4 of 5 clarification from comments

What dice mechanic gives a bell curve distribution that narrows and increases mean as skill increases?

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.)