U.S. grading system: Why discrete grades?

Question

Background

In the U.S., many colleges adopt an A–F grading scheme, topped with pluses and minuses.

Professors vary in the way they assign A to F grades, but my sense is most rely on discrete “bins”. For example, some professors use relative grading and (roughly) give an A to the top third or the class, a B to the second third, etc. Others will give an A to students who get between, say, 100% and 95%, and A− to students between 94% and 90%, a B+ to students between 86% and 89%, etc.

Using discrete bins necessarily implies some knife-edge cases where a student is right at the border with the next bin, and it would have taken very little for that student to jump into that next bin and improve their letter grade by one “level”.

That would be all fine (I guess) if letters where only honorary titles, and what mattered was the actual underlying percentage grade. However, in most places (i.e., all the places I know), it is the letter grade – and never the percentage grade itself – that is attached to the student’s academic record.

This could be an issue per se if a student wanted to show proficiency in a particular topic and got, for example, an A− that was in fact very close to an A. That student’s academic record will forever show very good but not perfect for that course, whereas it should really indicate almost perfect. This creates understandable frustrations from students who nearly made it.

Things get even worst when averaging through different courses and computing one’s GPA, since most colleges then use a scheme like the following one:

• A: 4.00
• A−: 3.67
• B+: 3.33
• B: 3.00
• B−: 2.67
• C+: 2.33
• …

Again, this means that the half of a percent a student needed to jump from an A− to A will have a significant effect on that student's GPA, where really, the effect of getting one more half of a percent should be negligible, regardless of the baseline percentage one is starting from.

In a world where GPAs are taken so seriously, this means that half of a percent can make a significant difference in a student’s life and career. Again, this can create understandable frustrations for students who nearly made it.

My question

I know that there are “palliatives” to deal with the discrete grading scale and avoid some of these knife-edge cases, such as rounding up decimals. In all fairness, I don’t think those are real solutions (rounding up decimals just moves the problem from one threshold to another), but that's not what I am interested in here.

What I would like to know is:

• Are there any arguments in favor of using a discrete grading scheme like the one above?
• Or is just a product of history and the difficulty to coordinate practices across the vast number of U.S. colleges?

In other words, is anyone claiming that the discrete scale has virtues of its own other than the fact that it is used in so many places – creating comparability –, and that it would be hard to coordinate a change at all those places at once?

Answer 1

I am in favor of using the US discrete grade system. Some thoughts I will address are as follows:

As the OP mentions, in the US, the ABC grade system is relatively standardized and well understood by a large variety of institutions. It makes comparisons among students from a variety of schools more compatible (although, yes, not perfectly comparable).

Beyond this, I feel that the ABC system delivers the appropriate amount of granularity. After all, you have to truncate grade percentages somewhere. In some sense every system is discrete.

One method that I have used for my students in math classes in the US is a clustering system guided by the grade cutoffs. The percentages are guarantees. After that, I cluster students into grade groups, with students who barely missed the cutoff still usually getting adjusted up. For example, let's say that we have the following percentages:

93.23%, 93.17%, 92.88%, 92.81%, 92.08%

I would give the first four students A's and the final student an A- under normal circumstances. (It is usually more nuanced than that). I feel that the grades I gave my students usually very accurately described their proficiency relative to the stated outcomes of the course. This was also vastly true when I was a student. In fact there were times where I would prefer to take my A grade than my raw percentage. An 'A' sounds much better than "I got 83% raw, but I was the second highest grade in the course."

In my opinion, the ABC system actually is a stronger system than a raw percentage system. Is a 95% in Dr. Haskin's Calculus I class better than a 97% in Dr. Bodrel's Calculus I course? Both would result in an A grade. Or how would a 97% in Calculus I compare to a 97% in Floral Design? By using the ABC system, we effectively run a sort of "low-pass" filter on the grades. In my opinion this actually allows for a fairer comparison between students.

