Here's My System
When looking at dice rolled values, there must be an understanding that probabilities for the lower and upper bounds are extreme (1 in 1296 and 6 in 1296 respectively). As a result, achieving the extremes of a value should be far more difficult than for those bounds in the point buy system currently in the game.
Here is a graph of the distribution for 4d6 drop lowest (Mean = 12.24, Standard Deviation = 2.85)
Take particular notice of the extreme rarity of 3, 4, and 5 values. The best way to accommodate this is to give values below the mean their own point system. Essentially there will be two parallel point systems. The basic weight of the point costs will roughly compare to the number of discrete rolls away from the medians (of 4,4,4,2... among others).
The modified point buy
Each of the six ability scores starts at 10. You get 19 points (Strength Points) to buy score values to be greater than 10, and you can get up to 10 more points (Weakness Points) from lowering certain abilities based on how much you lower them below the base score of 10.
Here is a table that shows the point values for each lesser ability score.
This demonstrates the extremely low probability that you will have multiple values less than 5 or 6. Each point of weakness you expend provides one point to your strength points.
Strength points reverse the effect of any attribute less than 10, and here is the table for increasing past 10:
As you can see, even if you expend all 10 weakness points, it is impossible to get two 18's in this system (or even an 18 and a 17). This is because the likelihood that you roll two 18's is 0.38%. To look at the system in action, here are some sample distributions (notice that moving along the curve in the negative direction has a bigger impact since there are less discrete rolls for the values in those regions):
- 16, 14, 13, 12, 10, 9 (this is the average result for a dice roll)
- 17, 16, 14, 9, 7, 5 (one standard deviation from above)
- 18, 16, 12, 9, 6, 5 (one standard deviation from above)
- 18, 16, 12, 10, 9, 3 (with one dump stat)
Note: This system may appear to take advantage of the fallacious statistical model commonly known as the gambler's fallacy. I am aware that rolling a low value (like four 1's) does not increase your chance of rolling a higher value; this system is designed to allow for adjustments above the standard system by accounting for probabilistic measures for the rolled abilities as a whole.