Friday, December 1, 2017: Calculating Your Own Grade for Physics 121

As we come into the last week of the semester the we find many students are asking if we can provide an estimate "where they stand" in the course grade-wise. Students also want to know "what they have to do" on the final exam or otherwise to earn a particular grade.

What follows is a set of three "prescriptions" for calculating and estimating of where you stand and some ideas that can give you the impression of plausible final letter grade outcomes. The three prescription let you answer these three three slightly different questions:

Below I provide three "do-it-yourself" prescriptions for answering three questions. These prescription are, by-definition, imperfect and approximate because of the fact that grade boundaries for exams can change and because a full 35% of your grade still depends on the last exam that you have not yet taken. Nonetheless, if you follow the steps below you will get a reasonably accurate picture of where you currently stand in the class and what you can expect in terms of your grade from the final. Furthermore, you can work out all of the details on your own.

"Snapshot Total" -- Where do I stand in the class? What is my current letter grade right now?

In Physics 121, all assignments, exams, and labs are scored on a point basis. Your letter grade in the course will depend on the total number of points you have earned. It's easy to work out the points, somewhat trickier to convert the points to a letter grade.

To start with, let's consider your "grade right now" -- in other words, assuming that you have in hand a sense of your performance on the three exams, the homeworks and the labs (everything except the final exam) where do you "stand right now". Our method is to develop a weighted total of the points from all assignments and then convert these into a letter grade using approximate grade boundary estimates. Here's the method:

You now have a number that should range in value from zero to a maximum of 650 points. This total number tells you where you stand in the course. Let's call this number your "snapshot total" To get an estimate of how this numerical score corresponds to a letter grade, we determine a "preliminary approximate borderline grade value" for each assignment. Here is how we calculate your grade: We use these boundaries to invent a set of imaginary students that earn the borderline scores on every assignment given in the class. These scores determine the grade boundaries for the whole class. We then compare your score to these boundaries. Here's how it works: A fictitious student, let's call her "Ms. A-Just-Barely" for example has earned an 85 percernt score on the first exam, an 80 on the second exam, and an 80 on the third exam. She also has a average homework score of 12.5 and a lab score of 90 percent. Her "snapshot total" therefore, according to the prescription indicated (rounding down) is 552 points out of 650. Every student who has a snapshot total of at least this APPROXIMATE value is right now at this instant earning an "A". We do the same for the other grade boundaries, e.g. "Mr. Just B-Just-Barely has earned 438 points. Here are the scores for all of the letter grade borderlines:


  PRELIMINARY APPROXIMATELY GRADE BOUNDARIES FOR SNAPSHOT TOTAL: 

       A-Just-Barely = 552 out of 650 or higher
       B-Just-Barely = 438 out of 650 or higher
       C-Just-Barely = 348 out of 650 or higher
       D-Just-Barely = 268 out of 650 or higher

So this tells you approximately where you stand in the class right now. It's a "snapshot" of what grade you have earned based on all the assignments for the course excepting for the final exam.

Now, immediately I need to make two huge caveats:

First, while the grade boundaries for the homeworks and the labs are fixed, the grade boundaries assigned to the exams given above are tentative, approximate, and certainly subject to change. The whole point of is that we tune the grade boundaries on an exam-by-exam basis to take into account the relative difficulty and importance of each exam based at least in large part on how students actually perform in detail on each problem. We always determine preliminary boundaries after we have taken a first look at the overall student performance on each exam, but we also always go back and make adjustments to all of the exam boundaries after the final exam when all of the scores can be looked at together. Usually we make adjustment to the exam grade boundaries, sometimes by several points relative to the preliminary assignment. Sometimes we adjust upward, sometimes we adjust downward. Therefore each of these preliminary estimated grade boundary numbers has "large error bars" with uncertainties of as large as ten percent or more either way. If you are close to a boundary, all you can say with certainty is that you are close to a boundary. You cannot predict with certainty that the boundary will not change and that you will on the other side in the end.

Second, the final exam counts a very large fraction of your score -- 35%. This means that any confidence that you have about "where you are right now" is less certain and less useful for predicting your final grade just because in fact a large fraction of the question about how you will do in the class depends on how you do on the final. In other words, the above calculation provides a "snapshot total" of where you are right now, but it doesn't really tell you much about where you are likely to be in the future. So we go to the next step.


"Extrapolated Total" -- Where am I heading? What is my likely extrapolated letter grade for the course based on my past exam performance?

