# Are course grade distributions supposed to be bell shaped?

I just got done teaching my first course, and my grades are heavily skewed:

I consider my assessments to be fair and have had them vetted by more experienced faculty. Also, the course averages are usually high in this course (mine was 83.5 or so). However, it seems like I have a strange distribution of grades. What might have caused it?

• To answer the titular question: no. And if you think your grade distribution is a problem, try asking senior faculty. Commented Aug 11, 2017 at 13:03
• It's not that any distribution in the world should be bell shaped, and grades can have any sort of distributions. The few times I had a look at the distribution in my exams they were either bimodal or loosely exponential. Commented Aug 11, 2017 at 13:07
• Note that there's a hard cap on grades at 100% - and that's where you seem to have a second peak. You might be essentially squeezing everyone that would have been on the high tail into one bucket. Commented Aug 11, 2017 at 19:23
• What @Walt says. This is a classical ceiling effect. A psychometrist's first impulse would be to add a couple more difficult items to the test, to better distinguish very good from stellar students. YMMV. Commented Aug 11, 2017 at 20:24
• Note that philosophically, a bell-shaped distribution also implies things like "As many students should fail my class as should get A's." Whether this is desirable at all is something you should probably decide. Commented Aug 11, 2017 at 20:45

The distribution doesn't have to be of bell-shape. In very large scale open exams it may be reasonable to assume a bell curve. In many other situations, the distributions can be affected by the class size, mix of the students, objectives of the course, validity of the exam questions, and difficulty of the exam questions, etc.

Class size: The smaller the class, the harder to observe any discernible distributions, such as a bell-shape normal distribution.

Mix of students: If you're teaching a quantitative class and there are two streams of students from i) art programs and ii) engineer programs, you may see some other distribution like a double bell-shape bimodal distribution.

Objectives of the course: Some courses can be designed based on a fixed and stringent set of standards. For instance, if you teach anesthesiology and the passing grade for the students to take the board exam is 90%, the end distribution is unlikely to be bell-shape.

Validity and difficulty of the exam: Invalid questions may lower the accuracy which prohibits you to see the true distribution; overly easy or hard exam can move the curve towards high- and low-boundary, causing truncated bell-shape distributions.

If I have to give an assessment I'd first suggest removing the 0% as it's a special case and yet tilting the impression of the curve quite badly. For the rest, I'd say if you're teaching an introductory course in which students are expected to gain a good foundation, this is not a bad distribution. If you're teaching a very advanced course, with nearly 20% getting close to full mark then the assessment may benefit from a re-tuning.

• Analyze the grades and your course objectives in tandem. Just grade distribution itself does not tell if you're doing a good job.

• Accumulate more data across cohorts of students. I find that after 3-4 times teaching the same course the patterns would start to emerge.

• Compare to historic grade distribution (just a few years before you picked this course up) to make sure you are not way off. Consult the appropriate dean if they are.

• If so inclined, try to analyze your exam items. There are special statistics to check if your exam questions are "good" questions. Most academic institutes should have an education affair office that can help you.

• To this very thorough answer, I'd add that a bimodal distribution such as OP observes is, in my experience, pretty usual. And, if you must use a single measure of central tendency, median, not mean, is the right one. Commented Aug 11, 2017 at 13:24
• The shape seems less important than the whether or not the shape can be explained. After all, what good are observations without a hypothesis? Commented Aug 12, 2017 at 7:57
• @corsiKa When I was very new to teaching I showed a senior member of the faculty my bimodal grade distribution and asked what it meant. "It means," he said, tapping the right hump on the graph, "that these kids get it and," tapping the left hump, "these kids don't." That has been my hypothesis ever since. Commented Aug 12, 2017 at 18:48

This looks to me like the sort of distribution you would expect from an exam that is simply too easy, and fails to distinguish at the top end.

You probably do have a roughly bell-shaped distribution of student abilities, but since in your exam the middle of the bell is at 85% or so, all of the high-ability tail inevitably get lumped together in the 95-100% bar.

• Interesting. Totally opinion-based follow-up - is that a bad thing? Commented Aug 11, 2017 at 14:57
• It depends what the point of the exam is. If you just want to test that students have attained at least a certain standard, this is fine. If you also want to be able to tell who is merely good and who is outstanding, then it is a bad thing. Usually in a university context you do want to do both. Commented Aug 11, 2017 at 15:20
• @A.W.Grossbard Yes, strong students often enjoy challenging exams. But exams that challenge the top students can easily lead to (literal) tears at the other end of the class. Commented Aug 11, 2017 at 18:42
• @A.W.Grossbard The point of an exam is not necessarily to challenge students. Especially in lower level classes it is to determine whether they have met a certain level of comprehension that is adequate to move on to the next course without being totally lost. A hard exam would be inappropriate in such a class. Commented Aug 11, 2017 at 18:52
• @A.W.Grossbard: As an instructor at a large urban community college, I've learned the following criterion: If I find any test question to be "interesting", then that's catastrophic (i.e., the entire class but one will fail it badly), and it must be stricken out. Commented Aug 12, 2017 at 18:07

Remember that your primary task is not to accurately tell better students from worse, but to make sure they learn what they are supposed to. This is also the role of the test: for students passing a test is supposed to be a fixed, specific goal to achieve, not a competition.

