How can I devise a scoring system for a competition that is more fair than straight percentages? 
I am trying to come up with a method for deciding the winner from among eight student groups competing for a prize.
The raw data and corresponding percentages measure participation per group in a campus program. 
The current rules say that the students who have the highest participation by percent of their population in the group win a prize.
However, I have received complaints that this scoring system unfairly benefits small groups because it is supposedly easier to coordinate smaller groups of people and get a larger percentage of successes.
I am not a mathematician, obviously, so I hope my description of the problem makes sense. 
As you can see, one of the student groups has 341 students and another has ony 11.
Any help you can give will be very much appreciated and may keep a riot from breaking out among the winners/losers.
 A: I don't think there is any theoretical answer to this one.  If there is an advantage to organising a small group it is a matter for the social sciences to investigate.  It is also possible there are economies of scale that make it easier to organise a large group.  So any penalty for small groups will have to be based on evidence that in fact they do find it easier.
In such situations it pays to look at the data.  Originally, I had some sympathy with your students view and assumed it would be easier to organise the smaller number, but if so it is not really evident in what you have shown us.  If you have previous year's data it would be worthwhile pooling it and seeing if a pattern emerges, but on the data you have (plot shown below) there is no evident pattern of higher participation by smaller groups.  The one influential point in the bottom right is a matter of some concern but is not supported by any pattern in the rest of the data.

So unless there is an evident trend I think you might be better of with the straight percentages.
However - if it is necessary to prevent warfare... you could negotiate a solution that everyone thinks is fair by arbitrarily down-scaling the small group numbers.  If you went the whole way, you would just use the absolute number of participating students, which would obviously unacceptably benefit the bigger groups.  One compromise would be something that penalises percentages according to the square root of their group size.  This effectively gives you something that can be regarded as mid way between the percentages you have, and just the absolute number of students participating.  So try:
$Adj_i=RawPercent_i\sqrt{\frac{n_i}{max(n)}}$
For the biggest group this will be the same as their raw percent.  For a group that is only 1/36th the size of the biggest group (which is roughly the situation of NE compared to EECS) their original percentage will go down by 5/6 - a very significant disadvantage, making the contest almost unwinnable I would have thought (not that they're doing very well even on the raw figures...).  The use of the square root here is arbitrary - I suppose I am just illustrating what you could do.  Instead of a square root you could try a third or fourth root, which does not penalise smallness as much.
Luckily for you, even if you did the ruthless square root penalty, you get the same winner CEE, although the procedure helps ME move up the ranks from fourth to second:

