Different...But the Same

There are two typical ways grades are set up in a course: total points and percentage weighted. These two seem radically different, but are really the same basic thing viewed from two different perspectives.

Total points says that, for instance, there will be 1000 points during the semester. So if you earn 900 points (or more), you'll get an A. If you earn 800 to 899 points, you'll get a B. And so on...

Percentage weighting would instead say that there will be 100% available during the semester. If you earn 90% of that (or more), you'll get an A. If you earn between 80% and 90%, you'll get a B. And so on...

The tricky part is in looking at the total semester layout:

Hmm... Actually, that wasn't that tricky, either. So where does it get confusing?

Well, most teachers don't come up with an exact multiple of 10 for their points. Instead, they just have whatever points were appropriate for that semester's activities. So, the reality of it might look more like this:

*shiver* Now the total points side looks quite scary. But it might also look like this:

Of course those percentages aren't exact, but rounded to the nearest percent.

Now the percentages look pretty scary, too. Yet, the two sides are still [essentially] the same!

How Do I Know What I'm Getting Along The Way?

That's the real ickiness! Let's look at a couple of examples:

Total Points

(Reminder of semester schedule for total points:

Total Points

)

Let's take a typical student, say Phong. Let's take a typical class, say Trigonometry (trig). As the semester progresses, Phong finds that she's earned a 65 on the first test, gotten an 8 and a 0 on the first two quizzes, and earned a 9 on the first chapter's homework. How's Phong doing?

    Phong earned:  65 + 8 + 0 + 9 = 82

    Total possible:  100 + 10 + 10 + 10 = 130

    Phong's got:  82/130 = 63%

Percentage Weighted

(Reminder of semester schedule for percentage weighting:

Percentage Weighted

)

In an alternate universe, Stu is also taking trig. Stu also gets a 65 (out of 100) on the first test. But he gets a 5 and a half (out of 7) on the first quiz and a 0 (out of 8) on the second quiz. And finally, he earns 8 (out of 9) on the first homework set. How's Stu doing?

    Stu earned:      65/100   5.5/7     0/8       8/9

    Percent weight:   10%     1.25%    1.25%      2%

    Stu's got:       9.25/14.5 = 63%

(Where'd those last numbers come from?! The 14.5 is just the total from the second line: 10+1.25+1.25+2=14.5. But the 9.25 is a little more weird. Stu got a 65 out of 100 on the first test, right? That's 65%. But that 65% only counts for 10% of his semester grade so it is really .65*10=6.5. So 6.5% of Stu's overall semester grade has been earned from that first test. (Or, similarly, Stu has lost 3.5% of his overall grade to munging up that first test.)

(Um...why did the 65% become .65 but the 10% stayed 10? Oh...er... We can do it the right way, but the numbers just become horribly nasty: .65*.1=.065. This still represents 6.5% of Stu's overall semester grade. Just it is also in decimal. *shrug* Anyway, back to the 9.25...)

Then Stu got 5.5/7≅79% on the first quiz which is 1.25% of his overall semester grade. So that represents .79*1.25≅1% of his overall semester grade.

Next came the 0/8 on the second quiz. Well, 0 out of anything is still 0% and 0% of anything is likewise 0%. So that quiz didn't do much to improve Stu's semester grade...

Finally he earned 8 out of 9 which is approximately 89% on the first homework set. That homework set counts for 2% of his overall grade, so that means his score gives him another .89*2≅1.75% of his semester grade.

Putting all of these pieces together: 6.5+1+0+1.75=9.25. So, Stu has earned 9.25 of the 14.5 percent of the overall semester grade which is approximately 63%.

Comparison

Whoa! How'd Stu end up in the same place as Phong?! It looked like she was skunking him on the quizzes and homework... But look at the breakdown side by side:

Now it is more obvious that Phong is only a slight bit ahead on each item. So how did Stu stay even in the overall? Let's add a column:

Wow! Things are worth a lot different portion of the grade than one another (well, the test and homework, anyway).

Can I Still Pass?

The next most popular question is, "Can I still pass?" (Or similarly, "What do I need to get a ____?")

Before we take a look, let's clear up a common misconception:

What Does Passing Mean?

