Bimodal distributions of student achievement

“There is nothing in the middle of the road but yellow stripes and dead armadillos”
Jim Hightower

There used to be a time, not so long ago, when exam scores of students taking college physics were distributed on a curve resembling the usual Normal distribution. This was almost invariably the case in large introductory physics classes, but the same observation also applied to advanced senior courses, typically attended by relatively few (e.g., ten or less) students per year (as one could ascertain by lumping together data from a few consecutive years).
This meant that the bulk of the students would perform comparably, as one would intuitively expect. In turn, this allowed an instructor to gauge the level of the material covered in class, as well as the presentation thereof, based on how well most of the students seemed to assimilate it (and rendered grading much easier too).
Increasingly, however, over the past decade I have been observing Bimodal distributions of student achievement, in the classes that I have taught, including at the graduate level.

A “Bimodal” distribution

Extensive discussions with colleagues, as well as with friends who teach subjects other than physics, have led me to believe that bimodal distributions are in fact rather common, these days. I have heard several theories purporting to explain why this would be happening. Whatever the explanation, though, bimodal distribution pose significant challenges to instructors.

The simplest example of a bimodal distribution (BD) is shown in the figure above (courtesy of Wikipedia). The difference with respect to the normal distribution is clear. In the case of a normal distribution, most of the students perform near the average (i.e., right in the middle of the distribution itself), whereas in the case of a BD the majority of the students perform either above, or below average. In fact, I have seen instances in which the “dip” in the middle was quite pronounced, i.e., there was a clear separation between the two groups of students.

What is a teacher to do, in the presence of a BD of student achievement, especially if both groups of students (i.e., performing above or below the average) are large enough in size that one cannot simply forget about either one ?
I can only think of three alternatives, none of which seem particularly satisfactory:
1) Keep the level of the course high, i.e., gear it toward the minority of “overachievers”. This will likely result in a large number of students failing the class. Is this a desirable outcome ? Admittedly, often times the purpose of a course (especially at the introductory level) is that of “weeding out” those students who may not have chosen the best academic path. Still, failing over half of a class seems harsh, especially if one wishes to encourage breadth, i.e., have students with diverse backgrounds and interests take also classes in subjects that are far from their academic focus [0].
2) At the opposite extreme, lower the level to the point where most students will pass. This seems to be equally unfair, as it seriously short-changes students with a strong background and interest for the subject matter, who wish to be challenged, in the process possibly learning something that they do not already know. Some of these students will be seeking admission to graduate programs later on, and expect to acquire knowledge comparable to that of their competitors, i.e., students from other institutions. Why water down the course material to a point where it may lose much of its meaning, only to keep afloat students who will often not learn anyway ?
3) Adjust the level to be somewhere in the middle. This would be the optimal solution in the case of a normal distribution, but in the case of a BD it is a surefire way to dissatisfy everyone, making the course either too difficult or too easy for just about every student.

I wish I had a better idea…

[0] It is often the case, especially in physics, that departments offer classes that are very qualitative, essentially only descriptive, aimed at attracting students who are not majoring in science. In my opinion, this type of course does not really give students a real taste of what doing science is like.

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9 Responses to “Bimodal distributions of student achievement”

  1. mareserinitatis Says:

    “failing over half of a class seems harsh”

    Gee, you might give the engineers the feeling that the physics professors of the world are a bunch of touchy-feely slackers.

    I had two profs who had no problem failing over 2/3 of the engineering majors in a class. I was personally of the opinion that it would be okay if they knew how to teach reasonably well…but, alas, it was not to be so.

    • Massimo (formerly known as Okham) Says:


      I had colleagues at my previous institution who did fail 2/3 of all students in a class. And because students called “fail” a D, which was below the minimum required of engineering majors in their physics courses, I have done something similar myself. Where I am now it is unthinkable, and simply not done — the chair himself would re-curve and re-grade, if I were to submit such a grade sheet.

      • mareserinitatis Says:

        So it’s harsh to the chair and not you?

        I guess that I would be disturbed if a taught a class where 2/3 of the students were failing because, personally, I’d be inclined to think that was a reflection on my (lack of) ability to explain concepts clearly.

