My other thread about comparing monk weapons got me to looking more
closely at the "accepted" method of comparing weapons in general, and
I'm curious where the formula comes from.

What I've seen everywhere is:

    (damage x 2 + bonus) / delay

The bigger this is, the better the weapon.  For secondary weapons the
bonus is always zero.  Fine so far.  Now, I've also seen this referred
to as the "modal damage", which feels like it must be a misnomer.
Mathematically, the "mode" of a set of data is the value that appears
most often.  What should be more interesting, I'd expect, is the
"mean", i.e. the average amount of damage per hit.  (But "average" is
actually an ambiguous term, which can in fact refer to the mean, the
mode, or even the median (the "middle value" in a sorted list of the
damage numbers).  But I digress.)

Being both curious and mathematically inclined, I wanted to go back
and understand where this formula comes from.  If a weapon is listed
as doing, say, damage 9 with delay 24, what does this really mean?
The "24" means it strikes every 2.4 seconds if autoattack is on,
right?  But what damage does it do when it hits?  Is it a random value
from 1 to 9, like rolling a 9-sided die?  (And if my damage bonus is,
say, +3, it would then be from 4-12 instead of 1-9?)  But if that's
the case, the average (mean) damage would be 5 (plus my damage bonus),
and the formula for comparing weapons would be

    [((damage + 1) / 2) + bonus] / delay

which isn't at all like the accepted formula.  I also found some
discussion giving a formula for maximum damage from a weapon, and
that formula involved STR and weapon skill.  But if those can affect
max damage, why don't they enter into calculation for average damage?
(The claim was max damage = the damage number x (STR + skill) / 100.)

Is it just that people's STR and weapon skills vary so much that
computing average damage is considered to be too complicated, and the
accepted formula is a simplification that "comes close enough"?  E.g.,
I could see the formula being pretty close for a character with STR +
skill = 400, But at level 30 my STR + skill is only about 270-280,
which could make a big difference in comparing weapons if my average
damage is being scaled by that amount and the formula thinks it's 400.

I could try doing some parses of my own, but my logs don't record
which weapons I'm using (other than the hints given by verbs like
"strike" or "crush"), which makes it hard.  (Somehow when I switch
weapons during a fight -- e.g. because I was trying to get some
skillups with 2HB but the fight's going badly so I'm switching to
my higher-skill 1HB or HTH weapons -- I'm too busy to spam a note
into the log to record the weapon change. :-)

So, the short form of the question is:  What is the formula for
finding the average (mean) damage per hit?  Alternatively, what is
the range of possible damage for a single hit, and are all values
in that range equally likely?

    -- Don.

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