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Let's say I have 2 sets of grades from students. Maybe I'm looking at gender differences or some such but the disaggregating factor could be continuous or multinomial.

Group A: 3 D's, 5 C's, 13 B's and 2 A's. Group B: 2 D's, 3 C's, 7 B's and 11 A's.

How do I perform a statistical analysis to determine whether the grades of B are significantly better than those of group A?

Would it just be a case of multiple chi-squared or Fisher tests, pivoting on grade X or better? Is there a Mann-Whitney-U equivalent that I could use?

It would be preferable that I could also include confounding variables. Like whether students had any additional tutoring on the side. But it's not necessary.

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    $\begingroup$ Note that any standard analysis will treat the students as independent, which in many real situations (for example if they study together) is inappropriate. In a simple two groups setup this can not even be detected from the data, as any difference you see could be explained by either difference in distributions or dependence (or a mix of both). So whetever you do will rely on the students behaving like an independent random sample from the population you wish to generalise to. $\endgroup$
    – Christian Hennig
    Commented Dec 7, 2023 at 12:08
  • $\begingroup$ @ChristianHennig Do you mean because student m is more likely to get an A because student n gets an A, kind of thing? $\endgroup$
    – Leonhard Euler
    Commented Dec 7, 2023 at 12:51
  • $\begingroup$ That's a possibility, for example if they learn together. In general one can't know for sure how dependence plays out, but teachers know there's interaction and that can often make a difference. (Of course only if there is in fact interaction in case of your sample.) $\endgroup$
    – Christian Hennig
    Commented Dec 7, 2023 at 22:34

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One solution is to convert grades to GPA and then do a t-test or U test. A = 4, B = 3. C = 2. D = 1. F = 0 is the usual way to do this. Or, if you have covariates, you could use some form regression.

If you want to keep the grades ordinal, then you could do a test that accounts for that, like Jonckheere-Terpstra, or, with covariates, you could use ordinal logistic regression.

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  • $\begingroup$ Both look like excellent suggestions. Many thanks. $\endgroup$
    – Leonhard Euler
    Commented Dec 7, 2023 at 11:37
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    $\begingroup$ Wilcoxon/Mann-Whitney (which I guess is what you call U) with correction for ties keeps grades ordinal as well, i.e., it doesn't use information other than the order. $\endgroup$
    – Christian Hennig
    Commented Dec 7, 2023 at 11:52
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    $\begingroup$ That said I wouldn't worry too much about a t-test here (violations of distributional assumptions will normally be harmless but see my comment under the question); in a regression setup it's a different matter because of floor and ceiling effects. $\endgroup$
    – Christian Hennig
    Commented Dec 7, 2023 at 12:10
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    $\begingroup$ The initial paragraph is a poor recommendation due to its statistically unsupported assumption that these grades ought to be coded numerically using the values given. Simple, better methods abound. What is needed here is a more thoughtful analysis of how the differences among the grades should be quantified. $\endgroup$
    – whuber ♦
    Commented Dec 7, 2023 at 14:56

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