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I will admit to being just a hair about a novice when it comes to stats, but I feel like I have a decent working knowledge of things such as normal distribution, linear curves, standard deviations, etc. That said, I have a colleague who is proposing something that just doesn't make sense to me, and I'd like some input from people who know better:

I work in education, and my department recently gave midterm exams. This year, we used new assessments, and as such, we didn't know exactly what to expect in terms of how the students would perform.

The grades for some classes were about what we might have anticipated, but for several classes, the students performed much lower than similar cohorts have on past midterms. So, as a department, we agreed to apply a curve to these scores to account for the different and give a boost to the groups that struggled.

My colleague took it upon herself to spearhead the curving. While I think applying a linear curve would have been best, she felt that applying a normal distribution curve made more sense. In explaining her methodology, she writes:

"I set the standard deviation on the curve to mirror how similarly each particular group performed on each exam. However, if the top score was pushed over 100, then I modified the number slightly until the top score was at or less than 100. This, too, can be altered. If we decide that the top student should always earn a perfect/nearly perfect score, then the standard deviation can be altered to push that top score closer to 100."

Am I missing something? I didn't think standard deviation was something that could be "set," as I understood it to be a reflection of the raw data available in the sample set.

I can't find evidence anywhere online of anyone changing or modifying a standard deviation to make a curve fit a desired outcome.

Can anyone clarify what might be going on here?

The Average Joe
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    Whether you're "applying a linear curve" or a normal one, in the end you're just making stuff up. Although your purpose might be noble and reasonable, you haven't articulated any objective criteria that could be used to guide your process. That will make it difficult for anyone to develop an objective or defensible response. – whuber Feb 02 '18 at 18:59
  • I essentially agree with you. Typically, I don't like to apply curves at all. I'm of the mind that if the numbers you get are bad, usually that means that either the assessment is bad, the students weren't adequately prepared, or both. (There is, of course, the chance that the students just didn't study, but in this case, the numbers don't really suggest that.)

    But again, this was a department decision.

    My question remains, can one "change" a standard deviation?

     – The Average Joe Feb 02 '18 at 19:51
  • One can only guess. A possibility is that your colleague is performing a traditional "curving" of the grades by forcing them to adopt a bell-shaped (Normal) distribution. Such distributions are determined by their location (average) and their spread--which is usually expressed as a standard deviation (SD). Narrowing the SD will cause all the "curved" scores to bunch up more. Perhaps that's what she's doing. If so, then it appears she has already determined what the average will be and is unwilling to change it. – whuber Feb 02 '18 at 21:51
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    Oh, that's definitely true. She's more or less unilaterally decided the average score should be 82%. So if that's the case, that might explain why she's manipulating the standard deviation? – The Average Joe Feb 02 '18 at 22:01
  • First come up with criteria to use on the re-grading function g that maps the old grade to the new grade. Presumably g should be monotonic. Also note that bell curving can violate g(x) >= x and actually lower a score. Other approaches not in that framework can also be considered. It might be that the lower scores are due to just one or a few questions. In that case re-score the test by re-weighting the questions. Another possibility is to let students re-do the questions they did not get and give them partial credit for the re-done questions to boost the scores. – G. Grothendieck Feb 04 '18 at 15:30
  • Shouldn't the average be C? A excellent, B good, C average, D needs work, F failing. – Chris Dec 11 '18 at 03:22

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While I have no complete picture of the problem at hand, I believe the following comments may be of help:

  1. It is reasonable to assume a roughly normal distributed shape for the marks, at least if: (a) the sample sizes are large enough to allow an assessment by eye that supports this assumption, or at least does not glaringly contradict it, and (b) if some outliers are ignored.

  2. You write: "While I think applying a linear curve would have been best..." An affine-linear transformation actually blends very well with a normal distribution, since if X is normally distributed, then so is Y=a*X+b for any constants a,b. Here the (random) exam mark is modeled by X, and the "adjusted" mark by Y. So in this application we need a>0. The standard deviation of Y is then a times the standard deviation of X.

  3. Since Y does not represent the raw data, its mean and standard deviation can be changed freely via the choice of the above affine-linear transformation.

  4. If your colleague only used a (that is, if she took b=0) to transform the data, it would not raise the average mark and thus not remedy the problem of poor overall performance; it would merely widen the gap between strongest and weakest results (if a>1).

  5. If the marks were to be raised by changing the constant b, this may necessitate reducing one or more top marks to below the maximum of 100.

Christian Rau
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