Prak wrote:Look, it should be common knowledge by now that I suck with statistics and probability. Hell, I have the thread you made to tell Ellenser about probability bookmarked for my own elucidation.
But I can figure out straight dice averages. So, yes, I know that the average of 3d6 is 10.5. I have no clue what dropping a die does to the average roll, so I stay away from such calculations.
But for Int in D&D, because 3 is the floor, the average orc int is 9.5, rather than the 8.5 the dice would suggest, because you're generating a number between 3 and 16.
Ah. I see where you're fucking up. Momo is right by the way, but I think you'll probably want a bit more explanation as to
why. The basic story is that the average of a die roll is not always the smallest possible result added to the largest possible result divided by 2. We use that as a
shortcut, but it's only actually accurate when dealing with a uniform distribution or centered bellcurve. The longform for determining an average is to add up every possible die result and then divide by the number of possible results.
So when determining the average of a d6, the "right" way to do it is to add:
1 + 2 + 3 + 4 + 5 + 6 = 21
And then divide by 6, getting a result of 3.5. However, since it's a uniform distribution, we know that the 1+6 gets us the same value as the 2+5 and the 3+4. So we can skip a bit and just average the beginning and the end - all the middle parts will average the same way.
When we determine the average of 2d6, the "right" way to do it is to add:
2 + 3 + 3 + 4 + 4 + 4 ... + 11 + 11 + 12 = 252.
And then we divide by 36, getting a result of 7. But we don't do that, because we can just average the 2 and the 12 because it's a well centered bellcurve and there are just as many 6s as 8s and just as many 10s as 4s.
But let's consider the average of something that is
not well centered. Like say, a White Wolf die. It has ten sides, but most of those sides are counted as "zero," with just a few of them counted as "one." Then there's one side of the die (the side which ironically
actually has a zero on it) which is either counted as 1 plus a bonus die or as 2 depending on which White Wolf product we're looking at. For clarity of the discussion, let's assume we're looking at the version that counts as 2. So the lowest value is 0 and the highest value is 2, but the
average is not 1. We instead go about doing this the right way, and add up the sides on the dice and divide by the number of sides:
0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 = 5
Now we divide by 10, and see that each die produces an average value of one half (note: average values on white wolf dice vary considerably from game to game, a nWoD die is worth one third, but this is a Scion die that is worth one half).
Now this goes back to our example Orc. We're rolling 3d6 and getting a well centered bellcurve, so we can skip the thing where we add up all 216 possibilities and then divide 2268 by 216 and get 10.5. But when we shift with a floor, it's no longer perfectly centered. There's only one result of 16, and three results of 15, but there's zero results of 1 or 2 and
ten results of 3.
We are still extremely lazy people, and we don't
want to add up all 216 results and then divide by 216. Because that seems like a lot of work. So what we actually do is ignore the floor entirely, work it out as a smooth bellcurve going from 1 to 16, then we add up just the differences from those values provided by the floor and divide
that by 216. So we upshit 2 on the one result that is a natural 3 and upshit 1 on the three results that are a natural 4. That's a total upshift of 5, divided by 216 is 0.02315.
Then we add that to our base average value of 8.5 and get the true average: 8.52315. But if you round it off to two significant figures, that's still 8.5 - because the floor value only comes up on one in 54 rolls and doesn't change the average by even one tenth of a point.
-Username17