When handing back exams, what should we tell our students about the distribution of exam scores?

Question

I've seen a variety of approaches to the day we hand back our exams.

1. Show every score listed in order, so every student knows exactly how they compare to others
2. Show an aggregated histogram, for example saying how many students got each grade
3. Give only descriptive statistics (mean, median)
4. Say almost nothing about the class's scores

How can we give information about the distribution of scores that can be productive and useful for students? I'd specifically like to hear about approaches you have tried, and critical discussion or anecdotes about how students gained or did not gain from the information.

Please assume that the class is not graded on a strict curve where a certain percentage of students will get an A; in that case it's obvious that we have to be open about the distribution.

Answer 1

Summary:

Favorite options

• "approximate" grade breakdown (no mean, median, or standard deviation, so there is no comparison to other students, this doesn't necessarily mean fixed grading)
• no information at all (note at many institutions this might actually be bad. Please see the answer to a related question by AndrewC for a full explanation about how this could work)

Least favorite

• full distribution (can create a competitive environment plus privacy concerns)
• limited statistics (special care has to be taken such that the ones you provide are not misleading, and again comparisons are being made to some central tendency, see Chris Cunningham's answer for a full explanation)
• no information at all (can potentially lead to grade anxiety if the students are at an institution with a grade obsessed culture - although this can work well in the right setting)

Explanation + Past Experience:

I have done three things is the past.

1. I gave just the median and standard deviation

2. I gave a histogram, with rough rough grade cut offs. See (3)

3. I gave the median and "rough grade cutoffs." I explicitly warned that these grade cutoffs are only guidelines for approximately where you stand in the class and not your actual grade on the test. For example I might give the following cut offs "A: >88, B: >75, C: >60, D: >40". I then say "These are not your actual grades, only approximate grades based on, roughly, how much I would expect an A, B, C, and D student to have learned the material." In addition, I explicitly say "The entire class will be curved, at the end, so note that there are ways that the average of your 'approximate letter grades' on these exams could potentially be lower or higher than your final grade (but this is rarely more than by a +/-). This is a very rough curve for you to gauge where you are at. No +s or -s are mentioned because this would give a false sense of accuracy." I also put this warning in the syllabus.

I found that students much preferred options (2 and 3 about equally) to option (1) because two and three give them a better idea of what your expectations are, and it also relieves some of their anxiety about not knowing what their grade is at. If you go this route, definitely don't forget to put the warning in your syllabus. This protects you with documentation in case a student were to complain to the administration about the average of their exam grades being slightly higher than their final grade. However, I have never had a student complain about this; they understand that the grade cutoffs are innacurate and are only presented as a service to them and that if they are close to a border line, they understand that they could be sitting at the lower grade.

The histogram is an option that could potentially add information to the grade cut-offs but while it does add information I think it has serious drawbacks. The reason why I prefer "approximate but slightly inaccurate grade breakdowns" to the histogram method is that I don't want to create a competitive environment. I like to frame the test results in language that emphasizes what I expected them to learn, rather than a class rank (even though these things are highly correlated). I did sense some added competitiveness when using the histogram (although it wasn't a huge effect), so I think I'm not going to do it in the future. Note, several academics often mention the competitive nature of math classes as a reason for the lack of women and other underrepresented groups in mathematics. Because of this, in the future, I am no longer going to tell students the median. I would, especially, not recommend giving out the full distribution, because in addition to the reason above, it might be possible to figure out who scored what with some investigative work on the students end. I also think the full distribution would convey a false sense of certainty.

One might ask, well given the last paragraph why don't you know the grade breakdown in advance (i.e. fixed grading). The answer to this is, when writing a new test every year, with new, sometimes creative problems, I never know all the possible mistakes students will make until afterwards. Weighing the mistakes I see and looking at the distribution, I then identify the "approximate cut offs".

Lastly, I'd really like to try not providing any information. I think if you create an atmosphere where grades are completely de-emphasized, and reinforce that the student's holistic performance will be used to determine a grade at the end of the course, this could work very well. If a student is anxious about where they sit in terms of grades, they could ultimately request a meeting with you to discuss what they need to work on and what they are doing really well. Unfortunately, the bureaucracy at my large University, and the general grade anxiety in a class of mostly freshman who earned 4.0+ GPAs in high school, makes creating this environment difficult. If you do go this route, you also need to explain it clearly, and in detail on your syllabus, and keep some form of documentation of student performance so that you can justify all your grades.

Answer 2

I've personally stopped giving the mean entirely, because it almost always gives an incorrect perception to students. Since the distribution is almost always skewed by very low-scoring outliers, the mean incorrectly tells more than half of the class that they are above-average.

I'd argue that giving a measure of central tendency at all sends a fundamentally incorrect message: "Here is the target value. Your target is to be about as good as the most mediocre student in your class this semester."

That's never my message.

Answer 3

I tell the students the mean, standard deviation, and maximum.

The mean and standard deviation, because they are the most statistically useful. (The median would be useful too, but it's unclear how it interacts with the standard deviation, and sharing both the mean and median might be confusing to some students.)

The maximum, because I think the students with good scores will be curious about whether their scores are the highest, and also in order to confirm or deny students' suspicions that the exam was impossibly difficult.

I don't include the minimum, or a histogram, because it might make the worst students feel bad. Also, including too much information would encourage students to obsess over scores even more than they already do.

Answer 4

Some people use a fixed grading scale, while others determine ad hoc cut-offs for an A, a B, etc.

I use a fixed grading scale: 80%=A, 69%=B, 58%=C. Since the scale is fixed, there is no reason that a particular student needs to know how the other students are doing.

