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Let's say a professor gives different versions of an exam to her two sections. Despite attempts to make the exams equal in difficulty, the minimum score, median, and mean are all significantly higher for section 1, while the standard deviation is lower. Is there a way to normalize the scores for section 2 to approximate how the students would have scored had they taken section 1's version of the exam? Other assessments (which were the same across sections) showed the two sections to have roughly the same performance.

Update

Here are the numbers:

Final 1:

  • Median: 92.5
  • Mean: 86.66
  • Standard deviation: 17.51

Final 2:

  • Median: 85.0
  • Mean: 74.3
  • Standard deviation: 26.24

About 45 students took each version.

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  • $\begingroup$ Taking a z-score? $\endgroup$
    – Alex J
    Commented Apr 28, 2023 at 1:02
  • $\begingroup$ Please see our posts about probability integral transforms $\endgroup$
    – whuber ♦
    Commented Apr 28, 2023 at 1:15
  • $\begingroup$ Do you use the term "significantly" in statistical standard manner for the results of statistical tests? If so, how did you test that the minimum score is "significantly higher"? You didn't use "significant" for the difference (or ratio) between standard deviations. Does that mean you tested that one for differences as well and found results insignificant? It may be helpful to explain exactly what you did for finding your (in-)significances. $\endgroup$
    – Christian Hennig
    Commented Apr 30, 2023 at 23:52
  • $\begingroup$ I'd be rather reluctant to do an automated normalisation/calibration in case the marking schemes are very similar/related, because this may imply that somebody in Section 2 can effectively earn more marks for doing exactly the same thing in some instances. This could be seen as unfair. The reason why I was asking about significance above is that personally in such a situation I'd want to see a rather strong indication that the difficulties were different indeed before messing around with a predefined marking scheme. $\endgroup$
    – Christian Hennig
    Commented Apr 30, 2023 at 23:56
  • $\begingroup$ @ChristianHennig No, I did not use the term in the statistical sense. I meant that the differences were large. $\endgroup$
    – Ellen Spertus
    Commented May 1, 2023 at 20:43

1 Answer 1

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One option would be to scale and then shift the scores for section 2 such that the standard deviation and mean are the same as for section 1.

Edit: So for the numbers in the question, you would multiply each section 2 score by $17.51/26.24$. This would make the standard deviations equal and would make the new section 2 mean $49.58$. So you would then add $36.08$ to each new section 2 score to make the mean $86.66$.

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  • $\begingroup$ In case standard deviations were not significantly different, I'd rather not do anything to make them equal, see my comments for the question. $\endgroup$
    – Christian Hennig
    Commented May 1, 2023 at 0:00
  • $\begingroup$ Thanks, but I'm afraid I don't understand what it means to scale and shift. I'm pretty ignorant (degrees in CS, not math/stats). $\endgroup$
    – Ellen Spertus
    Commented May 1, 2023 at 20:58
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    $\begingroup$ @EllenSpertus My apologies, I've edited my answer. $\endgroup$
    – Joe Mansley
    Commented May 2, 2023 at 12:11

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