Court Rules: 2.96 is Less Than 3.0

An excerpt from plaintiff's argument, which the court rejected.

 

From Penzo v. South College of Tenn., Inc., decided in 2023 by Judge Travis McDonough (E.D. Tenn.), but just posted on Westlaw:

South College requires its physician-assistant-program students to maintain a grade point average ("GPA") of at least 3.0 to remain in the program. This requirement and the consequences for failing to meet it are explained as follows in the South College Master of Health Science Physician Assistant Program Student Handbook ("Handbook"):

A student with a cumulative GPA below 3.0 at the end of any didactic quarter or clinical rotation will receive notification from the Student Progress Committee that they are on Probation. If the student fails to raise his/her cumulative GPA to 3.0 or higher at the end of the next didactic quarter or clinical rotation, he/she will be dismissed from the Physician Assistant Studies Program. If the student raises his/her cumulative GPA to 3.0 or higher at the end of the next didactic quarter or clinical rotation, he/she will be removed from probation. If a student's cumulative GPA falls below 3.0 for a second time, he/she will be dismissed from the Physician Assistant Studies Program….

After three quarters in the physician-assistant program, Penzo's cumulative GPA fell to a 2.96. In accordance with the Handbook, South College placed Penzo on academic probation and advised him that he would be dismissed if he could not raise his cumulative GPA to at least a 3.0 by the end of the next quarter. Because Penzo was unable to do so, he was dismissed for the first time from the program after completing his fourth quarter.

Penzo, however, successfully appealed his dismissal, attributing his underperformance to mental-health difficulties, and was allowed to return to the program for his fifth quarter. Though Penzo earned a 3.14 the next quarter, his cumulative GPA remained below 3.0 at a 2.97. As a result, South College again dismissed Penzo from the program, leaving him with "$140,000 in debt and [ ] no degree." Penzo appealed his dismissal a second time [but lost]….

Penzo's lawyer argued:

In regard to the termination of Mr. Perez's enrollment in the PA Program, the Handbook provides that students may proceed to the clinical phase if they have "a minimum cumulative GPA of 3.0." [The motion used Perez as a pseudonym for Penzo, but the district court refused to allow pseudonymity in this case, and thus used Penzo's actual name. -EV] In fact, all references in the Handbook to a student's minimum cumulative average present the minimum as a single-place decimal number, i.e., 3.0 rather than 3.00, and the only reference to a two-place decimal, i.e., 3.00, is on page 34 and is in reference to a course grade rather than a cumulative average. Moreover, the Handbook indicates via a chart on page 34 that South College applies standard decimal-rounding rules by showing that the letter grade "A" is applied for scores of 89.50% and higher, whereas 79.50% to 89.49% constitutes a "B" letter grade. Therefore, because Mr. Perez's cumulative average at the end of his fifth quarter was 2.97, when converted to a single-place decimal, 2.97 becomes 3.0

Points for creativity, I think. But not for legal success, held the judge:

It is unlikely South College reasonably expected students to understand the 3.0 requirement to be a rounded cumulative GPA (i.e., that a 2.95 GPA and above would meet the 3.0 GPA requirement) rather than a hard cutoff. The Handbook clearly states probation or dismissal is warranted "[i]f a student's cumulative GPA falls below a 3.0." There are no qualifiers or caveats modifying this statement. Students' cumulative GPAs are calculated to the hundredth's place; indeed, Penzo invariably refers to his cumulative GPAs with two decimal places. If that number is under 3.0, per the unequivocal terms of the Handbook, the student is subject to probation or dismissal. It is not reasonable to believe otherwise.

The Handbook, as Penzo points out, contains a chart suggesting grades awarded within individual courses may be rounded up (i.e., a grade of 89.5% constitutes an "A"). But there is no reference to rounding or any adjustment of cumulative GPAs in determining whether a student has met the 3.0 GPA requirement. Accordingly, Penzo's likelihood of success on this breach-of-contract argument is low and weighs against granting temporary relief….

Comments

[SRG2]

It's kind of the opposite of the Woronin problem. Marion Woronin, a Polish sprinter, once ran 9.997 in the 100m. This would make him the first white man to run sub 10s for the 100m, (This is still a rare event, relatively). However, IAAF (now WA) rules require that times are rounded up to the nearest 100th (always up, never down), and so his official time was 10.00s. Did he break 10s, then? And the answer is, factually, yes, officially, no.