Let me also add that having a raw percentage system would be awful from the perspective of an instructor. Students already anguish about their grades. Now picture if they had to anguish about not just getting an A, but getting 100%. You would end up having students with 98% in the course worrying about getting 99%. It just becomes excessively fiddly. As a former student myself, I would actually hate a raw system. I was grateful for being able to get an A with 94% in a class without having to worry about pushing it up to a 95%.

I will close by saying that school and grades are not always fair. Such is life. No matter what system you use, there will be drawbacks from one perspective or another. There is no system that is perfect.

---

Addendum: Several comments have mentioned thoughts on being graded relative to others. Note first that my proposed method never moves a student down a grade, only up. Several comments seem to overlook that.

@BartoszKP: "You are grading not 'their proficiency relative to the stated outcomes,' but 'their proficiency relative to the stated outcomes and relative to other students.' Did other students doing well make this student suddenly know less?"

How could raising a grade indicate a student "knew less"? It seems that raising a grade would actually indicate a student knew more.

I guess I could grade students solely on their raw scores, 100ths of a percent be damned. Then students would be graded with less relativity to their peers. But wasn't such a "zero-tolerance of rounding up" system one of the named drawbacks of the ABC system? It seems like we are trying to play both ends of the flute here. I can see that it would be patently unfair if I rounded students down a grade. I must say though, I have never had a student complain about being graded "relative" to their peers when I rounded their grade slightly up.

Secondly, to continue the thoughts on being graded relative to one's peers, we need to keep in mind that it is next to impossible to provide students from current semesters the identical exam experience as students from previous semesters encountered. By adjusting scores within semester, this can help control for slight variations in exam difficulty from semester to semester. Note that I do not always have complete control over exams, since they often must be written on teams with other professors. All told, which system seems more "fair," a system that does not control at all for variability in exam writing styles or a system that attempts to account for that?

Answer 2

This creates understandable frustrations from students who "nearly made it".

It’s worth keeping in mind that life outcomes are also discrete, and this also causes students (and everyone else) similar frustrations: you either get the fellowship/top grad school admission/job/promotion/whatever, or you don’t. So students would still experience the same frustrations even with continuous grades, and even under the completely unrealistic assumption that their grades are a perfectly accurate tool for measuring their level of knowledge, which obviously they aren’t.

I’m not a huge fan of the US grading system myself, but honestly I don’t think it matters very much whether grading is done on a continuous or discrete scale. Grades are a statistical tool and should be interpreted as such - when averaged over many scores they have some (limited) meaning, but any one individual grade doesn’t necessary mean very much.

Answer 3

For the overwhelming majority of classes, grading is at least partially subjective. In the case of essay-based classes, this is immediately obvious. Likewise for art classes. Even for math and hard sciences, though, there's an element where graders need to make their best judgements. When a student solves an equation incorrectly, but their only mistake was dropping a sign deep in the weeds of an entire page of algebra, do you take off 1% or 2% of partial credit? If a lab report didn't have the expected accuracy in its data but was within a single significant digit, how badly do you ding the student?

As a result, the exact, four-digit-long raw percentage grade you get in a class gives an impression of being an extremely precise measurement, but in reality, it isn't. Depending on how the graders happened to be feeling and thinking when they looked at your projects and tests, your grade could plausibly have gone up or down a whole percentage point or two with you submitting exactly the same work yourself. Assuming your raw grade is perfectly accurate is very similar to reading that the Earth is 25 thousand miles in circumference and assuming that number is accurate to the inch.

Because of this, I would argue that the discrete letter-based grading system is actually a more reliable measure of your performance, one that does not overplay its accuracy. The distance between partial letter grades is about the same magnitude as the margin of error I would give to raw percentage grades. So after your raw grade is converted to a letter grade, you can be confident that the grade you received is very highly likely the grade you deserve, even accounting for the subjectivity, fuzziness, and slight randomness of assigning percentage grades.

Answer 4

Consider the perspective of a transcript-reader. They're interested in scanning a school transcript with records of maybe 30 or 40 classes (for a bachelor's degree), with maybe half of those in the major, as quickly and efficiently as possible. The fact that there are a small number of different marks is a help in parsing the record.