For people who use ad hoc cut-offs, I can see why it makes sense to tell students what the grade distribution is, since there is an element of arbitrariness in setting the cut-offs, and they deserve to know what the cut-offs were based on. But to me this just raises the question of why it's valid to set ad hoc cut-offs, especially in a subject like math that has objectively right and wrong answers. If students are supposed to be able to differentiate expressions involving transcendental functions using the chain rule, then that should be the standard.

Giving out grade distributions suggests to the student that (1) the teacher wasn't really sure what standards were reasonable to apply; (2) the day when you get back the exam is a day to be spent in complaining and negotiating about grading standards; and (3) as Chris Cunningham remarks, it suggests that the goal is to achieve the same marginal level of competence as the average student in the class.

Answer 5

I say absolutely nothing about the statistics for the exam unless they specifically ask. All that info is available to them on Blackboard anyway. (Well, the mean is there, and I think the min and max are there too.)

Instead, I prefer to talk about what the results of the exam tell me. Saying that the average was a 72 and the standard deviation was a 20 will not convey much information to students (unless it's a statistics class). But this might:

Based on the results, it looks like we are doing OK but have some work to do. Specifically there was a pretty wide range of scores, so some people are doing very well while others are doing very poorly.

If I do a item-by-item breakdown, I will also focus on overall themes, like this:

Most of us did fine on the basic computations. But on the conceptual multiple choice items, the scores were not as good. This tells me we need to work more on conceptual understanding, so expect more clicker questions in upcoming classes and less time spent on basic examples.

Pretty generic, but importantly it gets the focus off numerical scores and therefore, hopefully, downplays the point-scoring mentality that students are so prone to fall into.

Generally speaking I want to view exams as assessments in the true sense of the word -- where "assess" literally means to "sit down beside". My role as the instructor is to take the raw results of the exam, process those into descriptions of strengths and weaknesses for students (both corporately as a class and individually) and finally into actionable information that tells me what to do in class and tells students what to do in class and at home. That takes a lot more work than just saying the average, but it's important and IMO the right thing to do.

Answer 6

Using Webworks allows for some more meaningful data distribution. I have given students a scatterplot showing (# HW problems solved, # exam score), with an obviously positive correlation. Then I point out the lower right portion (students who did a lot of HW and did poorly on exam) and the upper left portion (students who did little HW, but did well on exam) are nearly empty.

I have no data that this actually helped the students evolve their behavior, but at least I feel like I'm sending a useful message.

Answer 7

In addition to other good answers: on one hand, ideally students should not care how they compare against the other people who happen to be in the class, but should care about whether they've made the progress that was the presumed goal of the course. Education! Scholarship! Under that hypothesis, and under the assumption that the instructor is experienced in interpreting test performance, there would not need to be any "curving", much less reporting on how other people did, which is irrelevant.

In reality, in my experience, both positive and negative effects can be achieved by letting on some (privacy-preserving...) statistics about a given class's performance. People who feel they're doing badly but are outperforming many others can be cheered up. People who think they're doing fine but are underperforming... can be implicitly admonished. People who don't want to over-exert themselves can game the system. People who somehow get the idea that the instructor is unreasonably harsh may see (or not) that their standards are deficient.

Apart from the obvious distinction between required/weeder courses and electives, whenever I have to assign "meaningful grades" (meaning giving signals about adequate command of material, and whether requisite thresh-holds are met, or not), I try to obtain the best of all worlds by declaring that there are absolute thresh-holds, based on much past experience (so that everyone might get an "A", or a "C", or...), while also reporting rough statistics on actual scores... with commentary on my happiness with most peoples' good work... and/or disappointment with the general level of effort, thus, supposedly, squelching the subliminally-game-the-system-and-themselves reactions to rationalize far too much.

So, yes, there is a fraction of the population that worries too much, suffers from impostor syndrome, etc. Another fraction is confident that they're being short-changed. Another fraction is oblivious. Only roughly correlated with these traits are work ethics... :)

Answer 8

When handing back exams, I give the complete distribution, listing every score from highest to lowest, with names removed so that privacy is protected.

Not only that, but I also give the distribution of the total scores for all graded assignments, again ranked from highest to lowest and with names removed, so that students see their overall letter grade in the class at every moment in time.

The day I started doing this, student complaints about grades ceased immediately. Now, my students never complain that the test was too hard, because they can see that there is a healthy number of high grades.

Students are also now fully aware of how their letter grades are computed, and again there are no complaints.

Student evaluations improved when I instituted this policy. Grade grubbing stopped.

Students find this information on their class standing to be valuable. The good students try harder to stay at the top. The struggling students know where they stand and what they need to do to improve.

Some students see that they are so far behind that they need to drop the course. In most cases, I agree that they should drop the course, and try again next semester now that they realize how hard they need to work to get a good grade. For many freshmen students, this is the most valuable lesson they learn, that this is no longer high school and that they need to work harder.

I will never again keep the complete grade distribution secret. Students deserve and benefit from this information. It should be freely available.

Answer 9

As an adult student, I have taken some fairly difficult courses - two advanced chemistry courses come to mind. That professor, who was great, would give you the score "as is," but then tell you how exactly how many points the curve would give you if there was one, along with the "central score" in his model. No one could ever get him to fully divulge his process, but he did let slip that he discarded outliers - he probably used a truncated mean or trimean. On non-curved tests I think he gave the median. We all thought this method was fine, but it really all depends on the class environment you create. This was a highly interactive environment where none of us were really competitive, and in fact would openly share our scores with each other. The students that made an 'F' on one test would be making a 'B' on the next because of the extra "love" they would receive. Most of us were in our mid-to-late 20's with full time jobs, so we didn't have the time or energy for pettiness. We preferred to learn and help each other out.

In summary, we would receive our scores with some form of truncated mean and the extra points that slid the distribution over to where he wanted on curved tests.