Ever wonder why U.S. students publish their GPAs (Grade Point Averages) on their resumes but the colleges and universities that administer such grades do not show their GPA distributions? I don't have a good explanation for why they hold their grade distributions so tightly.

But this year, we got a glimpse of what's happened to GPAs at the top U.S. colleges. Harvard (link) and Yale (link) are currently giving out 8 As for every 10 grades across all courses and departments. This relevation proves the saying that the hardest part of the Ivy League is to get in.

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Let's dissect the numbers further. The headline of the Yale report says that 80% of the issued grades are in the A range. Since Yale doesn't give out A+, these A range grades are either A or A-. Guess what proportion are As?

Three out of four A-range grades are As, meaning that 60% of all grades given out to Yale students are As. The median grade at Yale is therefore an A. It is more likely to get an A than any other grade if one attends a class at Yale. (Not to pick on Yale or Ivies because the same grading policy is at play at most of the top universities in the U.S. Indeed, administrators justify this policy using the "just doing what everyone else does" reasoning.)

Twenty-percent of all grades at Yale are A-. Ten percent are B+. And the remaining 10% are B or below.

For those not familiar with GPAs, it is the average of all grades received by a student. Yale students must take 36 courses to graduate so Yale's GPA is the average of 36 letter grades. An A grade is scored as 4.00, A- 3.70, B+ 3.30, B 3.00, etc.

Since Yale doesn't give out A+, in order to graduate with a perfect 4.00 GPA, a student must have 36 As in 36 courses. If the student even gets one A- (and 35 As), the GPA = (35*4.00 + 1*3.70)/36 = 3.99.

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The Yale report also disclosed that the average GPA of a Yale graduate in 2023 is 3.70. That's an average of A-.

How do we interpret such a number? If we assume every student is the same, i.e. they receive the same GPA at graduation, then all students get a 3.70 GPA.

Given that each student takes 36 courses, each student therefore receives 22 As, 7 A-s, 4 B+s and 3 B or lower. That means each student scores As in almost two-thirds of the courses taken.

There is a ceiling on the GPA scale. No one can score more than 4.00. Imagine pushing half the graduates from the average of 3.70 to the maximum of 4.00. For each such student, another student must shift in reverse from 3.70 to 3.40 -- in order to keep the overall average at 3.70. If we put a fulcrum at 3.70, and weights at 4.00, we must put counterweights at 3.40 to maintain balance. As a first approximation, we can imagine that all student GPAs fall within the 3.40-4.00 range.

As I explained in Chapter 1 of **Numbers Rule Your World** (link), the "average" is this curious abstract thing that doesn't exist. The above scenario is unrealistic for several reasons: not all students are created equal; also, grades tend to cluster, i.e. someone who gets an A in one class is more likely to also get As in other classes.

The information released by these schools is about grades in courses but what we're really interested in are GPAs for students. We don't have data at the student level but we can still learn quite a bit from the data that have been made public.

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Let's imagine the median Yale graduate. This person's GPA is higher than half the class, and also lower than half the class; that is to say, the person is ranked right in the middle.

Is it possible that the median Yale graduate has a 4.00 GPA? In other words, is it possible that half the class graduate with 4.00 GPAs?

As mentioned above, to obtain a 4.00 GPA, a student must score perfect As, 36 out of 36 courses.

Based on the published distribution of grades, it is indeed possible for half the class to get perfect 4.00 GPAs. In this scenario, the bottom half of the class should still has an average GPA of 3.40 (better than B+).

This scenario is quite amazing! Half of all graduating seniors would be valedictorians.

I'm not saying this is reality, merely saying that the published distribution of grades makes this scenario possible.

We can take this to the extreme. Imagine that the same subset of students soak up all As, roughly speaking, the class is divided into those with As, and those without As, then 60 percent of the students would be valedictorians with perfect 4.00 GPAs. What would happen to the other 40 percent? Their average GPA should still be 3.25, just below a B+.

We know that 60% of all grades given out are As. The total number of As is (36 x cohort size) x 60%. In this extreme scenario, we assume that 60% of the students are 4.00 students, getting 36 As each. The total number of As accounted for by this group is 36 x (cohort size x 60%). Thus, the remaining students would score no As and they would be given all the other grades (A-, B+, etc.). The ratio of these grades are 20%:20%:10%, rescaling to 40%:40%:20%. So 80% of their grades would be either A- or B+, which is why their average GPA should still be in the B+ range.

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Now, in reality, it's not easy to get straight As. Even if all else equal, the chance of scoring an A in a specific class is 60%, and the probability of scoring 36 As out of 36 classes is still very low, about 1 in 100 million (assuming independence between courses).

At most top U.S. colleges, there has been remarkable grade inflation over the past decades. In the 1940s, only 40% of grades at Yale were in the A or A- range. Let's say 30% of grades were As, about half the proportion of 2023. What was the chance of getting 36 As out of 36 courses in the 1940s? It's not 1 in 100 million, but 1 in 10,000,000,000,000,000,000 (10 million million millions).

So, when the proportion of As is doubled from 30% to 60%, the chance of getting 36 As explodes by 100,000,000,000 times.

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After running through this train of thought, I realize that I can in fact tease out the student-level grade distribution. I just have to make certain assumptions. Here is one attempt to model the situation at Yale:

Not surprisingly, the distribution is very tightly bunched, with half the graduates earning GPAs between 3.64 and 3.75. The median GPA is 3.70. The top 1% starts at 3.87 while the bottom 1% graduates with a 3.48 GPA.

In the next post, I'll describe the assumptions required to get a student-level GPA distribution from the course-level grade distribution.

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Going back to my question at the start. Since college administrators (and current students) all believe that the As are legitimately awarded to students of the highest calibre, one would think that they should be proud of the grading curve! For reasons unexplained, it is impossible to find grading distribution information (either at the course or the student level) for any institution, other than when occasional studies are commissioned to study this topic.

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