A comparison of rolling attributes

In many role playing games (RPGs), attributes such as strength or wisdom are established at the time of character creation. In particular, Dungeons & Dragons, currently owned by Hasbro, and its offshoots (commonly referred to as Old School Games (OSG), and the Pathfinder RPG (PFRPG), currently owned by Paizo, use various systems for attribute generation.

In these systems, there are six attributes. The three physical attributes are strength (STR), dexterity (DEX), and constitution (CON). The three non-physical attributes are intelligence (INT), wisdom (WIS), and charisma (CHA). Each of the names are a generalization necessary for game play mechanics. For example, charisma is an approximation of the amount of power or influence one character will have over another.

In general, there are two types of attribute establishment. One is spending a certain amount of building points to determine in a non-random fashion what the attribute values will be. This is commonly know as a point-buy system. The other is rolling dice to randomly determine what the attribute values will be.

Point-buy systems commonly take the form of a pool of points which can add to a minimum level in each attribute. For example, if the minimum dexterity is 7, it may cost one point to bring it to eight. More points can be spent to bring the attribute number to where the player, creating the character, wants the value to be.

The point pool often determines the overall power of the individual character, and the companion characters. For example, in the PFRPG, a 15 point pool is often considered to be standard, while a 20 point pool is often considered to be high fantasy.

There are various dice rolling systems in these games. The most restrictive is commonly known as 3d6. In 3d6, the player rolls three six-sided dice. This will determine an attribute's value. Some versions make this even more restrictive by having each player roll all six attributes in order, not allowing changing one dice roll for another. Most versions of this, however, allow the player to assign each dice roll value to a particular attribute. Some versions allow players to roll more than six times, and eliminate the lowest values rolled.

Another common system is the 4d6 less one. In this system, the player rolls four six-sided dice, determines which die is the lowest, removes that die leaving three, whose summed values become the attribute.

Everyone knows that the 4d6 less one method produces higher attributes than the 3d6 method. Here is a simple analysis of the difference between the 3d6 and the 4d6 less one methods.

First, the average of the 3d6 method is 10.5. In game terms, this is only slightly better than the average person, whose attributes would be 10. The average of the 4d6 less one method is 12.2446. But, that only tells part of the tale. If we look at the distribution of both systems

 we see that the whole curve seems to be shifted to the right. But, since we are dealing with such varied numbers (the 3d6 method has at most 27 instances, while the 4d6 less one method has at most 172), we can normalize the chart by looking at the percentage of instances at each value.

Here, we see that although the upper range appears to be merely shifted to the right, the 4d6 less one method has additional advantages.

• While the average is only 1.75 points higher, the mode (that is the most common value) is two points higher. The mode for 3d6 is 27, which occurs at values 10 and 11. The mode for 4d6 less one is 172, which occurs at value 13, two higher than the value 11 from 3d6.
• We can see from the chart the likelihood of being average or below (that is a value of 10 or less) is 50% in the 3d6 method. If we look at a table of cumulative averages from the 4d6 less one method:

3	0.08
4	0.39
5	1.16
6	2.78
7	5.71
8	10.49
9	17.52
10	26.93
11	38.35
12	51.23
13	64.51
14	76.85
15	86.96
16	94.21
17	98.38
18	100.00

• We can thus determine that the chance of being average or below is less than 27%