Therefore, the proper question here is: do you actually believe all of your students except for those unlucky few deserve passing your class? Do they learned what they were supposed to learn? If so, this is fine. The scores do reflect good on you that you taught your students well. This might also be a sign that it's not the test that requires adjustments, but the curriculum—you could probably teach more material in that class, and it will also indirectly lead to the exam being more difficult.

If not, then you should definitely adjust the test.

Some time ago I collected graphs from some documents from Polish Ministry of Education on the high-school exam. This is a huge sample (around 300k yearly), and you can see that scores do not always take a bell shape. Some useful discussion on interpretation of these graphs is in a reddit thread, especially explanation on the peak around 30% (the passing point) of the language exam.

• That's great. Thanks so much for sharing that link. Commented Aug 12, 2017 at 18:10
• "Remember that your primary task is not to accurately tell better students from worse, but to make sure they learn what they are supposed to." This is clearly not the case in many circumstances. Commented Aug 12, 2017 at 19:08

Senior faculty here... Ask yourself if you are grading your students against each other, against the material, or a combination of both. If you are grading against the material then the shape of distribution is not relevant. For example, if the first exam is to write the "Hello World" program and everyone aces it, then you would have a very skewed distribution but you have the optimal result for the class as a whole.

If your learning outcomes are well-defined, as they should be, and your curriculum addresses those outcomes, then you are doing your job when everyone earns an 'A'. When I teach programming I am careful to grade students against the material, not against each other. I have a set of outcomes and I teach to those outcomes. On the first day I tell students "I hope you all earn an 'A.' I am not pitting them against each other, grade-wise. There are other ways to reward high-achieving students than imposing a bell curve or any other distribution on a class.

• Indeed, material learned should not be represented by a Gaussian curve. Heights of your students, yes. Commented Nov 13, 2017 at 20:41

Generally grading distributions are analyzed like this:

1. If too many students get above the bell curve, that means tests are too easy. The tests are not separating "outstanding" students from "average" students.
2. If too many students get below the bell curve, that means the tests are too hard.

However, I am not sure that looking at a grade distribution is the best way to get a feel for how students are reacting to the coursework. If you just look at the grades you are only really looking at the sum of all of the results, and it may be hard for you to identify the best thing you can do to improve your couse.

As an instructor remember that there are more resources available to you than just an overall grade on a test (or an overall class grade). I recommend breaking down an exam or class by topic and evaluating how well students did with each topic.

1. If a large percentage of students answered questions on a certain topic correctly, that is great! Your choice of textbook combined with your presentation is making it so that they can absorb the information very well. If you are in this scenario, and you have already covered the requirements of the course you may want to consider going into more detail on the material or bringing up additional related topics that you think would help the students. You have a huge opportunity to go beyond the basics, if you have the time. If you can't go into more details, you could also go into a more practical direction and talk about how to apply these concepts.

2. If a large percentage of students seem to be struggling with a topic, if it's not a crucial topic you may want to consider dropping it. If it is a crucial topic consider reviewing it again in class (possibly using a different approach), putting it on the next exam, or assigning a paper on the topic. You may also want to consider revising the slides on the topic for future classes or watching presentations by other people on the topic to give you ideas for other ways you can present it.

Some answers and comments have mentioned the competitive aspect of grades that is sometimes present. I would like to mention this in my answer as well.

• First, I am not convinced that every class needs to be competitive, carefully consider whether it will be useful for your students to add some healthy and constructive competition to the classroom experience before jumping into that.

• If you want to seek out and reward the outstanding students, I recommend doing so by assigning projects and research papers. Challenging students to do research on their own and present their work will naturally bring out the best in talented students, and it could be very valuable practice for their careers (and grad school too).

• Encourage students to publish their work if their work is at that level. This may also be a great way to find grad students if you are looking for them. At worst it will introduce students to what a career in research would be like.
• Please don't assume that students are not interested in research, there was one professor I had who assigned "realistic" research papers (intended to resemble the requirements of publishing in a journal as closely as possible). It was a great experience, and based on that I ended up doing a thesis instead of a pure "class based" master's degree.

NOTE: My experience is primarily in STEM fields.

The theory is, given a random sample of students who have met the course pre-requisites, you should see something resembling a bell curve in your grade results.

In practice, there are a number of things that skew this. Sample size is a big one. For smaller classes, the sample size of any one course is just too small. Self-selection is another. At the higher-ed level, students have self-selected for courses in their major, such that they take more courses to match their interest and (supposedly) ability. Additionally, the entire college admissions process should pre-select for students who have at least some academic ability. This is part of why many colleges and universities in the US have a graduation requirement that mandates students maintain a B average for courses within their major (or some variation of this).

Another flaw in the bell curve is that it only looks at final outcomes. The full bell curve should include a so-called "long tail" to the left for lower grades, indicating students that withdraw before completion of the course.

There are many other human elements that can also distort a "pure" bell curve. Teaching ability can be one of these. As a new instructor, you may need to develop some experience for how to better grade and measure the performance of your students. But with only one class under your belt, it's just too soon to say if this is the big factor for that specific set of data.

• And what theory says that one should see something resembling a bell curve in the grade results? Commented Aug 14, 2017 at 20:31

If you class size is large enough and you test isn't too easy or too difficult (with respect to the capabilities of your students), you should get an approximate bell curve.

Your distribution is bimodal. Either the students cheated(they had the questions before the exam) or the paper was too easy.

• I've seen bimodal distributions considerably more extreme than this one. In some cases, it just means that some students understood the material and others didn't. Commented Nov 13, 2017 at 19:58