Passing in college is almost exclusively a C (70%). A 'D' might have meant 'Done' before, but now it is just a synonym for 'FaileD'.

Now back to our parallel universes:

Phong (Total Points)

To find out if Phong can pass, we need to know what passing means for her:

    Total Points:   920
    Passing: 70%  *   0.70
    Points to pass: 644

Phong has 82 points and will need 644 to pass. That leaves:

   Needed to Pass:  644
   Phong has:     -  82
   Phong needs:     562

That means Phong will need to earn 562 of the remaining:

    Total Points:  920
    Points Done: - 130
    Points Left:   790

790 points. She'll have to get, essentially:

    Phong needs:  562
    Points Left:  790
    Needed %:     72%

72% (average) on all remaining material to pass the course. Since she currently has a 63%, that could be do-able. Looking at her scores she seems to do fine on home out-of-class work and small in-class work. Maybe she's just got test anxiety. A trip to Student Success Services might be able to help!

Stu (Percentage Weighted)

Stu, on the other hand, has:

   [Total Percent:  100]
    Percent Done:  - 14.5
    Percent Left:    85.5

85.5% of his grade as yet undertermined. He's earned 9.25 of the 14.5 percent he's completed (see above). So in order for him to pass, he'll have to get:

    Passing:     70
    Stu Has:   -  9.25
    Stu Needs:   60.75

60.75 of those 85.5 remaining percent. What's that? Well, that'd be:

    Stu Needs:     60.75
    Percent Left:  85.5
    Needed %:      72%

*gasp* Stu needs 72% (average) on his remaining material to pass the course as well! And, like Phong, since he currently has ~63%, that seems like a reasonable goal. He even has the same kinds of problems as she does and so might be amenable to similar help (i.e. a trip to Student Success Services for a Test Anxiety Workshop or Test Taking Strategies).

I Want to Do Better Than Just Pass!

Okay...what if Phong or Stu wants to get a B (80%) or even an A (90%)?

(920*0.70-82)/(920-130) ≅ 72% (do-able)

(70-9.25)/(100-14.5) ≅ 72% (do-able)

(920*0.80-82)/(920-130) ≅ 83% (real hard)

(80-9.25)/(100-14.5) ≅ 83% (real hard)

(920*0.90-82)/(920-130) ≅ 95% (*snicker*)

(90-9.25)/(100-14.5) ≅ 95% (*snicker*)

So getting a C seems to be reasonable. But the B is going to be real difficult to achieve. The A might be a little on the 'impossible' side given the current track record. These students are going to have to really embrace the lessons given by Student Success Services to make that kind of improvement.

Ongoing Grade Tracking

Let's see how things are going a little later, after our two students have gotten their help and begin to perform a little better on those tests:

So what does that bring them to? Well, for Phong we have:

    Phong Earned:     82 + 7 + 6.5 + 9.5 + 57.5 = 162.5
    Phong Possible:  130 + 7 + 8   + 10  + 80   = 235
    Phong Current %: 162.5/235 ≅ 69%

And for Stu we have:

    Stu Earned:       9.25/14.25    15/15    16/20    9.5/10    79.25/110
    Item Weights:       14.25%      1.25%     2%        2%         10%
    Stu Current %:   21.20/29.5 ≅ 72%

How did they separate?! They still seem to be running neck-and-neck... don't they? Let's see:

Now we see that, although their individual scores are progressing apace, the weights of their tasks to one another are not proportional (across the universe divide, as it were).

The instructor in total-points world has a chapter 1 test that is worth 100 points of the total 920 (≅ 11%) whereas their chapter 2 test is only worth 80 points of the 920 total (≅ 9%). Even though the total points tests taken as a whole are [approximately] 52% of the overall grade (480/920), they are not all equal to one another.

Over in percentage-weighted world, however, the instructor has allotted 50% of the overall grade to tests and since there are 5 chapter tests, that comes out to 10% each. Note that the points on the test are not relevant here! The chapter 1 test had 100 points just like the one in Phong's world but Stu's chapter 2 test had 110 points which is more than the other world's 80. Both students also scored approximately the same percentage on each test -- their performance on each chapter is nearly identical to one another. But at this stage, Phong's tests are not equivalent to one another whereas Stu's are -- regardless of their point total!!!

