D&D Ability Probabilities

D&D (3rd edition) rolls abilities by the best 3 of 4d6 six times and then arranging the results. It is a little tricky to calculate these probabilities in terms of the full distribution, so I am offering below the results of my analysis.

The following table shows the results, including both the six rolls and the two re-roll conditions. When all six rolls are made and arranged from highest to lowest, the following table shows the average result from each -- and the spread.

Rank	Average	RMS	Chance of Bonus / Result
			-4	-3		-2		-1		0		+1		+2		+3		+4
			3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
First	15.95	1.41	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	13.0%	23.6%	29.6%	23.2%	10.6%
Second	14.45	1.58	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.5%	5.0%	16.5%	30.0%	27.5%	15.5%	4.5%	0.4%
Third	13.23	1.69	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	1.5%	7.2%	20.2%	29.6%	25.4%	12.4%	3.4%	0.4%	0.0%
Fourth	12.02	1.91	0.0%	0.0%	0.0%	0.0%	0.0%	0.4%	2.3%	10.9%	21.7%	28.1%	22.5%	10.8%	2.9%	0.4%	0.0%	0.0%
Fifth	10.66	2.35	0.0%	0.0%	0.0%	0.3%	1.4%	6.1%	14.3%	23.6%	24.8%	18.2%	8.5%	2.4%	0.4%	0.0%	0.0%	0.0%
Sixth	8.72	3.53	0.3%	1.3%	3.2%	7.5%	12.8%	19.1%	21.0%	17.7%	10.9%	4.7%	1.3%	0.2%	0.0%	0.0%	0.0%	0.0%

There are some notable things to note from this chart. In general, the spread is fairly narrow for the top scores, but it widens out for the lower ones. You can see there is a 10% chance of getting an 18 in your top ability, and a 0.4% chance of getting two 18's (i.e. second is also 18). There is a 71% chance that your top 3 scores will be 13 or higher. This is important for feat prerequisites.

For comparison, here is the table for a single roll of best 3 out of 4d6:

Average	RMS	Chance of Bonus / Result
		-4	-3		-2		-1		0		+1		+2		+3		+4
		3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
12.24	8.10	0%	0%	1%	2%	3%	5%	7%	9%	11%	13%	13%	12%	10%	7%	4%	2%

Point Build

The CD utility included with the Players Handbook includes an optional point-build system. This can be found in the Dungeon Master's Guide (as "standard point buy", page 19), or reverse-engineered from the CD utility:

There are 25 points to distribute among the abilities, which all start at 8. Note that to buy the average single roll (12.25) for all abilities costs around 25 build points. However, to buy the average spread (16.0/ 14.5/ 1325/ 12.0/ 10.7/ 8.7) costs around 29 points.