[darynthe]

This is a statistical problem:

There is a contest for photographers that has a girl against boys. The sign up has been unequal, with 30 men vs 25 women. The idea was to use the average grade of each photo posted by team to say who won. Therefore it didn't matter that there were more boys than girls. I however mentioned (as a joke only but it was taken seriously) that since there are more men than women, this meant an unjust advantage to women's average.

The reason would be that the highest performers with 7/10 or more grades would less likely to sign up than the newbies after the initial phase. I have noticed that about 2% of entries posted to any given contest get over a 7.

Since it is usually the most devoted, disciplined and therefore skilleful photographers who usually sign up for tournements, it is my opinion that the more people these contests get the more likely the average will diminish.

Can anyone tell me if I am wrong or right and what would be in such a case the best way to measure the winning team?

(if curious here is the thread:
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The rules have been changed now and the winner won't be average but the sum of all entries by team. However I notice this has the disadvantage when someone omits to post an entry.

[Paul Siraisi]

I think the probability of being a low-skill photographer applies to groups of any size.

If 1/10 are good, and 9/10 are bad, then that first player will have a 1/10 chance of being good, as will the 2nd and the 3rd, etc. A team of 10 will on average have 1 good and 9 bad. A team of 5 will have 1 good half the time, and none good half the time.

So I don't see how it matters about group size.

[Monte314]

Somebody that knows math should do something about this...

*dog begins hyperventillating*

[roninpro]

	Originally Posted by Paul Siraisi To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I think the probability of being a low-skill photographer applies to groups of any size. If 1/10 are good, and 9/10 are bad, then that first player will have a 1/10 chance of being good, as will the 2nd and the 3rd, etc. A team of 10 will on average have 1 good and 9 bad. A team of 5 will have 1 good half the time, and none good half the time. So I don't see how it matters about group size.

I think that the group size does matter. For the moment, maybe we can assume that the skill level of each entrant is a random variable. (If you want, we can let it be random between 1 and 10.) And I will assume, for demonstration purposes, that the first group has 100 people and the second group has 10.

The law of large numbers essentially says that if you have a random variable X, the rolling average after n trials converges to the average of X, as the number of trials goes to infinity. That is,

(X_1 + X_2 + ... + X_n) / n -> average of X

when n -> infinity. (In fact, we can be even more precise: the rolling average converges to the mean with Sqrt[n] speed.)

For the purposes of this problem, this means that taking more trials (i.e. a bigger group size) will cause the average to stick close to the mean. Alternatively, a smaller group size will tend to deviate much more from the mean. In our example, the average of the skill rating is 5.5. In the first group, we have 100 people. Using Mathematica, I've computed the average for 5 random configurations:

5.82, 5.51, 5.5, 5.62, 5.92

For the second group of 10 people, here are the averages for 5 random configurations:

5, 3.7, 7.5, 6.8, 5.3

Notice how the smaller group averages tend to swing much more wildly from 5.5 than the larger group. It is a manifestation of the law of large numbers.

The conclusion is that the group size really does matter, but the issue is to what extent. At least this special case is easy: if the two group sizes are roughly the same size and are very large, their averages will be practically indistinguishable, basically rendering irrelevant the conditions of the contest!

[Latro]

The group size will affect the expected sample variance, but should not affect the sample mean by too much if the samples are reasonably close in size, especially if both are large. It must be assumed that the actual sampling is random, of course; otherwise this whole analysis fails.

[CrudeHypothesis]

I won't go into the maths, which is unusual for me, but it's certain that statistics can be biased depending on the sample. The larger the sample, the more accurate. Ideally you'd want to measure everyone, but that's almost impossible. Ergo, you are not wrong.

[12ax7]

roninpro's explanation made perfect sense to me. A smaller group will just have a larger chance of being off-center based on individuals.

Take the case of an extreme example: 1 woman signs up and 100 men sign up.
Her individual performance will determine the women's team score, while the men will wind up with a 5ish assuming you get an equal number of contestants at every skill level, voting reflects that, blah blah.

As for compensating for it... pick 5 men at random and don't count their scores??