To clarify, character creation by rolling ability scores does not involve finding the average rolls for a particular method and using them. It involves rolling (in our example) 4d6k3 six times. The average for those rolls might be interesting, but the average plays no part in the actual rolls you just made! Any discarding of an unsuitable set has absolutely no impact on the set you keep! Any discarding is done after those rolls (with the mathematical average associated with that method) has already been completed. Any later discarding does not alter the probabilities of that method to get those results.
Averages are measures of center, not measures of individual data points. That's a trivially true statement, so yes, you're absolutely correct. But modifying a data set at any point--before or after it is collected--
does, in fact, make a difference. It removes elements. By definition, you're changing probabilities--because, at the very least, you're changing either N or n (population size or sample size, respectively) which is almost always going to result in a change (as N or n-1 is used in the denominator of means, standard deviations, and proportions, for populations or samples respectively).
I mean, you might as well say that the average result of d6, or any platonic die, is impossible because you can't get half-values. Yes,
of course it's not possible to actually roll 10.5 on a d20--that doesn't, in any way, mean that the average ceases to be a meaningful descriptor of the probability space, nor that it is incorrect to make use of measures-of-center, before and after modification of the data set (population or sample), to talk about how those modifications alter the distribution.
Let's try an analogy: let's say that the average height of an adult human male is, say, six feet. Let's say that you have a six-foot tall adult human male standing in front of you. Do you have a male of average height in front of you? Yes.
Now, shoot every adult male who is less than six-foot tall. (note that I'm not actually advocating this in real life; it's just a thought experiment. Put the shotgun down and back away from the 2nd amendment!) Now that there are no adult males less than six feet, the guy standing in front of you can truthfully be said to no longer be 'average height'; he's now the joint shortest guy in the world! But his actual height has not changed one iota! He's still six foot tall, no matter the heights of other people alive or dead!
And if that guy has kids, the average height of those kids when they grow up will conform to the pre-culled population, not the post-culled population. Their heights will lie along the original bell curve of heights, not the bell curve of heights post-culling.
Ouch, bad analogy is bad.
First: height isn't a uniform distribution (as dice are) but rather an approximately normal distribution, and thus has a distinctively different shape. Making significant changes to a normal--like cutting out all values below the original mean--has
enormous impact on the information from the normal (it becomes not only non-normal, but extremely wonky, like an F-distribution with no front tail or something?). Cutting out values above or below on a uniform is distribution is a much smaller change.
Second: if you cull all members below the average from the entire population of human males, then the parameters
μ and
p almost certainly must change, because (1) you're changing the minimum value dramatically as well as the number of data points N, so
μ MUST become a different number for the modified population, and (2) the numerator and denominator change and we cannot say that the changes result in proportional differences (e.g. if there are 1000 males in the world, 50 are six foot and 500 are taller, removing the 450 that are not means the odds of being six foot are now 50/550 = 1/11 as opposed to the original 50/1000 = 1/20.)
Third: if
only those men over 6 feet tall have kids, there is a causative impact on the future generation--you have removed genes from the gene pool, applied artificial selection, which means important variables (genetics, socioeconomic status, and nutrition)
are no longer random and thus a normal distribution may not even describe the new data.
I agree that, for
actual living people, the shape of the distribution doesn't spontaneously change. But we're not talking about actual living people. We're talking about
generating a number of values, and discarding all of those that don't fit particular rules. If you continue to generate things based on those selection rules, you are rejecting some data and keeping other data...
which will change the shape of the distribution.
And when you discard the first (or second or third) set of rolls, the next set will be generated by the same mathematical rules of the first, and won't take any discarding (and that effect on the final average) into account. The next set will have the same bell curve as the original, and that curve will be unaffected by any discarding.
So the set of scores that you do keep are the same (and use the same bell curve) whether or not you are allowed to re-roll.
Okay, let's use a slightly different example then. Consider "4d6 drop lowest" vs. "4d6 drop lowest, reroll all 1s." That is a modification to the data set, and is largely equivalent to removing a certain slice of heights, so it even allows us to critique your previous assertions too. (Technically the most accurate statistical explanation of it is different, but not in ways that matter to the discussion at hand: we are still ignoring parts of the potential data set.) If you always reroll 1s until you don't have any 1s, then you're effectively rolling 4d{2,3,4,5,6} drop lowest. Here's the
AnyDice results for the two:
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As you can see, the distribution
is clearly different. It has a different center (mean), a different spread (standard distribution), and different probability values for every single point. All because we chose to remove--that is, re-roll--any dice that show a 1.
The parameters change when you apply selection rules that change the population from which data may be drawn. That is a statistical fact. Can you demonstrate otherwise, with actual
numbers rather than analogies?
Consider a hyperbolic example: "I will reject all values less than 18." What is the probability of having any given stat at 18, given this rule? 100%. That is
obviously different from the probability for 4d6 drop lowest. Then consider "I will reject all values less than 17." With only two options, we know
absolutely for certain that *at least one* of the probabilities--m_p(17) or m_p(18), where m_ means "modified"--
must be different, because m_p(17)+m_p(18) = 1, but the *unmodified* p(17) + p(18) is
much less than 1. Now consider a slightly more useful example, "I will reject all values less than 15." That's
easily programmed into AnyDice--it's the equivalent of rolling 4d{5,6} drop lowest. We can
clearly see that the probabilities are radically different...yet, according to your suggestion, they should look precisely identical.
Removing regions from the possibility space must by definition shuffle the probabilities, because p(x1)+p(x2)+...+p(xN) = 1
and m_p(x1)+m_p(x2)+...+m_p(xM) = 1, but N > M. *At least one* of the modified probabilities must be greater than the unmodified probabilities in order for these equalities to hold.
