Generating Stats: d6 Bell Curves

Rolling a handful of d6's to generate stats with probabilities on a normal distribution is the classic method. Effectively the oldest way of rolling stats in RPGs and the most common. What I'm interested in is all the variations on "roll Xd6" that have accumulated over the years.

When you pick up your fistful of dice chances are you're using one or more of these: 3d6, 4d6, drop lowest, drop highest, reroll 1's, in order.

*2d6 is used to generate stats too, but it's not technically a bell curve/normal distribution so I'm not including it.

Stats put a character in context with the game world, so tweaking the probabilities of your stats definitely has an effect on how challenging the game will be. Here's how each of these variations alter where the peak of the bell curve falls:

"In order" doesn't do anything to change probabilities, it just requires you to be more creative(?) about the character you're making. You've got fewer options when the stats you're rolling have to be used in sequence so you either run with it or reroll. It's a purely flavor modifier.

3d6 is the plain standard with the highest probability of rolling a 10 or 11. It's why 'average' in D&D is a 10. 4d6's distribution skews higher, with the highest probability rolls in the 13 to 15 range and a much lower chance of rolling single-digit values.

"Drop lowest" is almost always paired with a 4d6 roll and bumps the peak of the bell curve down a little to the 12 to 15 range. A 3d6 drop lowest has 8 to 10 as its most common rolls. It's not an extreme effect on the basic Xd6 rolls.

"Drop highest" on the other hand… It moves the peak of a 4d6 curve way down to the 7 to 9 range, and a 3d6 curve to 4 to 6. It's merciless. I've never played a character made with drop highest rolls, but it sounds fun.

"Reroll 1's" also has an extreme effect on the roll. It pushes the peak of the curve towards higher numbers. On a 3d6 roll the highest probability becomes 11 to 13, a 4d6 drop lowest is 13 to 15, and a 4d6 is 15 to 17. It does exactly what you'd think it would, just by more than I expected.

With those ranges in mind, the order of common d6 roll combos from average lowest to highest is:

1 3d6 drop highest (4-6)
2 3d6 drop highest, reroll 1's (5-8)
3 4d6 drop highest (7-9)
4 3d6 drop lowest (8-10)
5 3d6 drop lowest, reroll 1's (8-11)
6 4d6 drop highest, reroll 1's (9-11)
7 3d6 (10-11)
8 3d6 reroll 1's (11-13)
9 4d6 drop lowest (12-14)
10 4d6 (13-14)
11 4d6 drop lowest, reroll 1's (13-15)
12 4d6 reroll 1's (15-17)

If you want check out the probabilities, here are the calculations I used on anydice to visualize all this.

For 3d6 calcs y = 2, x = 3. For 4d6 calcs y = 3, x = 4.

Plain:
output xd6

Drop lowest:
output [highest y of xd6]

Drop highest:
output [lowest y of xd6]

Reroll 1's:
output xd{2..6}

Drop lowest, reroll 1's:
output [highest y of xdxd{2..6}]

Drop highest, reroll 1's:
output [lowest y of xd{2..6}]