Detouring further, when Thrust SSC broke the sound barrier, it was discussed whether if it failed to make the second run in the opposite direction required for official timings, would the first run count? And the answer - provided by an American competitor - was that it was a factual matter that Thrust SSC broke the sound barrier, regardless of whether it set an official record in doing so.

The college requirement of 3.0 was ambiguous because absent further context, 2.97 is 3.0 - that is, 2.97 is one of the set of numbers that round to 3.0, But the college provided enough evidence of context for the court to find that 3.0 meant 3.00. Factually, the kid scored 3.0. But contextually, the requirement was 3.00, and both factually and officially, the kid scored 2.97,

[Absaroka]

"The college requirement of 3.0 was ambiguous because absent further context, 2.97 is 3.0 - that is, 2.97 is one of the set of numbers that round to 3.0"

FWIW, I hold the opposite view. "Absent further context", to me, means we're talking a real number, and 3.0==3.00000000000, so 2.999999999999<3.0.

An example of context would be a transaction using physical currency where you are going to round to the nearest cent (or currently, it seems up or down, whichever favors the customer, if the till is out of pennies).

Might I ask what your math background is? TBC, I'm not suggesting your view is invalid if you don't have a math degree - it's just really interesting how people come to different views. FWIW, I have a stats degree, w/ extra math - IIRC one class short of a math minor or somesuch.

(as an aside, I have never quite understood '89.5 and up is an A'. It must be disappointing to get a B with an 89.6, but rounding up just moves the disappointment down to 89.499999.)

[SRG2]

My maths background is an A-level in Mathematics with Statistics - which is equivalent to 2nd year of a maths degree in the US, I think, It's hard to make direct comparisons, You can find the current syllabus here, and it's not too different from what I remember from what I did those many years ago.

https://www.cambridgeinternational.org/Images/597421-2023-2025-syllabus.pdf

We didn't do the mechanics component, fwiw.

[Absaroka]

Fascinating ... the 'is one of the set of numbers' made me suspect a math background. I'm curious, if you want to sketch it out, why you don't view this as a real number, infinite precision thing. In your experience, 3.0 and 3.00 imply rounding/precision?

That's the case, I think, in blueprints, where specifying a dimension as 1.0, 1.00, and 1.000 imply very different tolerances to the machinist.

[Stupid Government Tricks]

My non-math non-legal instinct says 3.0 in the handbook does imply 2.97 rounds to 3.0, but it's such a close call that it really depends on past practice; have they allowed previous 2.97 students to continue?

It doesn't matter in the big picture; if his GPA for the last 3 out of 5 quarters is below the cutoff, further coursework is likely to be even harder, and he should have taken the hint of that first 2.96 quarter and quit instead of running his student loan debt up to $140,000.

One reason for a GPA cutoff is to avoid saddling the other students with a poor performer. I've been in classes on both ends: with poor students who dragged the class discussion down, and with better students who I was holding back.

[SRG2]

I retained - and still retain - an interest in maths long after I stopped studying it academically. But also, I did New Maths from 12 to 15 and sets were part of that.

There's also the point that the college had the option of writing 3, or 3.00, and they did not, hence 3.0 was what they meant. If they had written "A GPA of 3 or above is required", 2.97 would not qualify. But the moment they bring in a decimal place, they open themselves up to a challenge that the actual GPA should be to the same precision as the hurdle.

[mse326]

3.0 wouldn't equal 3.000000000... in most science cases. BS in Chemistry and we always dealt with significant figures. 3.0 is two significant figures so any result we got would be rounded to two significant figures, thus 2.97 would indeed be 3.0.

I'm not suggesting that the handbook must be read this way, but it is certainly a reasonable argument to make. However, I do agree with the court that the handbook is likely using absolute terms.

[Rossami]

I could counter that if "3.0" was intended to mean the real number 3.00000..., then why wasn't "3" sufficient to also mean 3.00000....? You've rendered the ".0" part of the rule into surplusage. The choice to include a significant digit into your rule implies the level of rounding appropriate to the task.

And before you ask, my initial training was in physics/engineering rather than pure math, thus my bias to treat this as a measurement problem.

[jb]

This is my take as well. I can see arguments on both sides of the debate*, but the fact that they chose 3.0 as opposed to 3 or 3.00 seems at least plausibly meaningful. I imagine even the 2.97 and 2.96 values are actually rounded from some longer and possibly non-terminating decimal numbers, so why did the university decide that it's three significant digits that they care about as opposed to two? It seems likely that the university computer would round a value like 2.9999997 up to 3.00, even though it's technically lower by their argument.

