Let's take Frogreaver's example, for simplicity: 1d6.
We can agree that the average result of a roll on a fair 1d6 is 3.5. In this case, the average roll AND the average result we keep, are both 3.5.
What about 1d6, re-roll 1s? The only results we can actually keep are 2s, 3s, 4s, 5s and 6s. If we average the results that we keep, the average is 4.
But the average roll is STILL 3.5!
Okay, so you're employing a semantic argument. When I say "roll higher stats," I mean "roll
and keep higher stats." Because rolls you discard don't matter for actual play, by definition. For
generating stats, yes, you're always going to have a percentage chance of getting a stat roll you can't keep--because the generator doesn't care what your manipulations are. But if you generate 10,000
actually acceptable characters, their average is different, and within that set, there IS a higher probability of having good stats when compared to a set with no rejection rules whatsoever.
Let's take the nice, even bell curve of 3d6, averaging 10.5. If you discard every roll of less than 8, what's the average roll? It's still 10.5! You are not any more likely to roll an 18! Your bell curve is exactly the same, but the left hand part of the graph (up to '8') has been truncated.
Teeeeeechnically speaking that's not accurate. The probabilities--again, EXCLUSIVELY FOR KEPT ROLLS--have to go up, because the sum of the probabilities must be 1. They go up by an amount proportional to the number of options removed. For communication's sake, instead of talking about "rolling" (which seems to be the core of the bugaboo), I'm going to talk about
recorded stats--you never, ever "record" stats that don't match whatever rules you're using, so from the perspective of the "recorded" stats, it doesn't matter how many "rolls" you had to go through to generate a number, it only cares about those sets that play by da rules.
Like I said: by mathematical definition, if the average result goes up, EITHER the probabilities are different, or the possible results are different, or both. With your first example case, if the probabilities remained identical, the average result should be 2/6+3/6+4/6+5/6+6/6 = 20/6 = 20/6 = 3.333..., not 4 as you have claimed. The only way to get the 4 you have claimed is if we say that the uniform distribution is now p=0.2, not 0.166...; this gives us 2/5+3/5+4/5+5/5+6/5 = 20/5 = 4, just as you've said.
But that means the probabilities are now higher than they were before. They are, in fact, higher by exactly (probability of number of options removed)/(number of options remaining) = (1/6)/(5) = 1/30. 1/6 + 1/30 = 5/30 + 1/30 = 6/30 = 1/5. For a non-uniform distribution, the changes are harder to calculate, but they still exist.
It's true that, for simple rejection of a particular result value, you have to truncate off the rejected part when plotting the "kept" ("recorded") distribution. But the recorded distribution will, as a result, be (overall)
taller than the "rolled" distribution (given the same axes), as you cannot truly "remove" area from a complete histogram--that is, one where the probabilities sum to 1, which will always be the case for any dice-based distribution (since there is only a finite number of possible results, each with positive probability). This is the graphical analogue to the "probability increases from 1/6 to 1/5" statement. Some probabilities may go down, but the net effect will be most probabilities going up; a uniform distribution guarantees all probabilities go up.
When you roll your stats, the maths of the method you use to generate those scores (3d6, 4d6k3, 4d6k3 re-roll 1s, whatever) are not affected by any later choice to not use a particular set.
The data seems to say otherwise.
The probability of 6 on 4d6k3r1 is
much lower than on just 4d6k3--reduced by an order of magnitude, in fact. Your probability of
recording 17-or-higher has almost doubled. That sounds like a change to the math to me! Of course, if you aren't using a computer to automatically get rid of values you won't accept, you'll spend extra time, because you'll need to reroll all the ones that show up, but that's not the same thing. (Technically, you could just use d10s, dividing even numbers in half and adding 1 to each die--that would produce the same distribution WITHOUT any "rules for removal" parts.) That is: 4d6k3r1 is precisely equivalent to 4d5k3+3--yet with the former I have "rejection rules," while with the latter I do not. Seems hard to argue with that...
This doesn't make any actual set that you did roll any higher than they were before you took the low ones away!
I haven't been trying to say that it does, and given my repeated statements to that fact, I'm getting a little annoyed here. Rolling 4 dice--whether you keep 3 or not, whether you ignore 1s or not--has a particular distribution. Now, in my opinion, you are being
fantastically pedantic, since as I have repeatedly said, results you
could roll but which you won't (using my new terms)
record don't, and CAN'T, matter. So the difference between "recorded" and "rolled" is, essentially, semantics--characters end up having better stats, on the average, if you remove the possibility of characters having
, low stats.
By the way, you can check
the equivalence of 4d6k3r1 and 4d5k3+3 in AnyDice, if you like. The fact that a "rejection rule" can be equivalent to literally using a different kind of dice seems pretty conclusive, to me, that rejection rules can in fact be mathematically equivalent to a completely different distribution....which is the same as saying that the results are different (in this case, better; if we were rejecting 6s, it would of course be worse).