1. The A-F scheme is just a single glyph per class, instead of the two-glyphs (or three) for a 0 - 100 system.
2. The granularity of letter grades is entirely useful to the reader; it is doubtful that the reader cares about the difference between 85% and 86%, and so that ones-place digit is often just wasted space and eye-strain.
3. The A-F system seems compatible with "The Magical Number Seven, Plus or Minus Two" rule, which says psychologically people can juggle and assess around 5-9 discrete things in short-term memory at once.

Furthermore, many non-STEM classes have, by their nature, qualitative grading on assignments, performances, etc. Consider, e.g., the policies listed for Grading at Yale. The first possible protocol listed is "Letter grades for all assignments"; the second protocol listed is "Numerical grades on all assignments", and it asserts that this is the standard only for STEM courses. So in the first (possibly more common) case, presenting a number at the end of the course with two-digit precision would be vacuous.

Answer 5

There are a number of issues here. One, however, is that the table you give doesn't seem accurate to me - or at least, I'm pretty sure it isn't used by most US colleges.

Second is that your assertion that a student can "miss" a higher grade by an insignificant amount, while true in theory, is less true in practice. I never let that happen, for example, always giving students the benefit if a point total for the course was insignificantly different from that of another student who got the higher grade.

Third, it is a tradition that A means "finest kind", B means "good but not finest kind", C means "acceptable", and D means "needs improvement". Yes those are bins but I don't think I can make a statement that a person with 3.22 is in any way better than a person with 3.20. The means by which I assign grades are not that fine grained. The grade is an aggregate, not an absolutely precise measure. So I can't even distinguish that finely between students.

Fourth, if we compare different courses, even by the same professor, but in particular by different professors in possibly different fields, does a 3.22 mean the same thing. Is a 3.22 in a basic philosophy course exactly the same as a 3.22 in an advanced math course. It would be foolish, IMO, to assert that they were the same.

All of the above indicates that indeed, the grades are just bins. And the bins are accurate enough that, for example, an employer or graduate school admissions system can make valid judgements about the prospects of a student. There are exceptions, of course, but in the main, simple bins represent reality better than giving a precise measure to something that isn't.

Don't confuse accuracy with precision.

---

Also note a psychological effect of "bin grading". If a student realizes that with just a bit more effort they can raise their grade from B+ to A-, they may just be wiling to put in the effort. That won't occur with grades that are purely numerical.

Answer 6

half of a percent can make a significant difference in a student's life and career.

Here is where you are wrong. The difference between an A- and a B+, or an A+ or an A, etc will never make a difference in your life.

This is because:

1. That variation will regress towards the mean in any students career

2. GPA has arguably the lowest ratio of actual_importance / given_importance of anything you have ever encountered in your life. It is nearly entirely irrelevant once you no longer have to apply to colleges.

So why do discrete letter grades exist? Because they are easy to understand and compare, but most importantly because it doesn't matter. Grades do not define who you are. They are an inaccurate and minor indicator of what you could become, and useful for binning students into their colleges, no more no less.

Answer 7

I actually think the letter-grade system is very good. First, it makes the grades comparative in the sense that - in principle at least - the top student of a class can get an A+ even if the numerical grade is not >95. Second, while students may get unlucky in some courses and just miss threshold, they will get luck in other courses and just be above threshold: it tends to average out over the time of the degree.

Most important in my mind is that the binning system captures the reality that numerical marks are overly granular: there is no material difference in competence level between a student with 89% and one with 90%. The marking scheme is not that accurate and the difference in marks could be due to some trivial factor, such as fatigue on the part of the marker, or bias such as having marked a number of very good copies prior to marking a slightly less good copy, etc.

Students are IMO overly fascinated by numerical grades, and this binning - in particular the ability to assign A+ irrespective of the actual numerical grade - is a good overall way to assess students within a peer group.

Answer 8

Just to briefly reiterate other answers:

The information content (in an information-theory sense) of "precise numerical grades" is much lower than it might appear, even apart from complete lack of information about the purported meaning of numbers at a given level.

