A new RPG system

How to derive Skill and Difficulty from a Dice Mechanic

In the previous post. I discussed using a 2d6-2d6 dice mechanic, for the purpose of the examples in this article we’ll use that.  What we’re going to do here is explore how the statistics of our mechanics can inform our character traits and target number.  Again, we’ll be leaning heavily on that Standard Deviation.

So, what we have with a Linear (i.e. Roll and Add) dice mechanic is a distribution of numbers that gives us fixed intervals, and a standard deviation.  Working with 2d6 – 2d6 we know we have a range of -10 to 10, with  a standard deviation of 3.42.  From this we’ll first figure out how big each +! is worth relative to that, use that information to figure out interesting statistical information, and finally relate that to probabilities and difficulty.

Figuring Out a +1

+1’s are important in any game.  In Fate they are minor bonuses, in D20 they are very minor bonuses.  Yet, you’ll find them everywhere, either explicitly or in the systems avoidance of them.  And of course any bonus is really an accumulation of +1’s.  So where does that leave us in terms of what a +1 is valued at?  Objectively a +1 is a fraction of a standard deviation from the mean; specifically a +1 is equal to 1/STD.  And a +2?  You guessed it, a +2 is worth 2/STD.

So in the case of 2d6-2d6 we have a STD=3.42.  That means that every +3.42 stat points we will have gone up one standard deviation from the mean, and equivalently a +1 is worth (1/3.41=) .29 – which means a +1 is worth slightly less than a third of a standard deviation.  By contrast a +1 in d20 is worth .17 Standard Deviations or equivalently it takes a little over +5 to move up a standard deviation.

This is useful because we can discern all sorts of interesting stuff from Standard Deviations.

Working With Standard Deviations

Here’s what you really need to know about Standard Deviations (Sigma):

  • 0 Sigma – About 2/3 people everywhere are within this.  This is the point of “average”.
  • 1 Sigma – About 15% are above 1 Standard Deviation, this would probably approximate the population of people skilled at something.
  • 2 Sigma – About 2.5% of the population is above two standard deviations, a skill in this range would certainly represent a specialist.
  • 3 Sigma – One in 350 people are outside of 3 Standard Deviations, skill within this range represents the peak most specialist dream of reaching.
  • 4 Sigma – One in 15,000 reach 4 Standard Deviations, a person within this range certainly stands out from his peers.
  • 5 Sigma – Less than One in a Million reach 5 Standard Deviations, people hitting this mark are likely legends in their field.
  • 6 Sigma – Six Sigma is an important next step, because it is considered synonymous with “So unlikely we don’t worry about it.  There would be one per 500 Million people.  Therefor there is a dozen or so in the world.
  • 7 Sigma – This hovers around 1 in 500 Billion.  Anything in the realm is defacto supernatural.
  • More – Standard Deviations beyond 7 are so unlikely that they’re hard to describe.  Anything is this realm strains credulity.

Fitting the Standard Deviation into Stats

So using the information about Standard Deviations we can begin to describe our Stats.  Using the example of 2d6-2d6 I derived the following:

Score Stand Dev STD Norm Description
1 0.29 0 Apprentice
2 0.58 0 Apprentice
3 0.88 0 Apprentice
4 1.17 1 Skilled
5 1.46 1 Skilled
6 1.75 1 Skilled
7 2.05 2 Expert
8 2.34 2 Expert
9 2.63 2 Expert
10 2.92 2 Expert
11 3.22 3 Master
12 3.51 3 Master
13 3.8 3 Master
14 4.09 4 Doctor
15 4.39 4 Doctor
16 4.68 4 Doctor
17 4.97 4 Doctor
18 5.26 5 Grand Master
19 5.56 5 Grand Master
20 5.85 5 Grand Master
21 6.14 6 Legend
48 14.04 14 Godlike
49 14.33 14 Divine
50 14.62 14 Divine

(Middle Missing in the interest of space)

Here I’ve said that an Apprentice is within 1 Standard Deviation, A Skilled person between 1 and 2, an Expert over 3, etc…  6 and above are reserved for things beyond the normal scope of human ability.  That allows me to easily map in and get a sense of the capability of a score, and furthermore I know that in so much a “skilled” means 1 Standard Deviation, I know it will work out in play.  That being said “Skilled” “Expert” and the rest are to a certain extent arbitrary signifiers and could just as easily be replaced with “Good” “Great” “Fantastic” and the like.

Deriving A Difficulty Matrix

Most systems list difficulties Flat, and have to 50% for the unskilled listed as something like “Average”.  I think this is a mistake.  My question immediately is “Average” relative to what, or “Hard” relative to what.  Instead I work out relative difficulties, and I use the following as the guideline based on dice odds:

  • > 99% – Trivial
  • 99% to 90% – Easy
  • 90% to 50% – Challenging
  • 50% to 20% – Difficult
  • 20% to 5% – Formidable
  • < 5% – Daunting

Now, there are to a certain extent arbitrary, based on my own notions not objective measures.  Certainly if someone asked if I could do something and I only had a 50% chance of success I’d consider that difficult.  These words for these odds are simply the best I’ve described.  So, when I look at the probabilities relating to 2d6 – 2d6 I find the breakdown works out to something like this:

  • -10 to -8 – Trivial
  • -7 to -4 – Easy
  • -3 to -1 – Challenging
  • 0 to 3 – Difficult
  • 4 to 6 – Formidable
  • 7 to 10 – Daunting

Then using somewhere around the average of each of those and using an addition matrix relative to the listed skill ranks I am able to derive the following:

Trivial Easy Challenging Difficult Formidable Daunting
> 99% > 90 % > 50% 50% to 20% < 20% < 5%
Untrained -8 -5 -2 2 5 8
Apprentice -6 -3 0 4 7 10
Skilled -3 0 3 7 10 13
Expert 0 3 6 10 13 16
Master 4 7 10 14 17 20
Doctor 7 10 13 17 20 23
Grand Master 12 15 18 22 25 28
Legend 17 20 23 27 30 33
Super Human 22 25 28 32 35 38
Mythical 32 35 38 42 45 48
Godlike 37 40 43 47 50 50
Divine 40 43 46 50 50 50

This table allows me to look and say, “Challenging for a Master is 10″, or ” Formidable for an Expert is 13″.  Furthermore when coming up with a number for some random task during play on the fly I can say “I think that would be Difficult for someone that’s skilled” and come up with a 7.

Of course the matrix is hard for some people, so to derive objective ranking I start from the upper left hand corner and move to the lower right hand corner; on the chart you can track this with Bold Case. That give me a Objective Difficult chart like so:

  • Trivial – -8
  • Easy – 0
  • Challenging – 10
  • Difficult – 22
  • Formidable – 35
  • Daunting – 50

Of course this is assuming that everyone can grasp the fine details of scaling from someone unskilled to a god.  This of course is another reason why Objective Difficulty Charts are inferior to a Difficulty Matrix.

Conclusion

In this post we’ve seen how to work from a Linear Mechanic to Stats and Difficulties.  Other things can also be derived from here, like if you had an intelligence score you could generate an IQ.  However, while the does allow one to create the kernel, it is simply the first of a tiny step.  Unanswered questions like what to use for stats, how to handle contests, and so many other details remain.

Can you think of anything else we can derive from the Dice Mechanic alone?

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One response

  1. Pingback: Handling Scale | Living Myth Rpg

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