Blogs, Grading

Grade Distribution and Conclusion

As a final entry in the grading series, I wanted to take the time to analyze the grades that my students attained on exams last semester and provide commentary on whether I should make adjustments based on what we’ve seen thus far.  In particular, we saw in Low grades cause drug abuse. that giving lower grades can be detrimental to the students’ long-term mental and physical health.  We then noted in Grading policies. that the grades we give students should be representative of the amount of material that they understand from the class.  We now look at the current grading distributions of the exam grades in my classes.  With this, we can try to determine what an ideal grade distribution would be and how far we currently are from it.

Grade distributions

Last semester I taught four classes.  I had two section of calculus 1 classes and two sections of introduction to proofs.  I started by looking at the distribution of the average exam grades for both of my calculus sections.  In both classes, what I saw was actually a mostly uniform distribution of grades with nearly equal numbers of Bs, Cs, Ds, and Fs.  There were a few outliers on the low-end, as frequently happens due to students that stop attending but don’t withdraw from the course, but otherwise, this pattern held fairly well.  The mean and median of both classes was approximately 70%.  Below we have the histograms for both classes.  If you would like the full data I will link the spreadsheet at the bottom of the post.


In my introduction to proofs sections, I had a different outcome.  In particular, the grades were left skewed with more Bs than any other grade, with fewer Cs, approximately the same number of As and Ds, and fewer Fs.  Between the two sections, the mean was approximately 73% while the median was approximately 77%.  Again I provide the histograms below, and refer you to the bottom of the post for the spreadsheet link.


In every case above, we do not get the stereotypical bell curve that follows from a normal distribution.  This may be due to the relatively small sample sizes of approximately 30 students per class; however, this is generally a large enough size that the bell shape would start to take form.  In fact, I find I usually have a left skewed distribution when looking at my grades, primarily because there is a cap on the given grade that is lower than some people would be able to attain, where as the lower bound provides plenty of room for the variability of grades in the course.

Additionally, there was a noticeable difference between my calculus 1 and my introduction to proofs courses.  There are many reasons this may have occurred, so trying to exactly determine the reason is quite difficult.  I will attempt to name a few of these.

First, the student population of calculus is much more diverse with regard to mathematical experience as there are many freshman coming from many different high schools, whereas the introduction to proof class is aimed at students that have already been introduced to college mathematics and are on more equal footing with regard to prerequisite knowledge.

Second, the calculus courses are semi-coordinated, which means that I do write the normal exams, but the final is a shared final.  Therefore, I try to use language in the exams similar to the language on the final.  While I also try to do this during lectures, the wording is not always intuitive for students.  I have found this has little effect on good students, but the average and struggling students tend to have greater difficulties.  That is, I expect some of the lower Bs turn into Cs, Cs into Ds and Ds into Fs causing a shift in some students but not all, resulting in the corresponding difference in distribution.  In my introduction to proof courses, I am able to word the exams a little more naturally, which, especially in a proof based course, can help the students focus on answering the problem instead of what deciphering what is being asked.  (Side note, in the proof course the problems are often still worded awkwardly as the goal of the problem is in fact to decipher the meaning of a statement, but this is done with purpose).


In these last three posts, we have seen potential effects of students getting lower scores, explored the purpose of grading and how best to reach these goals, and in this post we’ve explored the current distribution of exam grades in my classes.  We have thus far seen that if we were to arbitrarily give everyone an A in every class, that this may indeed lower drug abuse in the population.  On the other hand, this would lead to the feedback from grading being completely useless since it does not give any helpful information to the students.  From current data, we see that I am not just giving everyone As.  In particular, if so desired, there does seem to be room to adjust the grading techniques without losing the utility of the grading systems.

So then, what would we ideally like the grade distributions to look like?  Is there a place were we can provide meaningful feedback, but still not crush the spirits of our students?  Or is it good to crush some students in the early stages in order to let them know that they should pursue other routes of study?  Can we do something about bringing the lower end outliers closer to the middle of the distribution?  Should we make tests more difficult so that we can have a truly normal distribution and assign grades based on number of standard deviations away from the mean?

I have more questions after writing this series of posts than when I began.  I find this is often the case when pursuing research topics, and often indicates a well-chosen topic.  I hope you agree and that you will provide your thoughts and opinions on the matter.  A discussion on grading should help everyone involved to find a better way to determine grades in their classes.

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