I think you really need to go into why they are separated as they are. If you have handicapped certain teams because they contain all the “winners”, then you have already determined that the teams have been fairly split. In that case, giving the smaller team extra points defeats the purpose of splitting them unevenly.
If the kids are all doing individual tasks, then all you really have to do is to allocate points to the individual kids, the calculate team scores by working out the average score per child.
So, if a team of 10 gets 10,275 points, and a team of 8 gets 8357 points and a teams of 11 gets 11,411 points, then you make the following calculations:
10,275 / 10 = 1027.5 points per child
8,357 / 8 = 1044.6 points per child
11,441 / 11 = 1040.1 points per child.
So Team B wins, Team C comes second, and Team A comes third.
Of course, if there are any scored tasks that are done in groups, and are made easier by virtue of multiple participants, then the large teams have an advantage that you can’t really offset using such a simple calculation. In such cases, you’d have to assign a sort of “difficulty factor” to the tasks, and incorporate it into the points allocation and calculation, in order to make things fair for the smaller teams.
Essentially, the teams are uneven because they self-select. The group of kids, at-large, is told to divide into teams of “about” 10 on their own. Because kids will know each other from school or church or the neighborhood or whatever, groups will congregate toward one another and some groups will be larger or smaller than others. Those who are left behind will be assigned to other teams by the adult supervisors.
Tasks are completely individual, though they’re done as a team, and are not based on any one skillset. So, for example, each team may be tasked with, say, eating a gross foodstuff (a can of sardines, for example) in a specified amount of time. If a kid does it in five seconds, they get 1000 points for their team. If 50% of the kids do it, then a small team (say, 8 kids) will get 4,000 points, whereas a large team (say, 12 kids) will get 6000 points. Hence, the inherent unfairness in scoring.
Is it reasonable to expect that a task will not usually get done by each team member?
If so, you could cap the max number of kids to do the task at whatever the smallest team size is (or maybe 1 or 2 fewer, to give them a chance to not have to do everything). So, 1000 points per kid, up to a max of, say, 6000 points.
May I also suggest that you worry less about what’s perfectly fair, and just make sure you explain exactly how things will be scored to the kids before they make their teams. Trust me, if the kids want to win, they’ll figure out how to make their teams effectively.
I remember being in a similar “team” competition that actually was a series of individual competitions where we did not find out until we had made the teams that, each round, each member of the team had to participate in at least one event. If we had known that earlier, we would have gone for a much smaller team (maybe split in two), because it would have let us compete much better without (sometimes) being forced to have a team member compete out of his specialty. I remember being sore at the contest organizers for not considering how much the rules were biased against larger teams, and for not telling us until after we’d made our team.
The idea is that each kid, regardless of which team he or she is on, will have the opportunity to do every task. And of course, they may opt out or fail at some of them, but that’s part of the fun. But the kids are encouraged to give it their best for the good of the team, not for their own score tally. But the challenges are designed so as not to rely on teamwork, but individual effort.
For a comparison, imagine the House Cup in Harry Potter. How well the individual Hogwarts students do in certain challenges affects the outcome of the team as a whole. That’s what’s going on in this scenario.
Also, in the case the kids will know the rules beforehand. And the wide variety of tasks will prevent the competition from being dominated by (and kids graviting toward) kids who have certain specialized skills (say, puzzle-solving, or athletic skills, or whatever).
Even though I’m sure friends teamed up with friends there was probably bias in selection where this was concerned. For example the nerd probably isn’t going to team up with the quarterback from the football team, he’s going to team up with this friends that are likely other nerds. Be sure to have events in which most get a chance to do an activity they excel at.
The problem with just dividing the scores by the number of kids on a team is that it’s not particularly sensitive to how strong the relationship between score and team size is. If the relationship is strong, it’ll work fine, but if it’s weak, you’ll end up strongly favoring smaller teams. If it’s somewhere in the middle, who knows?
I’m pretty sure that what you really want is an ANOVA model, but those are slightly complicated. I’ll think about it a bit more and see what I can come up with.
For each team size, compute the average score and standard deviation. For each team, subtract the average score for their team size, and divide by the corresponding standard deviation. Then multiply the new scores by the overall standard deviation, and add the overall mean.
It’s kind of a hack, but it makes rough intuitive sense. You’re figuring out how different each team’s score is from the average for their size in standard units, and then expressing everyone’s score in common units.