User:Cedges/Median Stats

= Median Stats =

The point-buy system included in 3.5 creates characters that are underpowered and bland compared to characters generated with the 4d6 method. This variant rule allows starting stats as long it would be at least as likely to get an absolutely better configuration as it would be to get absolutely worse configuration when rolling 4d6. That is, we accept starting stats below this generalized median.

Derivation
For multiple independent, ordered experiments, the ratio of results absolutely better than an outcome to results absolutely worse (R) is the product of the probabilities of outcomes better or equivalent less the product of the probability of the specific outcomes divided by the product of the probabilities of outcomes worse or equivalent less the produce of the probability of the specific outcomes.

R(x0, x1, x2...) = (product(G(x0), G(x1), G(x2) ...) - product(P(x0), P(x1), P(x2) ...) / (product(F(x0), F(x1), F(x2) ...) - product(P(x0), P(x1), P(x2) ...)

In the specific case of R = 1, one part over for every part under, the products of the specific probabilities can be removed from the equation, and R(x0, x1, x2...) >= 1 will hold as long as:

product(F(x0)/G(x0), F(x1)/G(x1), F(x2)/G(x2), ...) <= 1

Therefore, one could determine exactly if a set of stats were below this median by multiplying the number from the fourth line of the table for each stat and checking to see if the product is less than or equal to 1.

Simplification
The logarithm function eats multiplications and turns them into additions, which are easier to do in one's head. The median constraint will be met if and only if:

ln(F(x0)/G(x0)) + ln(F(x1)/G(x1)) + ln(F(x2)/G(x2) + ... <= 0

Allowed Scores
The following are the greatest allowed scores, with the highest score assigned to Str, the second highest to Dex, and so on. The attributes need not be assigned in the same order. Total is the total of the attribute scores and total mods is the total of their attribute modifiers. Comparable is the probability that a regularly rolled set of attributes could be compared to the listed attributes and be determined to be absolutely better or worse. Exact is the probability of rolling this exact set of attributes using the 4d6 method, before re-rolls. The attribute sets with a strength of 13 would be re-rolls under the 4d6 method.