* And not for nothing, if the policy is at all ambiguous, the university should have given the kid a break.

[A Japanese Student]

There was a similar case in Japan, in the context of patent infringement.

Claim 1 of the patent JP4974971 read in relevant part, "comprising an ultraviolet absorber having molecular weight of not less than 700". So what did the defendant do? Use a compound, formula C42H57N3O6, whose molecular weight is 699.9.

Intellectual Property High Court found no infringement, finding the plaintiff's equivalence argument to be barred by filewrapper estoppel.

[ah....Clem]

I'm surprised nobody tried to analyze it in terms of the 95% confidence level that is so prevalent in science publications.

[MollyGodiva]

GPA is calculated numerically without error, so the 95% confidence level with rounding to three sig figs is the range between 2.965 and 2.974.

[Area Man]

Haven't y'all heard? Math is racist; that's why 2 + 2 can equal 5. So are tests, grades, and GPAs. He should have just been taught the information and thereafter been trusted to perform.

[MollyGodiva]

That is a very MAGA perspective.

[Area Man]

It is? I thought it was a very woke perspective. The far left have been the ones pushing standard-lowering policies for the past decade (at least) in the name of fighting racism.

[MollyGodiva]

MAGAs are the ones who reject the idea of objective facts.

[Rossami]

The projection is strong with this one.

[Stupid Government Tricks]

Lack of a sense of humor is a very statist perspective.

[Dr. Ed 2]

My take on this is quite different, they had no problem, taking the money from a student who they reasonably knew wouldn’t succeed, and at the very least ought to be required to refund the fifth semester’s tuition and costs.

We need to start viewing education the way we view other transactions, with a vendor is only paid for the successful completion of the project. The semesters of education have no individual value, their value only consist in the context of the completed degree hence billing if a half of an education is like billing for half of a car, and that would be fraud.

[Stupid Government Tricks]

It was his choice to continue in the face of his poor GPA. His signature on those loan documents. His gamble. This is not the same as entrance tests being dumbed down for select students, or schools lying to prospective students about their graduation prospects or employment prospects after graduation. He came very close; this is not the case of the school accepting a piss-poor student and lying about his prospects.

[David Nieporent]

the way we view other transactions, with a vendor is only paid for the successful completion of the project.

In what back-assward place do "we" view other transactions that way? Vendors are paid for delivering what they've been paid to deliver. Home Depot doesn't give you a refund for the tools & materials they've sold you if the deck you've built with said t&m falls down. The car dealer doesn't give you a refund if you flunk your driving test.

[John 4]

If you bought a plane ticket and didn't make preparations to arrive in time to board the flight, are you entitled to a refund?

[MollyGodiva]

The court is right on this. The college is wrong to have a GPA cut off based on cumulative GPA, which is harder to raise with each passing semester. It should be based on semester GPA. It is not right to require a student to do better in school to stay enrolled and shackle them to their past GPA.

[John F. Carr]

I have seen similar arguments related to breathalyzer tests. If the machine said .08% on the display but the real measurement was 0.078%, is the defendant guilty? I forget whether the courts ended up going with math rules or traffic court rules.

[Dr. Ed 2]

Recollection is that the statute states “ in excess of”….

[twobaylorbears]

Seems like the court got it wrong. I think if you say "less than 3," it would be fair to think of it as "to the left or right of 3 on a number line." But the moment you say 3.0, you're talking about two significant digits, and 2.95 -> 3.0.

[Brophy]

The distinction you are making between a purely mathematical "less than 3" and the scientific "3.0" is an insightful way to look at the difference between exact values and measured values. In a pure mathematical sense, any number to the left of 3 on a number line is less than 3, even if it is 2.99999 followed by an infinite string of nines. This is a binary condition where a number either is or is not smaller than that specific point in space. However, as you noted, the moment you add a decimal point and trailing zeros, you change the conversation from abstract numbers to measurements and significant figures.

When you write 3.0, you are communicating the precision of your measurement. This value represents a range rather than a single point. In the world of significant digits, 3.0 implies that the value is known to the tenths place, meaning the true value could actually be anywhere from 2.95 to 3.04. Because of rounding rules, anything in that window effectively becomes 3.0. Therefore, while 2.95 is mathematically "less than 3," in a laboratory or engineering context, it is often treated as "the same as 3.0" because it falls within the uncertainty of that first decimal place.