        However, I should also condition that with the statement that I was speaking of a course for majors, not a general ed course or even a “weeder” course for the major. I do realize those have a very high fail rate simply because a lot of people bite off more than they can chew.

      • Massimo (formerly known as Okham) Says:

        So it’s harsh to the chair and not you?

        I am not sure I follow. I said that, if I, or anyone else, submitted a grade sheet with 2/3 of failing marks, it would be amended. I did not say that I routinely submit such sheet…

        I’d be inclined to think that was a reflection on my (lack of) ability to explain concepts clearly.

        Well, no instructor likes failing students, precisely for that reason. Consistency among different instructors is clearly one thing to look for, e.g., in multi-section courses.

      • mareserinitatis Says:

        I guess I was trying to say that my only experience in being in a class where 2/3 of the people failed were ones where the professor couldn’t teach his way out of a paper bag. I was unsure why your response to that was that you had done such a thing, especially after stating it was harsh. And I guess I’m still not sure…

      • Massimo (formerly known as Okham) Says:

        Well, if it had been only me, or a few of us teaching that course (introductory physics for engineers), failing 2/3 or all students, then clearly we would have been identified as outliers and someone might have regraded the exams. But the same outcome was observed with every instructor. It was actually considered pretty standard. Mind you, none of us ever set out to fail 2/3 of the class, it just happened.
        In my case, I remember failing many students who, at some point, would stop coming to class and doing the work, but never formally withdrew from it. And then there were those who thought that showing up for class and paying registration fees ought to be good enough for a pass — never really meant to do any work, and were flabbergasted at the end when they were told that they would fail.

  2. raudibert Says:

    I observed the bi-modal distribution from my earliest days of teaching at the university level. I assumed at first the two populations were separated by either capacity or experience. After a dozen years in the classrooms, I concluded the answer was elsewhere, that the two populations were separated by effort. The students with higher grades exhibited all levels of capacity, both low and high, but with effort the best approached perfect scores while the least prepared got Bs and Cs by dint of hard work. The second population had very bright students who made no effort and were satisfied with Bs because their priorities were elsewhere. The others got Ds and Fs, failing to either do their assignments or to ask questions. Their education was too low a priority to even attempt to pass some of their courses. In online teaching, I work for an organization that has free tutoring in the subjects I teach. I have never, in two years of work with them, seen any student who was failing my courses willing to work with a tutor to give him/herself a chance to pass the course.
    I completely reject the notion that some teachers “can’t teach their way out of a wet paper bag.” I think the metaphor says it all about this author. I know there are some teachers who are better than others, but no one — NO ONE – can stop a motivated student from learning. That was true before the advent of the computer and the internet, and it’s a laughable thought today. In the courses I teach there are online tutorials, papers, YouTube videos, and pages of scholarly articles by the tens of thousands.
    I think the biggest problem with the bi-modal distribution in this day of disappearing tenure is that adjuncts will pass students as a defensive measure since we all know high tight the correlation between teacher evaluations and expected grades.

  3. EinsteinEnvy Says:

    Just finished grading a physics exam. Only the second time i’ve seen a bimodal distribution. That may be because this is the first time in more than a decade that i’ve taught a freshman level (non-calculus) physics class. In my opinion something beyond separation of effort is going on here. I teach at a public, non elite institution in a state in which the education system administration is near incompetent (includes “bible thumping for politics” evolution deniers for example). My particular system believes everyone who wants a college education should be offered one almost regardless of preparation or academic inclination. There are students in this class who are comfortable with trigonometry and pre-calculus sitting next to other students who can’t do simple algebra or fractions.

  4. David Russell Says:

    When ever I hear the phrase “could not teach them selves out of a wet paper bag” I always look other professionals that deal with human subjects – diet counselors – addiction counselors – medical doctors –

    If your hired guide gives you a set of helpful instructions and you refuse to do any or most of the listed instructions – how is it your guides fault when you fail?

    Example – the public is famous for taking an antibiotic only until they feel better, contributing to a growing number of bacteria with antibiotic resistance.

    Example – dietician asks patient to exercise, eat healthy food, reduce caloric intake and get more sleep – sound advice – worthless if not followed.

    Just some thoughts –

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