Dropping a Lowest Score

Question

I have a lab class, where I drop the lowest score students receive, easy enough. Except, all items are not weighted equally, (lab A might be worth 10 points, while lab B might be worth 23 points, and so on).

Right now, what I do in an excel file is calculate the total points earned and the maximum points, except in each instance I remove a different assignment from the calculation. Then compare the various percentages and drop the one that yields the highest overall percentage.

TL;DR, here is the main question part ;)

Is this a problem that can be solved more easily, or can I answer the question of which to drop with a formula? I'd love to include the calculation in my grade book so it happens automatically. Right now, I have to go in and drop an assignment manually for each student, since my only option is to drop "lowest score" which doesn't work since 5/10 has a different impact than 3/30. (I realize I could scale everything and make them all worth the same amount, but that complicates other parts of the grade for me, so isn't ideal)

I've included a screen shot of what I do in excel for hopeful clarity.

The bottom three rows are what I look at.

Total Score w/o: Total earned points (from row 2) without the lab corresponding to that column

Total Max w/o: Total maximum points possible (from row 3) without the lab corresponding to that column

Combined Percent w/o: Just the values from $\displaystyle\frac{\text{Total Score w/o}}{\text{Max Score w/o}}$ for each column.

I have conditional highlighting which shows me the highest percentage, so in this case, I would drop the student "Hydrate" lab assignment from their grade.

*Note: I will confess I really wasn't sure what tags I should use. This stackexchange is way out of my comfort zone, so most terms were not familiar. Feel free to change them to whatever is most appropriate.

Answer 1

Let $t_i$ and $s_i$ be the possible points and actual points scored, respectively, for each individual assignment, and let $T$ and $S$ be the respective sums over all assignments. Assuming that the goal is to produce the highest percentage for the student, you’re trying to maximize ${S-s_i\over T-t_i}$. With a bit of algebraic manipulation, this can be rewritten as $\frac S T+{St_i-s_iT\over T(T-t_i)}$, so drop the assignment that maximizes ${St_i-s_iT\over T(T-t_i)}$.

For the small example in your comment to Travis’ answer, these values are $0.129$, $-0.017$, $-0.026$ and $-0.060$, so the first score should be dropped. For the slightly larger data set in those comments, this method selects the 15/35 score ($0.036$), just as you had originally computed. For the example in your question, “Hydrate” is the winner with a value of $0.045$, so it should be dropped, which also agrees with the method you’d been using.

Of course, if you’re doing this in a spreadsheet anyway, you can compute ${S-s_i\over T-t_i}$ for each assignment and have the spreadsheet find the highest value for you, as Henry points out in his comment to your question.

Answer 2

Figure out the the average value of an assignment, say 25 points. Leave all the scores intact, but before you average, deflate the denominator by 25 points. Truncate any score that happens to go over 100 down to 100.

This gives a break to students who do poorly on an assignment (or maybe two), gives a slight advantage to students who perform consistently, and makes your life a lot easier.

You can tell students you're "in effect" dropping the lowest score, and they will probably never be able to figure out that's not exactly what you're doing. Or tell them exactly what they're doing and they can do the math to see they're still getting a roughly equivalent break for a missed or botched assignment.

I've done this for the past 10 years with no complaints.

Answer 3

Let the total points be $S$, the amount of possible points on an assignment be $K$, and the score the student got on the assignment be $x$.

$${SCORE} = x\frac{S}{K}$$

You'll have to apply this to every equation, and drop the one with the lowest result. In your examples, the student has scores of $1/30, 5/7, 15/25$ and, $13/17$. This means the total score is $30+15+25+17 = 87$. You then apply this to the equation: for the first assignment, $$10\frac{87}{30} = 29$$ For the second assignment, $$5\frac{87}{7} \approx 62.14$$ The third; $$15\frac{87}{25} = 52.2$$ And finally, $$13\frac{87}{17} \approx 66.53$$

Because the first assignment yeilds the lowest result, it has the largest negative effect on the grade, and should be dropped.

This works because you're taking the weight of the assignment $\frac{Points}{Total Points} $, then multiplying it by the score, yielding the negative weight on the total grade.