Apparently many people believe that "letter grades" are "absolute", and resolve (!?) the ambiguities and context-dependence of numerical grades. Of course, this is not true in the first place, and, in the second, there is grade inflation, and so on.

But/and there are virtues to discretizing, as also observed, because it can reduce fixation on grades ... to some degree. Sure, edge cases are still an issue, but without discretization everything is an edge case.

Answer 9

All grades are discrete. I've never had a teacher use an irrational number for a grade. I also have never had a professor use more than ~5000 or so discrete "points" in a grade. Said another way all possible grades in a course can be uniquely described by the numbers 0-5000.

The problem is 3392 of 5000 captures absolutely nothing about a student's knowledge, skill or level of effort (why we care about effort when grading is a misery to me, but that's not important here).

Grades represent two or arguably one thing. First they represent how a student has done (in the graders opinion) against other students. Secondly (and far less importantly) grades represent if a student can accomplish the material in the future(the link between academic and "real life" performance is tenuous at best).

So given the goals and problems of grading all we can really do is separate students into categories and let the chips fall where they may. Humans are really bad at dealing with more than 10 or so categories/bins. So we created a system with ~10 bins and assign students to those bins.

Answer 10

Short verion:

Most of the answers here discuss the ABCDF grading scheme in the context of some numerical or percentage grading scheme. One strength of the ABCDF scheme is that it enables grading schemes decoupled from numerical score and instead focus on mastery of specific key content. Because the ABCDF schene is based on a commonly understood philosophy of student achievement, it allows comparison of student grade outcomes in courses that use a numerical or percentage grade basis to those that may use mastery, specifications, outcomes, or other non-numerical basis.

Consider a course with five learning objectives, each covered by an exam. If an instructor set a C as a 70% exam average, a student could achieve this 70% by a number of ways on a spectrum in between the following extremes:

• Earning an average of 87.5% on four exams and earning a 0% on the last one (say by earning 100%, 100%, 100%, 50%, and 0%) ,and
• Earning 70% on all five exams.

The former extreme could be trouble if the student earned a zero on the exam for a learning objective that is critical for success in the subsequent course. In fact, a student could easily earn a B despite failing an exam. This student would feel confident in their preparation for the next course, but might be no better prepared than a C or D student.

For example consider a General Chemistry I course that covers the following key topics: atomic structure, molecular structure, solutions, stiochiometry, and therochemistry. All five of these topics are foundational for many other chemistry courses. What if a chemistry department determines the strongest predictor of a student's grade in General Chemistry II is not their course grade in General Chemistry I, but their grades on the stoichiometry and thermochemistry exams. Their grades on the other three exams are not strong predictors. That is, a student who earns a C in Gen Chem I but does well on the stoichiometry and thermochemistry exams is more likely to earn an A in Gen Chem II than a B student who bombed the thermochemistry exam.

The ABCDF scheme allows a General Chemistry I instructor to set a grading scheme to require passing the exams on stoichiometry and themrochemistry in order to earn a grade of C or higher and qualify for General Chemistry II:

• Grade of A - Pass the exams for all five learning objectives (scoring 80% or higher is a pass), and pass the advanced project.
• Grade of B - Pass the exams for all five learning objectives (scoring 80% or higher is a pass) without completing the advance project; OR meet the criteria for the Grade of C AND pass the advanced project.
• Grade of C - Pass exams for three or four learning goals (scoring 80% or higher is a pass): stoichiometry, thermochemistry, and any other goal.
• Grade of D - Pass exams for at least three learning goals (scoring 80% or higher is a pass), but fail to meet the criteria for the Grade of C; In other words, you failed the exam for stoichiometry, thermochemistry, or both.
• Grade of F - Pass exams for fewer than three learning goals (scoring 80% or higher is a pass).

If the instructor allows students to earn the chance to retake an exam they almost passed, say by crossing some threshold on turning in homework or by completing a certain number of optional review quizzes, we virtually eliminate the "knife edge" issue. In my case there will never be a student who has 92.7% and gets an A- while a student with 93.1% gets an A. Students pass the exams or they don't. Students pass the advanced project or they don't. This specifications-style of grading emulates life to a certain degree. On the job, an employee completes a project by the deadline, or they don't.

