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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.

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    $\begingroup$ I might be wrong but by calculating Percents you've already ranked them against each other? Without percents you would have say in EECS 90 people participated and in IEOR 36. When left in raw numbers EECS wins, when converted to percents IEOR wins. Percents is a way of comparing groups with different N's already. Are you looking for a weighting system so that with lower N's it becomes harder to win? $\endgroup$
    – DanTheMan
    Feb 22, 2013 at 22:50
  • $\begingroup$ You have totally gotten my problem! We have 8 departments in competition for a prize at the end of the year for participation in a gift program. So, EECS has 341 possible participants, with 106 actual particants, and thus 31% participation. NE has 11 possible participants, with 4 actual participants and thus 36% participation. But, it seems unfair that NE could beat EECS with 4 individuals participating to EECS's 106 individuals. The kids say that NE has an unfair advantage, basically. Easy to win if NE gets all 11 students together over beers to decide to take the thing, right? $\endgroup$
    – Alison
    Feb 22, 2013 at 23:49
  • $\begingroup$ Oh, MSE won this competition last year with 68% participation - only 36 students! $\endgroup$
    – Alison
    Feb 22, 2013 at 23:50
  • $\begingroup$ Changing the title to reflect the fact that you're looking for a 'fair weighting', not really a normalisation. As DanTheMan has pointed out, percentages are a normalisation. $\endgroup$
    – Assad Ebrahim
    Feb 23, 2013 at 0:05
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    $\begingroup$ Interesting problem. On the other hand, if NE's 11 people get together over beers and have 100% participation, and you down scale their score so they don't win, they could rightly complain that literally there is no way they can win (as they can't do better than 100%). $\endgroup$
    – Peter Ellis
    Feb 23, 2013 at 4:52

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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.

enter image description here

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:

enter image description here

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  • $\begingroup$ An afterthought - if students made independent decisions to participate or not, you would expect smaller groups to have more variation in their percentages, both upwards and downwards. So you would expect the small groups to have more extremes, both high and low percentages. In fact, the standard deviation of the group's percentages in that case decreases proportional to the square root of the group size. This gives a sort of conceptual link to my idea of penalising them according to the square root of the sample size but I don't think a very good one. $\endgroup$
    – Peter Ellis
    Feb 23, 2013 at 4:50
  • $\begingroup$ A nice exploration of different issues. $\endgroup$
    – rolando2
    Feb 23, 2013 at 5:59
  • $\begingroup$ Thank you for all this work! I will share this with the students if they give me any more grief about the competition scoring! $\endgroup$
    – Alison
    Feb 25, 2013 at 16:47
  • $\begingroup$ BTW, EECS, the largest group, and NE, the smallest group, both participated at almost identical rates, even though at the beginning of the campaign, I told the 11 NE students that they had a chance to take the whole competition by simply getting together and all making gifts to the campaign. They didn't do that. So, yes, it looks like the advantages and disadvantages in terms of size of the group balanced out in the final analysis. People - what are you going to do? $\endgroup$
    – Alison
    Feb 25, 2013 at 16:50

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