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

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.

• I would love to hear other educator's opinions with research backing it up. As a student, I prefer to know how I compare to others in terms of grades, but I understand the desire for anonymity. Having highest, lowest, and mean grades listed would be really interesting to me though. – David G May 12 '14 at 0:52
• I'm glad I don't have this problem. The computer system producing the summary printout produces a table of grade distribution. I don't have a choice. I do publish the points earned -> grade function on the course www-page as well as the average point score on midterm exams. – Jyrki Lahtonen May 12 '14 at 6:54
• What research I'm aware of is all about how giving any overall data about their own performance is actively harmful in promoting further learning. They learn considerably more from instruction about what to do differently in the absence of a grade or numerical score. To repeat; individual scores discourage learning! (Explanations for this correlation is about strong students relaxing and weak students giving up, all students concentrating on an emotional rather than academic response and ignoring any pointers.) The data says don't grade if you want more learning, instead instruct. – AndrewC May 12 '14 at 16:34
• In my experience the earlier you do this correction/instruction-only feedback, the better. I've been delighted with the change in attitude its early use has resulted in, with students preferring to re-test than give up, and much higher pass rates (the course is a national externally marked test). Later in the course I've been able to mark tests with numbers, particularly near exams, but the foundational practice builds a culture of improvement post-testing rather than finality. Warning: the research shows that any grading nullifies correctional/improvement feedback - don't bother doing both. – AndrewC May 12 '14 at 16:40
• @MHH To separate research findings from my own experience: research says the presence of any summative grade/percentage/score nullifies the effect of any formative advice/feedback/instruction, and that such feedback is far more effective at improving performance. My experience says that building an improvement culture early on by providing advice-only feedback with resits for underperformers can have a lasting impact on approach to learning, which allows them to ride out our numerical common test scores with their "not there yet, let's fix it" approach. We got the culture right first. – AndrewC May 12 '14 at 21:28

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)

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.

• Also, I should point out that some of the methods I don't like could work depending on how the class is set up around them (I may edit my answer to reflect this). If you really focus on emphasizing the class is about learning, and are at the "right institution" for it, I think no information at all could potentially be the best option. However, in our grade obsessed culture, I don't think this will work at most institutions. – WetlabStudent May 12 '14 at 3:06
• When I employ option (3) I describe the grade intervals but I don't name them A, B, C, etc. I fear that as soon as you say a letter students forget everything else you tell them and take the letter to the bank. I just call them average, good, very good, room to improve, danger, etc. – Matthew Leingang May 14 '14 at 13:53

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.

• Do you provide any distributional information at all? A maximum or an expected cutoff for an "A"? – user173 May 12 '14 at 15:31
• I agree with this - set only targets that are properly aspirational, and measure students against what you consider good rather than where the mediocrity is. I like to tell them "You are not an arrow! If an arrow misses its target, it lies helplessly on the floor. You are a human - you have legs and a brain! If you missed the target, keep going. If you fell on your face, get up!" – AndrewC May 12 '14 at 16:45
• Indeed; they go from being the best of the best in high school to getting grades they see as abject failure. For some this change in feedback is catasrophic on a psychological level. – AndrewC May 12 '14 at 20:42
• @MattF I am currently running my classes with an A as 90/110, a B as 80/110, a C as 70/110, etc. so the letter grades are well-known. – Chris Cunningham May 12 '14 at 21:10

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.

• Interesting point about how it is unclear about how the median and s.d. interact. However, I would argue that the mean and s.d. could provide misleading information if your data is not unimodal or the mean is very far from 50%. – WetlabStudent May 12 '14 at 4:28
• Maybe lying, and giving a minimum score 2 points lower than the actual minimum, would spare a tiny bit of embarrassment? They will still know they did really badly but not "the worst". – Steven Gubkin May 12 '14 at 4:39
• Why not, also give the median, mode, skewness, kurtosis, and enough higher moments such that the best students will be able to reconstruct the entire distribution except for the minimum score. (Also, this will save you from having to teach these concepts :) – Evgeni Sergeev May 12 '14 at 12:11
• Rather than "lying" you could just say "all scores fell in the range X to Y" without explicitly saying that anyone actually scored those endpoint values. – R.. GitHub STOP HELPING ICE May 12 '14 at 12:41
• The median fits naturally if you give the 25th, 50th, 75th, and 100th percentiles. – user173 May 12 '14 at 14:12

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.

• Apparently your system works for you. But I think you overstate your points (1)-(3). Point (1) seems to rely on an implicit assumption that a non-fixed grading scale was used; I don't see why it applies in the generality that you claim. In my experience point (2) is easily addressed by telling the class that their grades are not subject to negotiation; they seem to understand this fairly well (even though some students seem to feel obligated to ask for better grades in every situation.) I don't see how (3) would apply if the average grade is (say) a B- and everyone's goal is a A. – Trevor Wilson May 12 '14 at 16:37
• @TrevorWilson "Apparently your system works for you" is a fairly information-free put-down and perhaps unnecessarily adversarial if you don't have more specific objections to fixed grading scales. – AndrewC May 12 '14 at 20:58
• It's fair to say that discussing the grade distribution with them could be misinterpreted by students to imply you took it into account while grading; they wouldn't necessarily see that you would calculate such data purely because you can, nor that you would tell them it alongside their grade if it wasn't a factor. Ben's point is that by using fixed grading scales you can reduce students' perception of subjectivity. (2) can be resolved as you say by appeal to authority, or as Ben says by using a system perceived as more objective. (3) does apply: giving data always results in comparison. – AndrewC May 12 '14 at 21:12
• @AndrewC I agree that sentence was "information-free," as you put it, so probably I shouldn't have written it. My intent was not to put anyone down nor to be adversarial. Actually the point that I intended (but failed) to make was the opposite: that this answer seems to consist largely of a polemic against non-fixed grading scales, when the question (as I read it) is not asking what grading scale to use. I shouldn't even have mentioned my specific disagreements, because now I'm afraid we are veering off-topic. I hope to address the points that you and Ben raise in another thread sometime. – Trevor Wilson May 13 '14 at 0:43

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.

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.

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... :)

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.

• My experience has in fact been the opposite -- that the top students realize they are spending too much time on the course, and realize they can get by spending much less! Still, thanks for the answer! – Chris Cunningham May 15 '14 at 11:49
• I do exactly what Mike Z does: I just tell them all the grades in numerical order. This is simple and informative. The idea that this might be a violation of privacy seems very dubious to me. I also agree with Mike that fully informed students seem less likely to "grub" for extra points. – Pete L. Clark May 18 '14 at 1:27
• Chris: maybe that is a rational decision. – guest Oct 31 '18 at 6:47

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.