Wednesday, August 25, 2010

Scaling of standard deviation of N for 4E at-wills.

Let's examine how the standard deviation (stdev) of N changes with level.

Recall the average number of hits to kill a monster by a non-striker player character (of the same level) using an at-will power with weapon damage dice [w] = level*[W],

N = [ROLE*(level+1) + CON]/[(average[w])/10 + 7*(level-10)/20]

where one of the egregious assumption made, was that the damage of at-will powers [W] increases each ten levels after level 10, in the pattern of:

level 21-30 --> 2[W]
level 31-40 --> 3[W]
level 41-50 --> 4[W]
level 51-60 --> 5[W]
etc ...

For weapon damage dice 2[W], it means that the number of weapon dice is doubled. For example if the weapon dice [W] is a d6, then 2[W] means 2d6, 3[W] means 3d6, etc ...

The average of damage dice "level*[W]", turns out to be equal to level*average[W].

But the standard deviation of damage dice "level*[W]", turns out to be [sqrt(level)]*stdev[W]. ("sqrt" is the square root). This can be found in any college statistics/probability textbook.

(For example, (stdev[3d6])^2 = (stdev[d6])^2 + (stdev[d6])^2 + (stdev[d6])^2 which gives stdev[3d6] = sqrt(3) * stdev[d6]).

Recall that for individual dice like d4, d6, d8, d10, d12, etc ... the standard deviation is:

stdev[dN] = sqrt[(N^2 -1)/12]

where sqrt is the square root. (A single die follows a discrete uniform distribution).

For example,

stdev[d4] = 1.118
stdev[d6] = 1.708
stdev[d8] = 2.291
stdev[d10] = 2.872
stdev[d12] = 3.452

To get the standard deviation of N, there's the formula:

stdev[N] = |dN/dw| * stdev[w]

where w = level*[W]. The term in between the | | absolute value brackets is the first derivative of N with respect to [W].

Doing the calculation of the standard deviation of N and scaling the level to infinity, we get:

stdev[N] = N^2 *(stdev[W])/[ROLE*sqrt(level/10)]

when the level becomes larger and larger.

Hence the standard deviation of the average number of hits "N" to kill a monster by a player repeatedly using at-will powers against a monster (of equivalent level), scales as:

stdev[N] ~ 1/sqrt(level)

This means the standard deviation of N moderately gets smaller and smaller as the level gets bigger and bigger. The number of hits to kill a monster (of the same level) deviates less and less from the average, as the level becomes bigger and bigger.

Basically while one is "always fighting orcs", the "orcs" are becoming more and more "predictable" as the level gets higher and higher.