Averages of groups of different sizes

[rizzy123]

Hello all,

We're going to be doing a fundraising competition, and I'm not sure as to the fairness. We will determine the winning group by just comparing the average amount raised by each group. Is this mathematically fair, or is their a better way? For some reason I can't help but think that it'd be unfair for a group of 10 people to compete against a group of 100 people. I think the group with 10 people will most likely win, but I don't know why. Any advice will be appreciated.

Thanks

 

[CRGreathouse]

There is some advantage to being in a smaller group. For example, consider splitting the group of 100 into ten groups of 10; unless all the teams so formed have the same average, one will have a higher average than the original group.

I suppose one way to handle it would be to treat everyone's actual fundraising total as a random variable based on the unknown effort -- this is fair enough, right? Then fix some confidence level (95%) and find for each team a confidence interval for their effort and award the prize to the team with the highest lower bound.

The exact math on how to do this depends on what model is chosen; I'm not sure which would be most appropriate.

The trouble is that it's hard to justify the method to the people, since they're probably not familiar with this kind of math. (If you have a bunch of statistics PhD's doing fundraising, this might not be a problem. But for most groups it would be,) So in that case you might have to settle for an alternate technique, like "take the average, let the small groups have an advantage" or "split the teams into big-, medium-, and small-size divisions and determine the winning group in each".