Most probable array of D&D ability scores

There's an element to the game Dungeons & Dragons that lends itself to a numerical analysis: the initial array of ability scores assigned to a character.

Some background:

A character in the game has scores assigned to each of six abilities (strength, dexterity, constitution, intelligence, wisdom, and charisma). The first step in creating a character is to generate an array of six numbers by rolling dice, using a method known as "4d6 drop lowest". This means, roll four six-sided dice, remove the lowest value, then add the remaining three dice together (the result ranges from 3 to 18). Do this six times to generate an array of six values, then assign these values to your character's abilities as appropriate for the character's role (fighter, cleric, wizard, etc.).

The problem

The question I want to answer here is: What does a "typical" array look like? More importantly, how would I know if the array I end up with is better or worse than average?

Each result is assigned to an ability of your choice, depending on the role you have chosen for your character (fighter, wizard, cleric, bard, and so on). Naturally a player will want an array with some high numbers in it, to assign to key abilities that are critical for the character to function in the chosen role.

The "quality" of the array is calculated as a sum of ability modifiers, that is, the sum of differences of each score from 10: $$M = \sum_{i=1}^{6}{\text{floor}\left ( \frac{S_i-10}{2} \right )}$$ where \(M\) is the modifier total, \(S_i\) is one of the six array values, and \(\text{floor}(x)\) is \(x\) rounded down to the nearest integer.

So an array having values of all 10 and/or 11, say {10, 10, 11, 11, 11, 11} would have \(M=0\), which would be considered average, neither good nor bad. And 10.5 is the average sum of three d6 dice. But we aren't simply rolling three d6 dice to get each value. We're rolling four dice and selecting the highest three! This biases the scores upward.

For the 4d6-drop-lowest method, the average sum turns out to be 12.2, not 10.5. An array like that, say {12, 12, 12, 12, 12, 13}, has a total ability modifier of +6. So that's what I'd expect from a "typical" array obtained from 4d6-drop-lowest trials.

In practice, one almost never gets an array with nearly equal values. The resultant array has a range. I still need to solve the problem of what constitutes a "typical" array.

Analysis and observations

I used a spreadsheet to generate 30,000 sets of six abilities using 4d6-drop-lowest for each score. After sorting the six values in each array, I then obtained distributions for the lowest value, the next lowest value, and so on up to the highest value.

At the peak of the probability distribution of total ability modifiers, the most common modifier total is 5 or 6. The probability of each is equal, suggesting that if we weren't rounding down in that formula above, we'd see the most common modifier total as 5.5.

To obtain the "typical" profile, from my 30,000 trials I extracted those adding up to 5 or 6, then calculated the statistical mode of the lowest ability scores, the 2nd lowest ability score, 3rd lowest, and so on up to the highest. This gives me a statistically typical array of 4d6-drop-lowest dice rolls. You can see the mode (peak) of each distribution here:

distributions of six 4d6-drop-lowest scores

Typical array: { 9, 11, 12, 13, 14, 16 } (modifier total = 6)

These are good to use for a "typical" character build. The game master could reasonably offer this array as an acceptable standard array for a character, if the player's own rolls don't work out as well.

Comments

Post a Comment

Popular posts from this blog

Syncing Office 365 Outlook to Google calendar using Power Automate

Image
When my employer switched over from the Google office suite to Microsoft's Office 365, it caused unexpected disruption in my family. I had been sharing my work calendar with my wife (sharing no details, just busy times). It became an integral part of her calendar, to help her coordinate (between her job and my job) who picks up and drops off my son from school, and to let her know when I was free to take a phone call during my work day. Since the switchover to O365, my ability to share a calendar externally was eliminated. Based on documentation I could find, it should be possible, but the option doesn't appear in my employer's implementation of Outlook. I can view my personal Google calendar in my work Outlook calendar, but I can't share it the other way. Microsoft's Power Automate provides a way to solve this problem. It lets you create flows of data, triggered manually or automatically by events, to move data around, not only among the Microsoft products but ...
Read more

Whose hands are biggest? You may be surprised.

Image
A recent project to create an ergonomic handle for 3D printing led me down a path that introduced me to anthropometric measurements of the human hand, which in turn revealed some interesting facts about hand sizes. I had no idea that hand size is dependent on nationality, but I found a number of research articles on this topic. For those who don't want to read further, the answer to the title question is: Based on the data found, Filipino males have the biggest hands, by a significant margin. In fact, the margin is large enough that this population may be considered an outlier from the general human population with respect to hand size. Even Filipina females have larger hands than most males of other nationalities. On the other end of the spectrum, Vietnamese females and Indian females have the smallest hands. Now you can read on for the whole story, or you can skip down to the chart at the end. Idea for a toy My unintentional investigation into hand sizes started while de...
Read more

New approach to screw threads in OpenSCAD

Image
This is an intellectual exercise with a practical end-result. There are better approaches to this problem, but the approach I describe here is interesting. A screw thread from a stack of discs Imagine a stack of 50 coins, stacked so that each coin is slightly offset from the one below it, with the offset moving around in a circle with each new coin. Here is what it looks like for fifty coins 20 mm in diameter and 0.5 mm thick, offset by 2 mm from a center of rotation, and stacked with a rotation offset of 30 degrees per coin ( click to enlarge any image in this article ): Obviously, this is a spiral stack of coins. But it also approximates screw threads. The thread profile looks like a sinewave rather than the trapezoidal profile of machine screw threads. Indeed, others have observed this, and used the freeware parametric CAD program OpenSCAD to make screw threads by extruding an offset circle while twisting it, as in this model on Thingiverse , for example. From a distan...
Read more