So I know all of the excel sheets/formulas for figuring out DPR in Pathfinder and 4e. However, I'm having some problems in figuring out what a formula for calculating average damage for a weapon would be in 3.5/PF.
We're messing with some home rules and want to flatten out/average some weapons. Essentially, I was hoping someone tell me how to figure out average damage for a weapon when taking into account critical.
I've heard that 19-20/x2 and x3 are (roughly) equal (if not taking enemy AC's into account). However, how does x5 and 17-20/x2 compare? How much would a property like Brutal from 4e add? (makes it so you re-roll damage on a 1). Hopefully someone better inclined toward mathematics can help me out!
The average value (technically "expected value" when it's hypothetical, and not based on an actual set of rolls) is equal to the (max value of the die + minimum value)/2
So the expected value for a d6 is (6+1)/2 or 3.5. Or a d20 is (20+1)/2 or 10.5. The Brutal property you mentioned simply removes "1" from the list of possible values, so a d6 would become (6+2)/2 or 4. It's only increasing your damage by 0.5 on average.
For the crits, assuming confirmation chance is equal, a x5 and a 17-20/x2 will come out the same. In the first case, you have a 1/20 chance of getting +400% damage. In the second, you have 4x the chance, but for 1/4 the extra damage (+100%). The problem is that a x5 crit will tend to give you massive overkill, and waste a lot of that damage. It's also less useful with crit feats that give things besides damage.
You can't really do it properly without taking AC into account since the chance to hit is more important than the weapon's base damage.
If you just want average base damage for a weapon with one die then such as 1d8 then divide by 2 and add .5 so a d8 would be 4.5
If the weapon has 2 die such as a scythe then divide each die by 2 and add them together. A scythe with 2d4 would be a 5.
Using the 1d8 as an example you add every number from 1 to 8, and then divide by 8.
If you want to account for the rerolling of a 1 then you add every number from 2 to 8, and divide by 7.
Base weapon damage is not really that important though. The damage that is added on at the end that comes from strength is what you need to focus on if you want more damage.
A simple way to visualize crits is this:
A 19-20/x2 weapon does weapon-average damage on a roll of 1 through 18, so if you total that up, you've got 18x average. Then on a 19 or a 20 it does double weapon average damage. So you've got 2x average and 2x average. Total 18 + 2 + 2 and you've got 22x weapon average.
A /x3 weapon does weapon-average damage on a roll of 1 through 19, so if you total that up, you've got 19x average. Then on a roll of 20 it does triple weapon average damage. So you've got 3x average. Total 19 + 3 and you've got 22x weapon average.
You can do that with all the threat ranges... literally count out what you're going to get. 15-20/x2? Well, 1-14 gives 14x and 15, 16, 17, 18, 19 and 20 give 2x each for 12 more, making a total of 26x weapon average which is equivalent to a /x7 weapon.
In any case, confirmation will be the same because your threat range doesn't play into it.