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Put these numbers in order: $\frac{x+5}{N+5}$, $\frac{x}{N-5}$, and $\frac{x+5}{N}$, assuming that $0 \leq x \leq N$.

Here’s the context: Suppose (hypothetically, you know) you’re grading exams on a percentage scale (93% and above is an A, 90-92% is an A-, etc) and after adding scores you realize that maybe because of an unusually difficult problem, the points don’t quite match your overall impression in the sense that the exams that indicate a deep understanding should be the ones that get an A. Now suppose that by adjusting the scores by 5 points they do seem to match up and this is simpler than going back and revising the partial credit schemes on the whole exam.

One way (which only works with foresight) is to add a question, like “What’s your name?” or “What was your favorite topic in the course?” that, in theory, everyone should get right (a freebie question, as it were). This adds 5 points to each person’s score (x) and also to the total (N), raising the overall percentage from $\frac{x}{N}$ to $\frac{x+5}{N+5}$.

A second way is to just drop 5 points from the “Points Possible” column, resulting in a score of $\frac{x}{N-5}$. Finally, a third way is to just give everyone 5 points extra credit, giving $\frac{x+5}{N}$. What’s a little surprising is that these have different effects, depending on the original score. Take a look:

$\frac{x}{N}$ $\frac{x+5}{N+5}$ $\frac{x}{N-5}$ $\frac{x+5}{N}$
x=0 0.0% 4.8% 0.0% 5.0%
x=25 25.0% 28.6% 26.3% 30.0%
x=50 50.0% 52.4% 52.6% 55%
x=75 75.0% 76.2% 78.9% 80.0%
x=100 100.0% 100.0% 105.2% 105.0%

For everything but the very highest scores (x=95 and above, it turns out), adding 5 points is the most generous option. But for the lowest scores (x=47.5 and below, it turns out) the freebie question is actually a better option than just dropping 5 points from the total. Indeed, the freebie question is almost as good as extra credit for people who otherwise missed all the points, and does nothing for those at the higher end, while dropping 5 points from the total does the reverse. (Which leads me to think that analyzing which scheme to use is no simpler than going back and adjusting the partial credit after all.)

Disclaimer: this year I think my exams are on target, and I won’t be adjusting any scores. I have, however, used each of these variations at one time or another in the past although I hadn’t looked closely at the distinctions before today.

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### One Response to “Finessing Grades”

1. jd2718 Says:

I have used a piecewise approach when I thought the center was out of whack. In words, I treat, for example, the first 50 points differently than the next 50 points, perhaps multiplying the first 50 by 1.2 and the next fifty by .8

0 -> 0
10 -> 12
20 -> 24
30 -> 36
40 -> 48
50 -> 60
60 -> 68
70 -> 76
80 -> 84
90 -> 92
100 -> 100

More often, I’ve used several pieces. I can recall one year feeling that those who scored 60-80 on a particularly hard test weren’t being properly credited for their success:

0 – 35 1
35 – 50 2
50 – 60 1.5
60 – 80 .8
80 – 100 .4

0 -> 0
20 -> 20
40 -> 45
50 -> 65
60 -> 80
70 -> 88
80 -> 96
90 -> 98
100 -> 100

For much smaller effects, I do the same as you and look at partial credit.

Ultimately, I make the tests, and then if I look at the work, and say, “you know, what I set as passing doesn’t represent what I am seeing on these tests” I need to rescale.