I am trying to get a raw score for my test grade, but I don't know anything about the distribution of test grades; all the information I have is the average and the highest score (and of course my own grade). Can I curve my grade based on this information? The method of curving doesn't matter, and the raw score doesn't need to be my "actual" grade. Here is the context:

I am moderating a competition between me and my friends, and we are split into different teams. The teams get points based on their test averages, but since we are all taking different courses with different difficulties I need a way to be able to fairly turn each grade into a raw score. Again, the curving method doesn't matter; all that matters is that it's constant for everyone. Does anyone know how to do this?

  • $\begingroup$ So to formalize your question, are you looking for a formula that will map the highest score to let's say 100, the average score to 50, and... a zero score to 0? $\endgroup$ – Rahul Feb 21 '16 at 21:06
  • $\begingroup$ I suppose so, yes. $\endgroup$ – Niko Feb 21 '16 at 21:09

You don't have the right kind of data to construct a uniform measure of difficulty that would normalize the different results on one scale. One needs the relative difficulty of tasks to be calibrated by a large number of cases where the same individual does multiple tasks, with enough cases and enough overlaps to quantify all the tasks in relation to each other.

If you pretend that every class has a test-result distribution that is from a 2-or-fewer parameter family of distributions, such as Gaussians, then you can use the two given pieces of information (and possibly also the number of students in the class, if that is available) to place everyone's results as points on one model distribution, and take those as the normalizations.

  • $\begingroup$ Could you give me an example of this? I'm not a statistics guy and don't fully understand your solution. $\endgroup$ – Niko Feb 21 '16 at 21:11
  • $\begingroup$ Imagine that every class is a Gaussian distribution of results from 99 students. The maximum test score will typically be around the 99th percentile of that distribution. From this and the average you know which Gaussian distribution is the one for that class. Then for the student's result for that class, normalize it relative to that distribution by computing z-score = (raw score minus average)/(standard deviation of distribution) and use that or something derived from it, such as percentile of that z-score within the distribution, as a normalized grade. $\endgroup$ – zyx Feb 21 '16 at 21:17
  • $\begingroup$ Ah okay so I'd need to know the standard deviation as well. Most professors don't compute that unfortunately :/ $\endgroup$ – Niko Feb 21 '16 at 21:18
  • $\begingroup$ But you can estimate it under assumptions such as what I wrote down. $\endgroup$ – zyx Feb 21 '16 at 21:19
  • $\begingroup$ This is just the "slightly more theoretically correct approach" than silly things like interpolating linearly between the average and the maximum. But in a more meaningful sense it is not possible to do what you want if the difficulty of the classes/tests varies a lot. $\endgroup$ – zyx Feb 21 '16 at 21:21

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