# Background

I'm the GM of a role playing group, improvised rule set.

I of course have stats for each character and roll dice to determine skill successes and so on. What I do different from the few other RPG rule sets I know, is that:

I roll three dice, and take the median value of them. This means I get a likeliness distribution where average numbers are a lot more likely to come up than numbers in each extreme end of the spectrum.

I prefer it this way, since the epic "Rolling a perfect d20" actually is a 5% chance and for me is not special enough to be special.

This gives me a distribution that looks like this:

### Likeliness distribution of 3d20 median rolls

(ex. the data below shows that with the three die, there is 1 chance in 138 to get a median value of 1)

      <=Value % <=Value Fraction >=Value % >=Value Fraction
Value
20.0    100.0 %          1 / 1.0   0.725 %        1 / 138.0
19.0   99.275 %          1 / 1.0     2.8 %         1 / 36.0
18.0     97.2 %          1 / 1.0   6.075 %         1 / 16.0
17.0   93.925 %          1 / 1.0    10.4 %         1 / 10.0
16.0     89.6 %          1 / 1.0  15.625 %          1 / 6.0
15.0   84.375 %          1 / 1.0    21.6 %          1 / 5.0
14.0     78.4 %          1 / 1.0  28.175 %          1 / 4.0
13.0   71.825 %          1 / 1.0    35.2 %          1 / 3.0
12.0     64.8 %          1 / 2.0  42.525 %          1 / 2.0
11.0   57.475 %          1 / 2.0    50.0 %          1 / 2.0
10.0     50.0 %          1 / 2.0  57.475 %          1 / 2.0
9.0    42.525 %          1 / 2.0    64.8 %          1 / 2.0
8.0      35.2 %          1 / 3.0  71.825 %          1 / 1.0
7.0    28.175 %          1 / 4.0    78.4 %          1 / 1.0
6.0      21.6 %          1 / 5.0  84.375 %          1 / 1.0
5.0    15.625 %          1 / 6.0    89.6 %          1 / 1.0
4.0      10.4 %         1 / 10.0  93.925 %          1 / 1.0
3.0     6.075 %         1 / 16.0    97.2 %          1 / 1.0
2.0       2.8 %         1 / 36.0  99.275 %          1 / 1.0
1.0     0.725 %        1 / 138.0   100.0 %          1 / 1.0


# Question

Are there other systems that incorporate this?

Is there a simpler way of getting this kind of distribution?

## 4 Answers

Dice pools are a classic way of generating a normal distribution.

3d6 is a classic, generating values from 3-18.

At least two systems, GURPS and HERO are based on the normal distribution of 3d6.

Fate / Fudge uses a pool of 4 dice valued at -1, 0, +1, which produces a range of values from -4 to +4 strongly centered on 0.

Here's an anydice graph that compares all three of them - the 3d6 producing by far the most recognizable bell curve: http://anydice.com/program/a9c0

• Which system uses the 2@3d20, and what does that notation mean? Feb 3 '17 at 21:12
• @user2357112: I've never seen the notation, but anydice appears to be interpreting that as 'roll 3d20, and use the 2nd one'. In other words, 'roll three dice, and take the median value of them'. Feb 3 '17 at 21:24
• Ah, it's the system from the question. Feb 3 '17 at 21:28
• Technically a normal distribution is a continuous one, not a discrete one. N dice produce a Binomial distribution which approximates a normal distribution as n becomes large - 3 isn't large. Feb 4 '17 at 0:06
• @DaleM The “bi” in “binomial” is for two, so Nd2 is a binomial distribution. But for other dices it approaches the normal distribution as well, so whatever. (Also, 3 is huge. I rarely encounter numbers larger than 1.) Feb 4 '17 at 0:30

The distribution is highly dependent on the number of dice that you roll. A single die gives a flat distribution, two dice are linear, three dice give a quadratic bell curve, four dice yield a cubic curve and so on.

The operation you perform on the dice change the character of the curve somewhat, but the center-weightedness can be adjusted by the number of dice.

Here are a few operations you can perform on dice and their distribution curves. I'd go for the simpler ones. "Midpoint" is my favourite.

Have you considered using Percentile Dice?

They solve the 5% critical success feeling too common, as you can easily just set the threshold to 2% or 1%, either high or low depending on your system.

While they give you a uniform distribution between 1 and 100 as opposed to your normal distribution, depending on your exact system they could work; you would just need to adjust the thresholds accordingly.

Worth noting that reading percentile dice is slightly easier than reading the medium value of 3d20, and you can easily just purchase a single polyhedral set and get both dice. To get 3d20 you are generally looking towards purchasing single d20s or multiple polyhedral sets.

• Hey @Matt, thanks for the answer. I actually bought a pound-o-dice and got a huge load of d20. So no worries there. Feb 3 '17 at 13:39
• What exactly is a/are Percentile Dice? Feb 3 '17 at 15:25
• @JonasWielicki they are essentially either 1) a set of 2d10 - one that has 10, 20, 30, etc. and one that has 1, 2, 3, etc. so that when you roll them both you'll get 32% (30 + 2), 45% (40 + 5), etc. with 000 (00 + 0) being 100%; or 2) a d100 with the obvious correlation. I almost always see 2d10 used, though I also have a novelty set of "odd" dice that includes a d100 Feb 3 '17 at 15:36
• @AnonJr Since there is no such thing as a 100th percentile (but there is a 0th percentile), I always read percentile dice as 00-99. Given an X% likelihood of success, the roll simply must be less than X to succeed. No need for special handling of 00 that way. Once people get past the idea that "bigger=better", it's a far simpler approach. Feb 3 '17 at 17:28
• @MontyHarder I've seen both used, and run in games where both tacts have been taken (0% - 99% / 1% - 100%). Thank you for rounding out my comment. Feb 3 '17 at 17:31

Runequest - the version of old - had the percentile system (see Matt answer).

This works very well IMO, as you can define - various degrees of success (critical, special, normal, ... - the shape approximated, by making somes ranges larger than others

For instance, in RQ with a 50% proficiency

• critical success : 1/20 of normal --> if dice <= 2%
• special success : 1/5 of normal --> if 3% <= dice <= 10%
• normal result : 1/1 of normal --> if 11% <= dice <= 50%

Vampire had a D10 dice pool system with a nice twist : The result is a count of dices under / over a threshold. This is definitely quicker than adding the dice.

The result quality is a nice small scale easy to understand:

• 1 marginal
• 2 good
• 3 excellent
• 4 extraordinary

This is a cool base system to expand on if you like distributions, because you can play on the success threshold as well as on the number of dices, or the number of successes required.

• I like that dice system from Vampire, might use that for things myself! Feb 3 '17 at 19:21