By the time the semester is over, the two will work out close to the same if they keep improving their performance at the same rate. However, at different stages of the semester, Stu might be ahead of Phong (as now), Phong might be ahead of Stu, or the two might be tied (as they were before).

Moral: Comparing yourself to friends in other sections isn't necessarily going to work out very well...

Continuing Chances At Success

Okay...what if Phong or Stu wants to: pass (C; 70%), get a B (80%), or even get an A (90%)?

(920*0.70-162.5)/(920-235) ≅ 71% (fair)

(70-21.2)/(100-29.5) ≅ 70% (fair)

(920*0.80-162.5)/(920-235) ≅ 84% (tricky)

(80-21.2)/(100-29.5) ≅ 84% (tricky)

(920*0.90-162.5)/(920-235) ≅ 98% (yeow!)

(90-21.2)/(100-29.5) ≅ 98% (yeow!)

So getting a C seems like it won't be a stretch. But the B is going to be fairly tricky to achieve. The A seems to be getting even more 'impossible' than before.

The reason for this is that the students are incorporating those Student Success Services lessons as best they can, but changing long-standing negative behaviors is really difficult for most people. If they'd perfectly learned how to Avoid Test Anxiety or employ Improved Test Taking Strategies, they'd have gotten not 72% on that chapter 2 test, but more like 88-96%. (This estimate is based on looking at their other assignments for that chapter and realizing that an in-class activity probably involved a group who may or may not have been of equal caliber to our focal pupils.) Had that happened, their current performance would have risen farther than ≅70% and they might still be in reach of that A everyone truly desires. But it just isn't going to happen for most people. Sometimes the reasons we start a semester poorly are amenable to changes/fixes...sometimes they aren't or are more difficult to change/fix.

Moral: Don't beat yourself up because something has happened, just do the best with what you have. (You know, that whole lemons ⇒ lemonade thing...)

If these students take the lessons given by Student Success Services to heart, they might make it out of trig not just alive, but smelling like violets or petunias (if not roses).

One More Time!

Let's take a last look at our students after the chapter 3 test is done to see how they are getting along:

So the two continue to improve at the same rate and they now have:

    Phong Earned:     162.5 + 7 + 10 + 10 + 9  + 70 = 268.5
    Phong Possible:   235   + 8 + 9  + 12 + 10 + 90 = 364
    Phong Current %:  268.5/364 ≅ 73%

    Stu Earned:       21.20/29.5   8.5/10   13/12    12.5/15    9/10   58.25/75
    Item Weights:        29.5%      1.25%   1.25%       2%       2%       10%
    Stu Current %:    34.85/46 ≅ 75%

See, now that Phong has another point-heavier test she's starting to catch up... What about their prospectives for the future?

(920*0.70-268.5)/(920-364) ≅ 68% (easy)

(70-34.85)/(100-46) ≅ 66% (easy)

(920*0.80-268.5)/(920-364) ≅ 85% (do-able)

(80-34.85)/(100-46) ≅ 84% (do-able)

(920*0.90-268.5)/(920-364) ≅ 101% (hey!)

(90-34.85)/(100-46) ≅ 103% (hey!)

So getting a C now looks like an easy ride -- just maintain what they are doing.

The B is still getting a little further away, but it is now within reach of our grasp. Looking back over our recent track record, our students are nearing the 80% mark on their tests and keeping their A-B scores on other portions of their grade. Their test scores are going up about 7-8% each time so they should be there by the next test (and they still have two chapters left so that is definitely a possibility.)

That A is now scooting beyond nasty. In fact it's getting harder for Stu than for Phong. Now they'd each have to not only do everything pretty much perfectly, but even hand in a bunch of extra credit! *sigh* Who's got time to do that with other classes and family issues and working to pay for school, insurance, gas, car payment, etc. *shakes head*

*shrug* At least that B came down to Earth. *bounce* (Again, lemons ⇒ lemonade! Don't look at the goal you can't reach without minor mental/physical trauma! Look at the awesome goal just over that ridge. Isn't it beautiful? A B for someone who flunked the first exam and missed a quiz is pretty good. And they didn't even have to put in much extra effort -- just show up so they didn't miss any more quizzes and get some tips on Studying and Anxiety Management from the folks over at Student Success Services...)

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