# Dances with GPAs

The trick to dancing with dragons is to never lose your grip on the tail.

If we are going to try and summarise a complicated, long-term process with a single number, and I don’t see such shortcuts going away anytime soon, then it helps to know:

• Exactly what the number represents.
• How it can be used.
• What the processes are that go into its construction.

We have conventions as to what things mean but, when we want to be precise, we have to be careful about our definition and our usage of the final value. As a simple example, one thing that often surprises people who are new to numerical analysis is that there is more than one way of calculating the average value of a group of numbers.

While average in colloquial language would usually mean that we take the sum of all of the numbers and divide them by their count, this is more formally referred to as the arithmetic mean. What we usually want from the average is some indication of what the typical value for this group would be. If you weigh ten bags of wheat and the average weight is 10 kilograms, then that’s what many people would expect the weight to be for future bags, unless there was clear early evidence of high variation (some 500g, some 20 kilograms, for example.)

But the mean is only one way to measure central tendency in a group of numbers. We can also measure the median, the number that separates the highest half of the data from the lowest, or the mode, the value that is the most frequently occurring value in the group.

(This doesn’t even get into the situation where we decide to aggregate the values in a different way.)

If you’ve got ten bags of wheat and nine have 10 kilograms in there, but one has only 5 kilograms, which of these ways of calculating the average is the one you want? The mode is 10kg but the mean is 9.5kg. If you tried to distribute the bags based on the expectation that everyone gets 9.5, you’re going to make nine people very happy and one person unhappy.

Most Grade Point Average calculations are based on a simple arithmetic mean of all available grades, with points allocated from 0 to an upper bound based on the grade performance. As a student adds more courses, these contributions are added to the calculation.

In yesterday’s post, I mused on letting students control which grades go into a GPA calculation and, to explore that, I now have to explain what I mean and why that would change things.

As it stands, because a GPA is an average across all courses, any lower grades will permanently drop the GPA contribution of any higher grades. If a student gets a 7 (A+ or High Distinction) for 71 of her courses and then a single 4 (a Passing grade) for one, her GPA will be 6.875. It can never return to 7. The clear performance band of this student is at the highest level, given that just under 99% of her marks are at the highest level, yet the inclusion of all grades means that a single underperformance, for whatever reason, in three years has cost her standing for those people who care about this figure.

My partner and I discussed some possible approaches to GPA that would be better and, by better, we mean approaches that encourage students to improve, that clearly show what the GPA figure means, and that are much fairer to the student. There are too many external factors contributing to resilience and high performance for me to be 100% comfortable with the questionable representation provided by the GPA.

Before we even think about student control over what is presented, we can easily think of several ways to make a GPA reflect what you have achieved, rather than what you have survived.

1. We could only count a percentage of the courses for each student. Even having 90% counted means that students who stumble a little once or twice do not have this permanently etched into a dragging grade.
2. We could allow a future attempt at a course with an improved to replace the previous grade. Before we get too caught up in the possibility of ‘gaming’, remember that students would have to pay for this (even if delayed) in most systems and it will add years to their degree. If a student can reach achievement level X in a course then it’s up to us to make sure that does correspond to the achievement level!
3. We could only count passes. Given that a student has to assemble sufficient passing grades to be awarded a degree, why then would we include the courses that do not count in a calculation of GPA?
4. We could use the mode and report the most common mark the student receives.
5. We could do away with it totally. (Not going to happen any time soon.)
6. We could pair the GPA with a statistical accompaniment that tells the viewer how indicative it is.

Options 1 and 2 are fairly straight-forward. Option 3 is interesting because it compresses the measurement band to a range of (in my system) 4-7 and this then implicitly recognises that GPA measures for students who graduate are more likely to be in this tighter range: we don’t actually have the degree of separation that we’d assume from a range of 0-7. Option 4 is an interesting way to think about the problem: which grade is the student most likely to achieve, across everything? Option 5 is there for completeness but that’s another post.

Option 6 introduces the idea that we stop GPA being a number and we carefully and accurately contextualise it. A student who receives all high distinctions in first semester still has a number of known hurdles to get over. The GPA of 7 that would be present now is not as clear an indicator of facility with the academic system as a GPA of 7 at the end of a degree, whichever other GPA adjustment systems are in play.

More evidence makes it clearer what is happening. If we can accompany a GPA (or similar measure) with evidence, then we are starting to make the process apparent and we make the number mean something. However, this also allows us to let students control what goes into their calculation, from the grades that they have, as a clear measure of the relevance of that measure can be associated.

But this doesn’t have to be a method of avoidance, this can be a useful focusing device. If a student did really well in, say, Software Engineering but struggled with an earlier, unrelated, stream, why can’t we construct a GPA for Software Engineering that clearly states the area of relevance and degree of information? Isn’t that actually what employers and people interested in SE want to know?

Handing over an academic transcript seems to allow anyone to do this but human cognitive biases are powerful, subtle and pervasive. It is harder for most humans to recognise positive progress in the areas that they are interested in, if there is evidence of less stellar performance elsewhere. I cite my usual non-academic example: Everyone thought Anthony La Paglia’s American accent was too fake until he stopped telling people he was Australian.

If we have to use numbers like this, then let us think carefully about what they mean and, if they don’t mean that much, then let’s either get rid of them or make them meaningful. These should, at a fundamental level, be useful to the students first, us second.

### 4 Comments on “Dances with GPAs”

1. lenandlar says:

This post has changed my views slightly on the GPA issue. We’ve a best student prize here and the winner this year accumulated 2 fail grades over the 5 year but with a higher GPA. The second place student received no failed grades but got a slightly lower GPA. Some are of the view that the student with no failed grades should be the winner. Some argue that failed grades should be included in GPA calculations.

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• nickfalkner says:

Thanks for the comment! I’m genuinely convinced that we are beyond the mathematical here and, once we’re in that space, it becomes a much harder measure to rationalise.

Your example just illustrates how challenging the issue is and how many problems exist already, we just don’t often see or talk about them.

I know you didn’t ask but my thought is that the ‘best student prize’ needs to be clearly defined so everyone knows what it means. If there’s a no fail requirement (which I’d not support, as you can probably guess) then, if that’s advertised, everyone knows what’s going on. However, a standard measure of “best GPA” assumes total inclusion implicitly.

Explicit statement of meaning is always good and thank you so much for letting me know that this post was useful to you!

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• lenandlar says:

Thanks for your response too. And yes there is a clear beat student criteria but this year it got complicated because the student accumulated the 2 Fs one each in the associate degree (first 2 years) and one in the final 2 years. They treat the associate year as an exit qualification but then include it for overall calculation. It’s a case that could possibly only be dealt with in hindsight.

I’m rethinking my views on failed grades ever so slightly because of the arguments and examples you’re sharing. Thank you.

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