I got to thinking: How did the Standard Array come about?

I figured they did some sort of simulation. Take 4d6, drop lowest, sort them in order, and take the mean, median, or mode of each row.

I wondered, though – what that would look like, and how much variation would there be. I mean, just in my recent character generation forays, I’ve had some really good rolls, and some bad ones. Also, Order of the Stick style, what if 4d6/drop lowest was a metagame universal rule? So that each person that lived basically had that die total. So if you looked at “the upper 20% of adventurers,” you’d see one stat block, “lower 25%” would be the underperforming bandit cannon fodder of the world, etc.

Let’s hit it one thing at a time.

*4d6, Drop Lowest: The Simulation*

Not being a random number purist, I used RANDBETWEEN(1,6) in Excel to make my die rolls. I took four “dice,” added them together, and subtracted the smallest one.

I did this about 160,000 times. That should force the system to get close to constant values as possible. To the right is what AnyDice says about the distribution from which we’re picking:

So, what does that look like? Interestingly enough, it’s not *quite* what the book gives you.

Mode | 9 | 11 | 12 | 13 | 14 | 16 |

Median | 9 | 10 | 12 | 13 | 14 | 16 |

Average | 8.5 | 10.4 | 11.8 | 13.0 | 14.2 | 15.7 |

DnD 5e | 8 | 10 | 12 | 13 | 14 | 15 |

**Mode:**This gives the number that appears most frequently out of 160,000 rolls. Compared to the 5e norm, it is one point better on the bottom two rolls, and you pick up a 16 instead of a 15 for your best. The mode of the full distribution is 13 (from the graph).

**Median:**Line up all 160,400 trials, and give me the average of trial 80,200 and 80,201. The median is also the 50% percentile. This matters a bit because 4d6 drop lowest is going to be weighted slightly to the higher numbers. The median of the distribution itself is 12.

**Mean (Average):**Sum of everything divided by number of trials. The mean of 3d6 is 10.5, the mean of 4d6 is 14. The mean of 4d6 drop lowest is 12.24. You can reproduce the standard array if you assume you round up numbers that are 0.75 or higher, and round down lower than 0.75. Strict truncation would give you 8, 10, 11, 13, 14, 15. Strict rounding would be 9. 10, 12, 13, 14, 16.

*Percentiles*

First, why do we care? It’s a tool for GMs to tweak their campaigns. If you think the standard array is too stingy or too rich, you can adjust it. If you don’t want the girl who really lucks out on her rolls (it’s no surprise that out of 160,000 rolls, you can find an array like 16, 16, 16, 17, 17, 18) to overshadow completely the poor schlub that rolled 6, 9, 10, 11, 11, 12, then you can give the same array – for good or ill – to everyone.

10th Percentile | 6 | 8 | 10 | 11 | 12 | 14 |

25th Percentile | 7 | 9 | 11 | 12 | 13 | 15 |

40th Percentile | 8 | 10 | 11 | 13 | 14 | 15 |

DnD 5e |
8 |
10 |
12 |
13 |
14 |
15 |

50th Percentile | 9 | 10 | 12 | 13 | 14 | 16 |

60th Percentile | 9 | 11 | 12 | 13 | 15 | 16 |

75th Percentile | 10 | 12 | 13 | 14 | 15 | 17 |

90th Percentile | 11 | 12 | 14 | 15 | 16 | 17 |

**Parting Shot**

*3d6 Pick Your Order*10th Percentile | 4 | 6 | 8 | 9 | 10 | 12 |

25th Percentile | 6 | 7 | 9 | 10 | 11 | 13 |

40th Percentile | 6 | 8 | 9 | 11 | 12 | 14 |

45th Percentile | 7 | 8 | 10 | 11 | 12 | 14 |

50th Percentile | 7 | 9 | 10 | 11 | 12 | 14 |

60th Percentile | 7 | 9 | 10 | 12 | 13 | 15 |

75th Percentile | 8 | 10 | 11 | 12 | 14 | 15 |

DnD 5e |
8 |
10 |
12 |
13 |
14 |
15 |

90th Percentile | 9 | 11 | 12 | 13 | 14 | 17 |

*4d4+2 – Request from a Redditor*

Because it’s easy, I was asked to drop in 4d4+2. That one will cluster around 12, with more dice providing slightly more variation, and of course a hard minimum of 6.

10th Percentile | 7 | 9 | 10 | 11 | 12 | 13 |

25th Percentile | 8 | 10 | 11 | 12 | 13 | 14 |

40th Percentile | 9 | 10 | 11 | 12 | 13 | 14 |

DnD 5e |
8 |
10 |
12 |
13 |
14 |
15 |

45th Percentile | 9 | 10 | 11 | 12 | 13 | 15 |

50th Percentile | 9 | 11 | 12 | 12 | 13 | 15 |

60th Percentile | 10 | 11 | 12 | 13 | 14 | 15 |

75th Percentile | 10 | 11 | 12 | 13 | 14 | 16 |

90th Percentile | 11 | 12 | 13 | 14 | 15 | 17 |

I wonder how unplayable 3d6 in order is under 5e. Under earlier editions, I can see the problem: there's a real chance of a character who is fodder in the first encounter. But under 5e, you get attribute bonuses from both your class and your race, so I'm not sure anything would be that unpleasant anymore.

I think it'd be pretty fun to try.

Note that there is no minimum attribute for classes (only for multiclassing). You can have the Wis 3 Cleric in 5E… he prepares only 1 spell, until 5th level (if he raises Wis at 4th,) or 6th level (when (StatMod + ClassLevel)>1 without raising wis).

It will, however, make the encounter balance system less accurate..