Simple random attributes lead to average characters?

Jon Spengler says in Rolling D&D Stats is Bad For You, a Reprisal that standard rolling methods generate a lot of mostly-average characters. I.e. many chars with all their attribute close to their overall mean. In contrast, assign-an-array methods tend to give extremes — characters who are strong at one thing and weak at another. And thus the latter is usually better for play.

My instinct was that he was right, but I decided to put some numbers on it to check it, and so that we can measure how much difference the various creation methods make.

Some statistics

The obvious measure to use for this is Standard Deviation applied to an individual character attribute set. For example, (11,11,11,11,11,11) has SD 0 (zero), (13,12,11,10,7,6) has SD 2.8, and (18,18,18,3,3,3) has SD 8.2.

Rolling 3d6 per attribute and taking what you get, the SD is 2.83. The Immergleich attribute-rolling method (3d6 per attribute, reject character if modifiers don’t sum to 2) leads to a mean SD around 2.85.

(Does taking the mean of an SD make sense? Maybe. Maybe not.).

In contrast, the default Dungeon World statline of 16, 15, 13, 12, 9, 8 has SD 3.2 and the default 5e statline of 15, 14, 13, 12, 10, 8 has SD of 2.6. An “extreme statline” of 18, 16, 13, 8, 5, 3 (i.e. one designed to use the whole range) has an SD of 6.1.

Of course, most D&D-like games use the attribute modifiers much more than the raw value. In Immergleich I’ve removed them entirely – raw values are only used internally by the attribute rolling program.

Using the BECMI modifier rules (as in Immergleich, Lamentations of the Flame Princess, and most OSR games):

  • (11,11,11,11,11,11) is still zero
  • (13,12,11,10,7,6) is sd(c(1,0,0,0,-1,-1)) –> 0.75
  • (18,18,18,3,3,3) is sd(c(3,3,3,-3,-3,-3)) –> 3.3
  • 3d6 no filter –> 0.86
  • My method averages –> 0.86
  • DW statline is sd(c(2,1,1,0,0,-1)) –> 1.05
  • My “extreme statline” is sd(c(3,2,1,-1,-2,-3)) –> 2.37

I’ve omitted the 5e line as its not designed for BECMI modifiers.

What about point buy?

Point buy methods leave the choice up to the player – they can be all-average, or they can be extreme. What this leads to in practice is an empirical matter. WotC may have data, but I don’t.


It seems like the difference is less than initially assumed — the apparently-quite-extreme DW array is not that different from a simple rolling method. In theory an assigned array can lead to more extreme characters, but given the arrays that are out there’s not much in it.

Now, it’s possible that SD is a poor measure of this, or indeed that we need a different approach that takes account of how attributes are used in play. (In the latter case, it could be very game-specific). I’d be interested to hear ideas for this.

It’s also possible that if I repeated this with 3e/Pathfinder/5e modifiers (“subtract 10, divide by 2, round down”), or with the 3e/Pathfinder/5e rolling methods (some variation of “4d6 drop lowest”), there would be a more pronounced difference. But my personal concerns are with my game, that rolls as I have done here. I will leave the analysis of other games as an exercise for the reader.

Of course, one can have random assignment from an array (or a set of arrays). See  for a particularly developed example.

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