One of the drawbacks of the above "snapshot total" is that, because it does not include the final exam, it "de-weights" your Hour Exam scores relative to Homework and labs in terms of how much exams matter for your overall course grade. And just to this extent, it does not really tell you clearly what you can expect in terms of where you might end up in the course after you have taken the final. Therefore, we are motivated to consider an alternative calculation to estimate your final letter grade for the course by extrapolating your past Hour Exam performance to estimate your final exam performance. The theory here is that your lab and homework scores are not a good predictor of future final exam performance, but your hour exams are. To do the extrapolation, we repeat the above sum, but this time increasing the weighting factor on the exams by a factor of 2.4. Here's the method: This is what we call the "extrapolated total" score. Now as before, we use the grade boundaries to invent total scores for imaginary borderline students. Here are the borderlines for "extrapolated total" scores:


   PRELIMINARY APPROXIMATE GRADE BOUNDARIES FOR EXTRAPOLATED TOTAL: 

       A-Just-Barely = 836 out of 1000 or higher
       B-Just-Barely = 632 out of 1000 or higher
       C-Just-Barely = 493 out of 1000 or higher
       D-Just-Barely = 378 out of 1000 or higher

This tells you "where you are heading" based on the assumption that your Final Exam performance can be accurately predicted by your past performance on Hour Exams.

But there are drawbacks and perils here as well:

First, you have only taken less that half of the total exam points (25% taken, 35% yet to take) so extrapolating your future based on what you have done in the past is rather uncertain activity. Roughly speaking, those ten percent error bars for "snapshot total" method above are probably at least doubled in size for the estimated extrapolated grade.

Second this extrapolation assumes that the grade boundaries for the final exam will be proportionally matched to the grade boundaries of past hour exams. This is really just a wild guess. There is no way to know what the actual grade boundaries for future exams will be until after the Final Exam has been given.

Still, in certain ways the "extrapolated total" method gives you different information than the "snapshot total" method. In particular it tells you where you are "heading" grade wise if your Hour Exams are a good predictor of your Final Exam performance.

"Scenarios" -- How well do I need to do on the Final Exam? What do I need to do to get a particular grade?

Now, finally, there is a question we get over and over this time of year which is this: "How many points do I need to earn on the final exam to be sure of earning a grade of X?" Here "X" is "A" or "B" or whatever.

Again, this is a perilous and uncertain activity for all of the reasons indicated above, only even more so since we are now talking about calculating "margins" instead of averages. So let's quadruple the error bars on this exercise to at least 40% uncertainty. With this extreme caveat we proceed:

Let's start by noting that in this point system, the final exam is worth 350 points. So for any given final exam score which we represent as a percentage (0 to 100), you can divide by 100, multiply this by 350, then add this to your "snapshot total" to get your putative "scenario" grand total points. For example if you earn an 80% on the final exam and your "snapshot total" calculated above is 530 points, then your "scenario total" would be 0.80 * 350 + 530 = 810 points.

This "scenario total" is a number from 0 to 1000 that you can now compare to the "extrapolated total" grade boundaries above to see your estimated "scenario" final letter grade associated with this score. (For example, in this case of a "scenario total" of 810 total points, this corresponds to a grade of "B").

Indeed you can work this backwards from any "extrapolated total" boundary to estimate the points you need to achieve on the final exam to earn a given letter grade. In other words, you can answer the question "What do I need to earn on the final to get a grade of X" by finding the final exam score that is needed to give you a "scenario total" that matches the corresponding "extrapolated total" grade boundary. For example, if your "snapshot total" was 530 points (corresponding to a current grade of "B") and you wanted to estimate how well you need to do on the final to squeak out an "A", then you see that you need to get 835 total points for an "A", which means you need 305 points for the final exam corresponding to a percentage score of not quite 88%.

Now again, I need to emphasize that this exercise is fraught with huge uncertainties. In particular, if you go through these steps you might somehow conclude that "it is impossible to earn a grade of X no matter how well I do on the exam". Or, conversely, you might conclude something like "it is impossible for me to earn a grade of less than X no matter how poorly I do on the exam." I would very much urge against any inclination to draw any such conclusions . This is because at present we do not have fixed grade boundaries for the final exam. And we cannot assume that the grade boundaries will be simple extrapolations of the grade boundaries from the hours exams. In looking at past years, we find that the grade boundaries for the final exam are extremely variable from year-to-year. It is entirely possible that a strong performance on a difficult exam might boost your grade up beyond where you might otherwise imagine you can reach. It is also possible that a disastrous performance on a relatively easy exam may injure your final letter grade much more than you might estimate here. Grade boundaries for the final exam will only be set once we have had a chance to look at the overall performance of students on the final exam, including careful consideration of how students performed on each of the nine individual problems. And remember: I am very likely to adjust any and all grade boundaries for all of the exams, including previously taken hour exams, based on overall student performance levels after all of the exam scores have been tabulated..

The bottom line is this: You can only approximately estimate where you stand in the class but because so many points remain to be earned for the final exam, any prediction that you might make about where you are headed and/or how well you need to do on the next exams is wildly uncertain. On the other hand, the fact that there are so many points to be earned still means that most students still have a large opportunity to improve their scores, no matter where they stand now. I've designed the point system so that it is at least possible in principle for almost every student in the class to move themselves up or down a full letter grade based on how they do on the final.

In other words, no matter where you stand, it's not too late, and there is work to do. Yours,

-Corbin Covault P.S. Students ask about "Bonus Points". See Document #2, Section 15 page 26 for details on pilicies and application of Bonus Points for PHYS 121. In a nutshell, any bonus points you have earned will be automatically applied as insurance against a low score on any one of the nine final exam questions.