I was pondering the concept of balance for a bit, and tried to input some calculations into a discussion I was in to make the ones already presented a bit more accurate and clean looking... but I ran into some issues.
The standardly accepted formula for taking a melee weapon and finding out the average damage it puts out is (max-min)*0.8+min=average. The 0.8 represents an assumed 80% balance, and can be adjusted by the user's specific balance with the weapon on a case by case basis, but most discussions tend to be about ideal builds, and the comparisons are for users strong enough to have capped balance, crit, and so on.
In some guide I read, an explanation of exactly how balance works details the following: all possible damage values of a weapon's range spread over a curve, and the top of that curve is the balance. the half of the possible values above the curve happen 50% of the time, while the values below the curve happen the other 50% of the time. A high balance means the portion above the curve (the right side is what I mean above) is a small number of possible values, so those numbers will individually show up more often than the ones below the balance point. example, a weapon with 0-100 damage and 80% balance, the 20 possible damage values from 80-100 will show up half the time, while the values from 0-79 will show up the other half of the time.
Kakashi7's formula is Average damage=Max*balance+min*(1-balance). In his example, a weapon with 50-100 damage, and 30% balance is used. He arrives at 65 average damage. Plugging these numbers into the wiki's formula, we get (100-50)*0.3+50=65 which matches.
Question one: Does the above formula accurately represent the above curve?
In regards to magic, it is possible to get 100% magical balance. This doesn't mean your magic attacks will always hit max damage, but will actually hit max damage half the time, while all other numbers in the possible range evenly making up the other half. Now, lets say we have a hypothetical spell that does 0-100 damage, and has 100% balance. The formula would look like (100-0)*1+0, and result in 100 average damage. This is clearly not accurate, as half the attacks will produce damage values anywhere from 0-99. The average damage should therefore be something closer to 75. Using kakashi7's formula, we we get 100*1+0*(1-100)=100, which is still clearly not 75.
Question 2: what is wrong with the formulas avg=(max-min)*balance+min and/or avg=max*balance+min*(1-balance) then? What sort of formula would accurately recreate the actual effects of balance, to give a more accurate average damage?