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?


4 Answers 4


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).

  • 19
    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, 2013 at 22:48
  • 3
    @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, 2013 at 22:54
  • 1
    @aeismail - A test where only 23% of the questions are answered perfectly is still a lousy test. So unless the scoring system is viciously nonsensical (you lose all points for question 4 if you misspelled anything in question 2), it was not appropriate for the students' level of ability. Also, note that I recommended revisiting the scoring standards to try to salvage the situation.
    – Rex Kerr
    Feb 27, 2013 at 14:36
  • 4
    @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, 2013 at 15:58
  • 6
    @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, 2013 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.


Take the sqrt of the raw grade (present raw grade as a percent) and you will get something that looks ok.


Do exactly what he said! Not from slavish obedience but because it works surprisingly well. IOW add 49% (= 72%-23%) to each person's grade. [Actually now that I think about it...even simpler is to just add 50%. Very easy to understand.]

New scale: Mean: 73% Min: 54% Max: 118%

The average is now a "low C". (Assuming normal 60-70-80-90 F-D-C-B-A boundaries.) This mean is probably a little lower than normal but is reasonable enough that you don't get cries of murder for the tough exam. (Especially since they are getting thrown a bone versus no change.) And probably the class needs some remnant of a "spanking" for doing poorly. So low C is a good mean for this individual test.

The min will be an F now but not such a killer that semester is lost for those who failed it (if they do well rest of course can pass the course, maybe even eke out a gentleman's C). And there will probably be a handful of Fs but not that many. Some failures out of a group of 84 is normal.

The student who got 68% will have an over 100% mark now. Which she probably deserves. It quells some of the complaining from the Hermione Granger types that feels slighted when the standards are lowered after they perform well and the class gets a break.

I think if you try it, you will find a very reasonable grouping into F/D/C/B/A set. Plus it has the added advantage of doing what you were told (merely as a bonus). Plus...you don't need to totally overthink this. It is one test out of a few.

One further added benefit is the simplicity. Instead of having to explain some gawdawful transform (like binning each quintile or square root of arctangents). You just have a quick difference. Easy to process for you. And easy to understand for the class. Sometimes simple better beats complicated perfect.

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