Exam statistics

After removing two outliers, the correlation between scores on exam 1 and 2 in the class I’m teaching was 0.24.  This seems surprisingly low to me; has anyone with more experience seen similar numbers before?  The correlation between p-set scores and each of the exams separately is higher (about 0.3); the correlation between p-set scores and the average of the two exam scores goes up (to 0.35), which is not surprising to me.

This is not a completely silly question, since exam statistics do reveal something about exam construction and grading (if not necessarily about the quality of teaching and learning going on).  The most common example of this I’m aware of is bimodality: it’s extremely common for math exams to have bimodal score distributions.  This is because a typical first draft of a math exam contains questions from a variety of areas, but at a relatively uniform level of difficulty (nothing super hard, nothing really easy).  Thus, one tends to pick up a signal associated with the ability to solve math questions at a given difficulty level, independent of subject matter, in addition to the signal from content mastery.  Giving questions at a wider variety of difficulty levels tends to smush the two peaks together.  But I don’t know a similar story for inter-exam correlation.

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2 Responses to Exam statistics

  1. Toby says:

    The correlation between my midterm and final scores the last time I taught linear algebra was about 0.7, so your numbers do look surprising to me. I think I would have gotten less correlated scores if I had an earlier midterm, though, before we hit all the hard stuff.

  2. JBL says:

    Thank you for the confirmation — I don’t think I’ve ever computed correlations before, but this seemed really weak. (A friend says the statistics teacher at Stuyvesant has found that the first exam is an extremely accurate predictor of final letter grades, but didn’t have numbers.) Your comment suggests one possibility: because this is a Calculus 2 class, in which students all have learned some calculus before, Exam 1 might have had signal from “how much calculus did you learn in a class other than this one”. Exam 3 is coming up soon, and I’m looking forward to running the numbers on it afterwards.

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