This creates a fascinating overlap where 2.95 occupies two roles simultaneously. It is strictly less than the integer 3, but it is also a valid representation of the measurement 3.0. If you were told to keep a temperature "less than 3.0 degrees," a reading of 2.95 might actually be considered a failure to stay below the limit because, within the precision of your instruments, 2.95 rounds up and "is" 3.0. This highlights why significant figures are so vital in science; they tell us how much we can trust the "less than" or "greater than" relationship when dealing with real-world data.

[TwelveInchPianist]

We're talking about discrete units here, not continuous measurements that can be made with a limited about of precision, so I don't think the rule that 3.0 is 2.95-3.04 applies.

[TwelveInchPianist]

In any event, I don't think the rules apply to specifications, because there's no imprecision in a specification, it's not a measurement.

[Rossami]

The grades may have been discrete units but the average of those grades is no longer discrete.

And unless the rule is articulated otherwise, the precision used in the specification implies the rounding to be used in the measurement. Thus, the question is not whether the specification is 2.95-3.04 but whether the measurement of grades received, when calculated in accordance with the specification, arguably including the implied rounding rule of the significant digits, meets the specification.

[Incunabulum]

Which is why you would use significant digits and not rounding.

2.97 - but you're only going to the tenths in precision so you drop the .07, because you don't know how accurate/precise it is because it exceeds the limits of your measurement.

[jb]

Huh? Why would you always round down? At the extremes, your approach would say 1.9999999 is the same as 1, even though it's basically double that and only very slightly different from 2.

You should indeed drop digits that exceed your precision but you do that by rounding to the nearest digit you have precision to support.

[Rossami]

That is not how significant digits works.

[TwelveInchPianist]

"And unless the rule is articulated otherwise, the precision used in the specification implies the rounding to be used in the measurement."

In my experience the precision available in the measurement determined the rounding used in the measurement.

[Stupid Government Tricks]

It certainly does; the handbook itself rounds 89.50 up to A and 89.49 down to B.

[Rossami]

Not applicable. That part of the handbook is defining the rule for converting class performance on tests, homeworks, etc into the individual letter grade for that class. Your GPA is created by averaging those letter grades. The rule for calculating a GPA does not have to be (and in this case, isn't) identical to the rule for generating the underlying grades.

You do inspire an interesting question, though. Should GPAs be calculated based on that average of averages or should the original detailed scores be passed through and only converted into categories (letter grades) at the end? Everything we know about measurement theory and data analysis say that we should keep the data in its raw form until the last possible step. The only good counter-argument is that it would be far more complex - and complexity is often perceived as unfair even when the answer it produces is objectively better.

[Stupid Government Tricks]

Your applicability test is a quibble. They are both grades, one being an average of the others.

[O Tempora]

I would have loved to have written the opinion. Just this: “2.96 < 3.0.”

[arbitrary aardvark]

The court was wrong. I'm a literalist. We should go by the text of what the parties agreed, not what we think they meant to say.
textually, he was at 3.0, although not 3.00, and 3.0 is what they had agreed to. a text is construed against the drafter if ambiguous, but there's no ambiguity here.

Once upon a time, I had a job grading state standardized tests, which determined if the kids would graduate. The official answer to the question I was working on was 36.4, but about 5% of the kids got the correct answer, 36, because of the way the question was worded.i was instructed to give those answers a zero. I either quit or was fired when I wouldn't.

[Dr. Ed 2]

This is why we should demand that old test questions be published.

[Stupid Government Tricks]

I wrote this elsewhere about how hard it is to be literal about this rounding confusion.
===============
When someone says "the 1910s" they can only mean 1910-1919. But that's actually shorthand for "00:00 on Jan 1st 1910 to "23:59:59 on Dec 31st 1919", whereas with house numbers, which in the US are mostly even and odd on separate sides of a street, does 1910-1919 include both sides of the street while 1910-1918 would only cover one side? "The 1900 block" is perfectly clear ... except for not knowing about corner lots and whether it includes both sides of the street. I had to deal with calculating sales taxes for several years, and was surprised to find city limits running down the middle of the street, so even/odd made a difference in what tax was calculated. When a speed limit sign is in the middle of a block, where is the actual boundary, and is that where you can get a ticket or just where you have to start slowing down?

Then there's the question of what "the 1900s" means; is it 10 years or 100? Is "the 2000s" 10, 100, or 1000 years? You have to go by context. I've tried writing such intervals as "the 190x" or "the 19xx" and that just looks silly.

