The big complaint about rolling stats is usually the large variance between characters. That arises naturally from the roll used to determine each ability score, but it is also enhanced because each ability score is rolled independently. We can clamp down on the variance for each score (e.g. using 2d6+6 instead of 3d6) but we can also limit the (actual or apparent) independence between scores in order to shape the distribution, yet we rarely see the latter explicitly in rolling variants. Compare to point buy, where both the randomness and independence of ability scores is removed.
I just mused for a few minutes and came up with what might be a new mechanic based on the idea of Markov chains, which the mathematically inclined might recognize. The basic idea is that the roll used to determine the next ability score is determined by the total ability modifier of scores already rolled, but we don't care about how we got that modifier or what it was at previous steps. There are lots of ways to go about that process, but here is one that creates an average character similar to 4d6 drop lowest but with much smaller variance between characters.
Let i be the number of ability scores already rolled, and k be the total ability score modifier of the existing ability scores. For the next ability score you roll 3d6 plus bonus dice equal to (i-k) if i-k>0, and keep the best 3. (For people who have traditionally liked to roll scores in order, which roll corresponds to which ability score should be randomized so that Strength, for example, isn't always just a 3d6.)
Here is a typical example showing (i-k) and the rolls.
Roll 1: (0-0)=0, so roll 3d6. Get 11.
Roll 2: (1-0)=1 so roll 4d6. Get 13.
Roll 3: (2-1)=1 so roll 4d6. Get 17.
Roll 4: (3-4)=-1 so roll 3d6. Get 8.
Roll 5: (4-3)=1 so roll 4d6. Get 10.
Roll 6: (5-3)=2 so roll 5d6. Get 16.
This is a perfectly playable character with a total ability score mod of +6.
Here's a list of 100 characters generated using this method.
[sblock]
Code:
7 18 12 9 11 10
6 12 15 14 14 12
8 15 15 12 11 11
16 9 16 17 10 10
10 10 13 12 11 13
13 13 7 11 14 11
11 11 14 15 8 10
8 15 10 13 10 13
12 8 14 14 13 10
12 10 12 11 11 13
8 16 12 12 11 15
10 11 15 15 17 13
12 9 15 13 13 15
6 15 16 6 14 14
11 13 12 13 16 12
10 15 11 8 17 13
6 15 11 17 8 14
10 15 10 12 4 16
14 13 14 13 9 8
10 17 14 9 10 14
8 12 16 13 15 15
10 15 9 10 13 15
5 14 14 12 10 14
12 6 12 13 16 12
13 13 14 6 13 16
10 13 7 16 14 14
9 13 15 11 11 15
13 9 15 16 9 15
11 13 12 12 14 9
8 6 14 17 13 15
17 9 9 16 9 13
11 8 13 9 17 6
10 10 10 12 17 15
8 16 15 12 14 8
15 10 11 14 7 17
11 13 14 9 10 14
12 14 13 12 12 12
9 11 15 15 13 13
7 14 11 17 7 15
9 14 12 12 12 14
9 10 15 14 14 9
8 17 9 13 12 16
14 9 12 7 14 18
14 13 10 10 18 16
14 9 7 16 17 8
12 12 13 13 10 12
14 3 10 16 16 6
7 18 9 14 10 16
4 12 14 14 8 16
14 4 18 9 16 7
13 16 9 8 11 12
10 10 13 15 12 10
10 18 13 7 15 15
6 11 16 17 11 9
15 17 8 9 15 7
11 7 18 10 12 13
7 14 14 15 7 11
11 11 16 7 16 12
6 16 15 11 12 12
12 14 11 11 14 6
10 15 9 15 14 12
15 12 9 14 10 11
9 16 14 12 5 15
9 11 14 16 3 18
15 10 13 11 10 9
15 12 6 18 8 17
8 13 16 7 17 13
17 11 10 9 16 9
14 6 9 18 12 13
8 7 11 13 17 15
8 9 13 15 15 6
10 14 14 17 14 13
9 16 9 13 14 12
11 11 16 11 13 8
10 13 15 9 17 11
9 16 9 9 16 15
7 16 16 17 16 8
8 16 5 14 16 13
9 12 14 9 17 17
16 8 11 5 16 17
8 11 12 15 16 9
10 10 14 15 15 13
15 5 14 12 17 11
10 8 13 13 10 13
9 15 13 13 14 10
17 11 13 13 16 10
8 11 13 12 10 18
4 15 7 16 14 13
5 15 15 16 10 14
12 13 7 13 15 9
11 9 17 12 15 10
5 12 14 16 12 10
6 15 13 11 18 4
11 15 11 14 15 11
7 18 9 14 13 9
13 12 10 9 17 14
12 14 7 12 12 10
7 16 13 9 16 9
9 16 10 11 17 13
10 15 9 14 13 11
[/sblock]
As you can see, the first roll is always 3d6, but the second roll could be as high as 8d6 if the player rolls a 3. A player rolling decently will almost never see any bonus dice, while a character rolling poorly is likely to end up with a few high scores by the end. The conceit is that the average character will average about +1 in each ability score, similar to 4d6. Strictly speaking a character could have 3s in every stat, but to do so the player must roll 3+8+13+18+23+28=93 ones in a row, and it will just never happen. Despite this suppression of low scores there really isn't any enhancement of high scores compared to straight 3d6 rolling. This method has an average total ability modifier of about 4.75 with a standard deviation of about 1.7. Compare that to 4d6 drop lowest, which has average total ability modifier of about 5.25 with a standard deviation of about 3.5. The distribution of maximum score minus minimum score (a quick and dirty measurement of how "interesting" a character is) is basically the same for all 3 methods. So this method doesn't really address the problem of rolling "boring" characters with no real strengths or weaknesses.
(Math nerds might appreciate this, though: the standard deviation of the ability modifier for a single ability score is about 1.5 for 3d6, and 1.44 for 4d6 drop lowest, and we can simply multiply by sqrt(6) to find the standard deviation for the sum of ability score modifiers if all scores are independent and rolled using the same method. That is, about 3.7 and 3.5, respectively. The Markov chain method above gives a standard deviation on the sum of about 1.7, meaning we have suppressed nearly all the variance introduced by the summation of 6 random variables using traditional rolling methods to the variance of just slightly greater than that of a single 3d6 or 4d6 roll.)
Lots of variants are possible, or course, and one can target different power levels (or if you prefer, the "population" from which the character is assumed to have been selected from) by changing how bonus dice are given out. One could even penalize characters that are doing too well (e.g. roll 4d6 keep lowest 3), although I think keeping things bonus-oriented feels less coercive. Any method that tweaks the probabilities back toward the mean, however, will tend to suppress the variance of traditional rolling methods. The approach introduced above, however, keeps the familiar flavor of roll nd6, keep 3.
