Saturday, January 4, 2014

The Case for Narrow Ability Scores

Ability Scores, What's Up With THEM???

In OD&D, ability scores only do two things:
1. Roll-under mechanics (roll under your Strength to succeed).
2. Generating ability bonuses.
Since the ability score --> ability bonus conversion is handled by a separate table, we can pretty much engineer that separately.  Tables like this:

4-      : -2
5-8    : -1
9-12  : +0
13-16: +1
17+   : +2

As long as higher ability scores give higher bonuses, we can rejigger that table however we want.  So, we can engineer each independently to get whatever sort of scores/bonuses we want.

Throwing a Sacred Cow on the Grill

Everyone freely modifies the ability score bonuses.  Some go from -3 to +3, others never give any ability score more drastic than -1 or +1.  This is an old tradition.

But rarely does anyone mess with the 3d6 spread for ability score generation.  That shit's sacred.

When I DM, I call for a lot of roll-under mechanics.  I use them extensively: jumping over a gap, handling drunkeness, decoding some ancient language on the wall.  

And in my (admittedly limited) experience: I think that the 3-18 spread is a little too large. A player with a 17 in a stat can pull off roll-unders with remarkable consistency, while a character with a 7 struggles.

So, I would like to narrow the ability score spread.  The average will still be ~10, just the standard deviation will be smaller, and the highest and lowest scores won't be so drastic.

Digression: In 3.5e and onwards, they do a lot of stat checks that amount to simply the ability bonus (score-10/2) added to a d20 roll and then compared against a static DC.  This is analogous to what I'm talking about, since an Str 18 becomes d20+4 vs DC 11 (or whatever), which is a 70% success chance instead of a 90% success chance with a straight roll-under mechanic.

Digression: It's sort of interesting to note that with with newer editions of DnD, roll-under mechanics never get used, the only thing that they are good for is generating ability score bonuses.  It would be trivial to prune out ability scores altogether and just play with the bonuses.

Narrowing the Spread

So, how about this:  Instead of rolling 3d6 to generate a number from 3-18 (avg = 10.5), we roll 4d4 for each stat to generate a number from 4-16 (avg = 10). 

Bonuses are calculated like this:
5-      : -2
6-8    : -1
9-11  : +0
12-14: +1
15+   : +2

The nice thing about this method of stat generation is that it produces pretty much the same stat bonus distributions compared to the 3d6 method printed at the top of this post.  Behold.

4d4 3d6

Score Percentage Score Percentage
-2 5- 1.95 4- 1.85
-1 6-8 23.83 5-8 24.07
0 9-11 48.45 9-12 48.14
1 12-14 23.83 13-16 24.07
2 15+ 1.95 17+ 1.85

So if you were worried that using 4d6 for stat generation would make all your attack bonus, dex bonus math go all wonky, calm yo' tits.

Happily, it does succeed in my goal of bringing ability scores closer to 10, and making the roll-under mechanic a bit more reliable.

Using 4d4 even remains eerily similar to 3d6 when you modify for racial ability modifiers.

4d4+1 3d6+1

Score Percentage Score Percentage
-2 5- 0.39 4- 0.46
-1 6-8 13.28 5-8 15.74
0 9-11 44.93 9-12 46.29
1 12-14 35.55 13-16 32.86
2 15+ 5.86 17+ 4.17
Using the "roll an extra die, drop the lowest" during chargen also lines up pretty well.

5d4h4 4d6h3

Score Percentage Score Percentage
-2 5- 0.59 4- 0.39
-1 6-8 11.33 5-8 10.1
0 9-11 41.01 9-12 40.74
1 12-14 40.63 13-16 42.98
2 15+ 6.44 17+ 5.79

To convert from 4d4 to 3d6, add your ability bonus to your ability score.
To convert from 3d6 to 4d4, subtract your ability bonus from your ability score.

Closing Thoughts

I've made a few sample characters using this method, and rolling 4d4 just isn't as fun as rolling 3d6.  It's not the d4's fault that it's such an unfun die to roll, but it is.

Maybe you like the power and consistency of having an 18 in a stat.  Or you don't use roll-unders much anyway.  But if so, this probably isn't for you.

If you want to adapt a 4d4 bell curve to a different spread of bonuses (maybe -3 to +3) it's pretty simple to crack open the probabilities and assign thresholds for different bonuses wherever you want.

The average of 4d4 is 10, which is slightly lower than 3d6's average of 10.5.  

If you want to see the actual probability curve, it looks a little like this:

Thanks for the lovely graphic/computations.


  1. Initially I like this a lot, but my one big problem with it is that it weights ability scores even more heavily towards center of the curve.

    It's fairly common to roll a character whose ability scores are all in the 9-12 range, which is fine. But rolling outside of that range is fun and exciting. I can't see it as a positive thing to make that excitement even less likely than it is already.

    Further, as another GM who calls for a lot of roll-under checks, I can't say I've ever thought characters with high or low ability scores cause a problem. One in ~200 characters will have 18 strength. It won't happen often, but when it does, won't it be fun to overcome nearly every strength challenge the GM throws at you?

  2. Wouldn't rolling ability checks on 3d6 accomplish your objectives?

    1. True! But I'm trying to unify mechanics when I can, and also want to use stat checks more often (replacing a few things that saves are normally used for).

    2. I hope I'm not misunderstanding your point, but rolling ability checks on 3d6 would make people with high ability scores even more overpowered because of the difference between linear and bell probability curves.

      Lets say you have a strength of 17, and want to lift a heavy portcullis. The GM asks for a roll-under check.

      If the roll under is a 1d20, failure is on an 18, 19, and 20. Each number has an equal chance of coming up, so there's a 15% chance of failure.

      On 3d6, you fail only on an 18. And because probability is weighted towards 10/11, the actual probability of rolling an 18 on any given 3d6 roll is less than 0.5%.

      Or I'm completely misunderstanding your point and everything I just said sounded really condescending.

    3. Wait, no, yes, LS is totally right.

  3. You could also roll 6 d6 drop 2 lowest and highest or 7 d6 keep 3 middle