I noticed a lot of legal documents which use 00:01 and 23:59 instead of 00:00 and 24:00, and a lawyer told me that's because some people think because 00:00 and 24:00 are both midnight, they are the same instant, not 24 hours apart. Are people really that stupid? I wonder how much is just lawyers looking for more quibbles to pad their billable hours. Then I wonder what happens in the first and last 60 seconds of a day, or whether that would be 59 seconds.

That's why I think context matters more; too many people think 3, 3.0, and 3.00000 are the same number. I've met doctors who seem to be unaware of basic mathematics is similar ways. I don't have any better expectations from medical students.

What is the school's history in similar situations? I generally despise common law precedent as locking in bad decisions, but I don't classify this as legal precedent, just the history of how the school and its students have treated similar situations.

[Incunabulum]

You're not wrong but nobody uses 2400 though. 2359:59.xxxx is the end of the day.

0000 is the start of the next day. This is a standard worldwide.

You could certainly use 2400 as the end of the current day and then roll right into 0000:00.xxxxxxxxxxxx. Nothing wrong with that. But I've never seen a place that does it that way.

What would be wrong is using 0000 *and* 2400 - its redundant *and* its confusing. In reality midnight is the end of the old day and the start of the new day, conceptually in timekeeping you are tracking the moment that you switch from one day to the next.

[Stupid Government Tricks]

No. It is not wrong, it is not unused, and using both 0000 and 2400 is not redundant, because they are 24 hours apart. All you are doing is proving that lawyer's contention that people are too stupid to distinguish 0000 and 2400 as meaning different midnights of the same day.

[Dr. Ed 2]

The 1910s were 1911-1920 — think 01 to 10, and why it was the 20th century and not the 19th, and why we’re currently in the 21st-century.

[Incunabulum]

Textually, he was 2.9 though.

[Brophy]

Yes, the court was wrong. I learned the rounding rules when I was in high school but the widespread ignorance of the people on this thread shows that a minority take chemistry or physics where these rules apply. Everyone else gets the wrong answer.

[Jmaie]

Reminds me of a discussion I had with my boss about review scores. We had weighted averages on a number of indicators, and the difference between an overall score of three and four meant a significant difference come bonus time. The first year I scored 3.65 and that was rounded down to three. The next year I scored 3.97, and that was rounded down to three. The year after that, I scored 4.18. That was when my boss told me he was only allowed to give a 4 to 10% of the group, and so 4.18 = 3…

[Stupid Government Tricks]

Grade inflation strikes again!

[Dr. Ed 2]

And the next year you were with a new employer?

[Incunabulum]

Penzo is assuming the school is required to 'round' it to a single decimal rather than 'significant figures'?

The school only resolves the GPA calculation to the ones place, the 1st decimal is still considered significant while the .07 is considered too imprecise and thus dropped.

Leaving a GPA of 2.9;)

[Rossami]

Again, no that's not how significant digits work.

Let me see if I can put this is simple enough terms. Let's say I want to measure the length of my shoe but I don't have any measuring device more precise than my thumb which I remember is about 1 inch from knuckle to tip. I get 11 inches. But I'm talking to a Brit who only understands metric. I know that 1 inch = 2.54 cm so that means my shoe is 27.94 cm long. But implying that I measured it to that level of precision is insane since I only used my thumb. The significant digits rule says I round it using conventional rounding rules (plus a special tie-breaking rule when the digit after the significant digit is exactly 5 - something not applicable here) to 28 cm. There is no discipline that uses an 'always round down to 27' rule.

[Incunabulum]

Honestly, the sort of person who would bring a law suit about this is exactly the sort of person who should not be a doctor. Or lawyer.

[Gasman]

"Honestly, the sort of person who would bring a law suit about this is exactly the sort of person who should not be a doctor. Or lawyer."
We can be certain that even if he graduated with honors, he would not be a doctor. Or a lawyer. A physician-assistant is a masters degree program that prepares one to perform various para-professional components of health care, which varies by state licensing laws, usually with requirement for close direction of a physician. PA's are similar to advance practice nurses in many ways. And neither is a physician.
His sunk cost of $140,000 for what might represent only half of his training time seems steep. Depending on how many more quarters would have remained in his education, he might be paying more for his masters level education than what a real doctor would pay. Seems like he was destined to be broke whether he finished or not.

[questioner7]

Court is wrong here. 3.0 is the hard cutoff, but 2.97 _is_ 3.0. It’s not 3.00, but that was not the cutoff; 3.0 was.