However, even if I use a grading scheme like this, I am still basing my grading philosophy on the common understanding that the grade of A is reserved for the highest achieving students, the grade of B is reserved for students who have mastered most of the courses, the grade of C is reserved for students who have learned enough in my course to be successful in the next course, the grade of D is reserved for students who have learned something in my course but not enough to be successful in the next course, and the grade of F is reserved for students for whom I cannot be certain they learned anything in my course. This philosophy makes my A comparable to another instructor's A despite the radically different means of determining that A.

In a system where I was required to note a student's grade using a numeric grade of some kind, be it percent or percentile, I could never run a grading scheme like this one. And I like this scheme better because it forces my students to focus on learning and mastering key material instead of playing a mathematical game with a point-based system.

Answer 11

The 0–100 grading system is problematic in several regards.

To begin with, let’s assume the true knowledge level of a student can be meaningfully represented by a number in [0, 1], and that the grades are professors’ best efforts to estimate that figure. One professor’s function will end up being convex upward while another one’s would be downward, one will convert 0.5 to 40% while the other would grade the same student at 60%.

Not that it even makes sense for professors to try to define such a function in their minds. If the problem was solved correctly but with an inefficient method that suggests the student failed to master a more appropriate approach, is that worth 100%? 90%? 80%? 84.2%? What weights to apply to each of thousands of possible factors? Not to mention the grading will likely drift as the professor goes through students’s papers because it’s so vague.

Additionally there are some perverse incentives, such as the professor feeling free to deduct a point here and 5 points there for things that annoy that professor personally, where most would have refrained from turning an A into a B on petty grounds.

A discrete system can be formalized much easier. Like math olympiads: you solved it correctly? You get a 7, no variance.

Professionals in most fields take pride in their ability to estimate parameters of complex things, but study after study (I think Kahneman quoted some in his famous Thinking, Fast and Slow) shows that many such professionals are consistently outperformed by simplest methodologies akin to the APGAR score.

Answer 12

I am not an expert on the US system, but the UK system used to do that as well, and it sort of lives on in degree classifications or "honours" (first, 2:1, 2:2, third).

The idea as I've always understood it is that

• An A shows that the student has understood the subject material clearly beyond what was taught in the course.
• A B shows that the student has achieved what we set out to teach, no more and no less a.k.a. "ticked all the boxes".
• A C shows that the student has understood basic concepts, but hasn't got as far as we'd hoped.

So the discrete grade bands are based on qualitatively different understanding of the subject material. I've also been told by one of our industrial partners that there were engineers who you could give a problem to and they'd solve it, and engineers who'd come to you with a fully worked out solution before you even realised (as project manager) that there was a problem somewhere - and that this distinction correlated remarkably well with getting a second or first class degree.

So my answer would be that discrete bins do make sense if they're used to encode qualitative information, like which students will go beyond what you teach them without much prompting.

Another aspect of the UK system is that we're explicitly told (at least at my university) not to mark on a curve: the proportion of firsts / "As" should only be determined by the quality of the work, and can (and does) vary quite a bit from year to year.

That said, several employers that I've written references for have an online form where I have to select whether the student I'm writing the reference for is in the

• top 1% of their cohort
• top 5% of their cohort
• top 20% of their cohort
• top 50% of their cohort
• not in the top 50% of their cohort

and you get the distinct feeling that they wouldn't ask this if they didn't use it somehow to determine who to invite for an interview or not. So you can try as a university to not "grade on a curve" but employers will react to that.

Discrete bins can help giving information beyond your rank in a cohort, especially with small cohorts where the meaning of a particular rank changes from year to year, if they are coupled to particular competence standards and you're allowed to vary the percentage of "A" grades you give out each year.

Answer 13

I think it depends partly on the nature of the class. As a (former) teacher of electronics, I could argue that ABCDF is too granular:

• Can student identify a clipped signal or not?
• Can student measure the voltage on a test point or not?

But an art class would be very subjective.