top a score of 93)

Otherwise, you're probably best off using a nonlinear transform; `arctan` is a nice function for that since it will squash down the tails; and you probably want a SD of at least 10 not 5, so you could do something like

``````new score = 72 + 50/Pi * arctan((score - 23)/10)
``````

which would give a max score if 93.5, a min score of 54.7, and one standard deviation up and down would give 79.4 and 64.6 respectively.

But again, the better options are to devalue the test as a bad test, and to change the grading scheme (an awful lot of work, but it will give fairer results).

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Great answer, saved me from typing something very similar. – JohnnyO Feb 26 '13 at 22:09
You can't have a top score of more than 100% — Why not? Once you acknowledge that the grade is not even an approximation of the percentage of work done correctly, why does it need to be a percentage at all? – JeffE Feb 26 '13 at 22:48
@JeffE - I see your point, but some schools like grading out of percentiles and may not be amused that some student has a score of 220%. Then again, whoever got that 68% did an awesome job; ten standard deviations above the mean! – Rex Kerr Feb 26 '13 at 22:54
@JeffE - That's why you make some questions harder than others. Getting them stuck on every single question is not a good way to help students demonstrate their proficiency. (It is a boon to those students particularly adept at jumping between questions when stuck.) – Rex Kerr Mar 1 '13 at 15:58
@JeffE - Making a majority of the questions an exercise in near-limitless climbing means that every test is about management and prioritizing as much as or more than the subject matter. – Rex Kerr Mar 2 '13 at 9:58

There are several ways you could scale the tests. You could take the top score, make it 100%, then add the same amount to everyone's scores. You could take the average, add a certain amount to everyone's score so that the new average is some value you desire. You could have a sliding scale so those with lowest scores get most added, while those with highest scores get least added.

Without knowing the desired outcomes or what constitutes an "A", "B", or "C", it is hard to say what you should scale your scores to.

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