Game Design Dice – What Dice Mechanic Gives a Bell Curve Distribution That Narrows and Increases Mean as Skill Increases?

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I'm looking for a dice mechanic that produces a bell curve at all skill levels. I also want it to increase accuracy and precision as a character's skill increases — this basically means better average and smaller standard deviation. The simpler the system is, the better, of course.

  • Skill + 2d6 increases accuracy as attribute/skill increases but precision remains the same.

  • Count 4+ in (Skill)d6 increases accuracy with skill but precision decreases as skill increases.

  • (Attribute)d(Skill) keep 2 where skill is 12, 10, 8, 6, 4 increases accuracy and precision as skill improves (gets smaller) but means there are only 5 steps for a given skill (or possibly attribute). It's also a system that requires a bunch of dice in each size. And has the "issue" of lower rolls being better. And that an increase in skill means a smaller dice while an increase in attribute means more dice.

  • (3 + Skill)d6 keep 3 increases both accuracy and precision but stops being a bell curve around 9+ dice (skill of 6) so provides few gradations in skill.

  • Skill + d20 is not a bell curve at all.

What dice mechanic would give me the distribution I need for this system?

(Note that this mechanic isn't important enough to me to want to add a lookup table to achieve the desired distribution.)

Best Answer

(SKILL+CONSTANT) dX, keep highest CONSTANT

I don't know exactly the behavior you're going for, so there'll be a few arbitrary numbers in my example:

  • skills can be ranked from 0 to 5
  • dice rolled are d12 ('cause I think they don't get enough love)
  • we're going to keep highest 3 dice.

In this case we're looking at rolling (3+SKILL) d12, keep highest 3. It gets you the following features:

  • increasing mean
  • rate of increase of mean tails off at high skill levels (which is a feature I like in skill systems)
  • variance drops with level
  • only requires one type of die (for simplicity)
  • skewed distribution, which you said would be okay
  • pretty simple execution of the method--no crazy amount of table-time spent on it

Here's an image of the sort of distributions that result. (Click to get to and play with AnyDice program.)

anydice output of (skill+3)d12 keep 3

As an alternative, @Ilmari Karonin points out you can get pretty-similar looking results from d6: check out this anydice program which used d6 and a constant of 4, rather than my d12s with a constant of 3. For the "cost" of adding one more die you can use a much-more widely held die-size.

(Ilmari is one of the very fine mathematical minds on RPGSE--you should read his stuff. It's worth your time.)

Now let's be clear: you mentioned something like this in your original post, and say it "provides few gradations of skill" before not being enough of a bell curve. But in a comment you say "if it's necessary, only a smaller number of skill gradations may be supported if it simplifies things." Not sure how to balance those, so I ignored them.

I'm throwing this out there with graphics and a program you and readers can play with. You've given an underconstrained and loosely-defined problem; I think this tool and this method might get you what you want.

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