Test grade

#1
I took a test 2 weeks ago and the professor handed back the grades today.

The average was 48
Low 18
High 84

my grade was 60

I'm not sure what the standard deviation was. Let's assume 7


The professor sent out an email saying

"On Monday I will hand back the tests, where you can correct your mistakes and get half the points back on the ones you missed. Due Tuesday"



So the people who missed more points will benefit more from something like this. The person who got 20 will have the potential to get 40 points back vs. the person who got 80 will only have the potential to get 10 points back.

Does this skew the original data?

Does it condense the standard deviation?

Ultimate question..for those who scored above the standard deviation..will their grade ultimately be lowered due to this... given the professor bell curves the grades at the class?
 
#2
There are many possibilities, and they depend on:
(1) who takes advantage of the teacher's offer
(2) how many points those who do take advantage of the offer actually get back by doing so

For example, check out the attached pictures. The four variables here represent:
GRADE: roughly fits your description of original grade distribution.
NEWGRADE: the new distribution if everyone in class takes advantage of teacher's offer.
NEWGRADE2: the new distribution if only those who originally scored lower than the mean take advantage of the teacher's offer while everyone else doesn't.
NEWGRADE3: the new distribution if only those who originally scored higher than the mean take advantage of the teacher's offer while everyone else doesn't.

(All of these make the unrealistic simplifying assumption that each person who takes advantage of the teacher's offer manages to get back every possible point.)

As you can see, the standard deviation has substantially decreased in the first two scenarios, but increased in the third. As far as skew, the distribution (at least with these numbers I've made up) seems to get less skewed (more normal) in the first two scenarios, and severely less normal, essentially bimodal, in the third.

So it is impossible to say exactly what the teacher's offer will do to the grade distribution; it depends on the students' behavior. Speaking as someone who has taught, I would (sadly) not be surprised if NEWGRADE3 represents the most realistic of the three possibilities I've analyzed here.

As far as how you would fare in each of these distributions if each one was graded on a curve, calculate your z-score on each distribution.
 
#4
Since your professor said this is due on Tuesday and there's an urgent practical conclusion to be drawn, I'll just add the z-score info here myself.

Here's the particular z-score comparison you were probably most interested in:

your z score in original distribution:

(60-48)/21.457 = +0.559

versus

your z score in NEWGRADE (where everyone, including you, takes full advantage of teacher's offer):

(80-74)/10.728 = +0.559

In other words, the teacher's offer won't change your standing in the distribution if everyone, including you, takes full advantage of it.

If everyone else does and you don't, though...yikes:

(60-74)/10.728 = -1.305

Urgent practical lesson: do the rewrite, you have a lot to lose if you don't.