The Best Way to Roll Your Stats

[Meepo_]

I've always wondered what the best way to roll your D&D/d20 stats was. I've played with the first edition rules; you roll 3 six-sided dice (3d6), add them up, and those are your stats. You used to even roll them order: Str, Dex, Con, Int, Wis, and Cha. It was brutal: I thankfully rolled a 17 for intelligence (and bumped it up to an 18 by sacrificing a few points of strength) to make a bad-ass wizard as my first character. But the variation between power levels of characters was way too much: you were just as likely to be an absolute genius (18) as you were to be only slightly smarter than your average bear (3).

Now I'm all for point-buys. You get exactly what you want, and your stats aren't any better or worse than the next guys'. But after I read 5e, and noticed how it attempts to return to the roots of D&D where fun > optimization, I also realized that rolling your character stats is just about the most D&D-like way to build a character. Later editions attempted to make characters a bit better. All editions from Advanced on have allowed the player to choose which numbers go where. Alternate stat rolling methods began to surface to make adventurers better than normal people. The first alternate way I learned how to roll stats was to roll normally, and re-roll any 1s (3d6r1). The more widely accepted method is to roll 4 six-sided dice, and to add the best 3 (4d6b3). The first removes tiny stats, while the second makes all one's stats better on average. I was faced with a question: which method is the best?

It probably doesn't matter that much, but I just had to know. Mathematically speaking, the average roll on a d6 is 3.5, so the original method yields average stats of 10.5. Re-roll the 1s and the new average per die is 4, so the average stat would be 12. But what is the average stat when you take the 3 best of 4 rolls? If the first 3 rolls are all 6s the method doesn't matter. All 4s and there is a 33% chance of improving the stat with another roll: all 3s and the fourth roll boosts the total 50% of the time. But if you get three 1s, the fourth roll has an 83% chance of boosting the total, whereas if you re-roll your 1s each die has a 100% chance of improving. The 3d6r1 method makes terrible rolls nearly impossible, and the 4d6b1 method makes all your rolls slightly better. But I know no way of mathematically calculating the average of 4d6b1. So I did the only logical thing a D&D nerd would do: I built an experiment.

It takes hundreds of die rolls to produce accurate, average results, and I had no interest in spending my weekends rolling dice [alone and without an awesome story to be a part of], so I coded a nifty program to do it for me. It uses Java (please don't hate), and lets me roll any number of stats, using all three methods, and then calculates the mean of each almost instantaneously. I won't post the code unless anyone cares to see it, and I'll spoiler the results of the first trial so you don't have to scroll past a wall of numbers. But without further ado, here are the results:

Trial # and roll amount	Trial 1: (100)	Trial 2: (100)	Trial 3: (100)	Trial 4: (1000)	Trial 5: (10000)	Trial 6: (1000000)	Trial 7: (1337)	Average of Trials 1 - 3
3d6 (Control)	10.04	10.76	10.29	10.54	10.4957	10.498284	10.477187733732237	10.363333
3d6r1	11.81	12.31	11.77	12.099	12.0322	11.998569	12.051608077786089	11.96333
4d6b1	12.1	12.14	11.73	12.129	12.2332	12.248802	12.100224382946896	11.99

If you really want to see the actual numbers rolled, this spoiler contains the full output of Trial 1:

Spoiler: Trial 1:

Show

3d6: 3d6 reroll 1: 4d6 best 3:
11 14 14
11 14 15
8 12 9
9 14 13
11 17 12
9 13 15
11 10 15
12 12 8
6 13 15
9 11 14
16 12 13
16 12 10
14 11 11
11 9 13
10 11 10
8 11 12
6 8 3
4 15 13
7 12 11
8 11 6
7 13 16
10 14 15
10 6 11
9 10 10
6 15 8
11 15 11
9 11 17
11 11 10
11 12 11
7 14 10
8 7 15
9 10 13
9 17 16
12 13 13
11 12 13
11 8 8
17 9 17
10 8 11
18 7 17
9 10 15
10 13 13
7 13 15
9 14 12
10 10 17
9 12 13
9 10 13
8 13 8
13 13 8
12 13 14
14 15 16
9 12 11
8 9 14
13 7 9
8 10 14
5 17 15
6 13 12
7 13 9
10 9 12
9 13 13
10 11 12
14 8 11
4 13 11
9 9 10
8 18 13
13 12 15
12 10 11
11 13 10
7 11 11
8 12 10
11 12 11
11 13 9
12 13 14
13 9 12
11 10 14
11 16 12
17 12 16
7 12 13
8 13 13
7 11 11
7 14 13
14 11 9
7 12 14
8 9 14
10 11 9
9 8 8
15 12 10
10 9 13
10 12 10
13 11 11
11 13 14
9 9 12
11 14 12
8 13 11
11 12 15
10 15 10
11 12 15
11 14 14
12 13 11
11 11 5
14 15 14
Mean:
10.04 11.81 12.1

I started out with three trials of 100 rolls each and obtained the average, but the results were inconclusive. Sometimes 3d6r1 would be better than 4d6b1, and other times it would be the reverse. To get more accurate results I rolled the stats 1,000 and 10,000 times. I then got bored and decided to see what 1,000,000 rolls would be like (took a few minutes, which is a lot by computer standards), and what 1337 would give me (besides more decimal places). It seems to me that the two rolls are not really that different, but the 3d6 had an overall average close to 10.5, the 3d6r1 had an overall average close to 12.0, and the 4d6b1 had an overall average closest to 12.25.

I guess rolling four times and taking the best three is in fact slightly better than rolling three times and re-rolling ones. The superior method is included in the core books, after all. I can't say this wasn't a fun little project- its pretty cool to watch all those numbers scroll by on my screen faster than a CN Rogue to a dragon's hoard. I'll appreciate anything anyone has to say about my experiment: where I screwed up, what I could have done better, whether there are other rolling methods that I should test, or that I should get a new hobby.

Anyways: I bid you all farewell, and may you be blessed with many a powerful character!

[Solaris]

I often have them use 3d6+6, drop low. In essence, it's like the regular 4d6 drop low with one of them automatically being a 6. It produces a lot of 14s and 15s.

[Meepo_]

I often have them use 3d6+6, drop low. In essence, it's like the regular 4d6 drop low with one of them automatically being a 6. It produces a lot of 14s and 15s.

Huh, never heard of that one.

Ran it 300 times, mean is 14.63- I could see this in some highly high-powered games.

10,000 rolls later and the average is 14.4907.

[Eldan]

I don't like a lot of the higher-rolling methods. I guess I don't see why PCs have to be above average in everything and even one-in-a-thousand in one or two things. I even less see the absolute panic some players seem to have over every stat under 10, much less stats under 8. Remember: a 7 is as "bad" as a 14 is "good". I've played quite viable characters with a 5 in unexpected places, it can work. (There was the wizard with 6 dex, and the druid with 5 charisma...).
Now, I see why too much randomness can be bad and ruin a build, but on the other side, I just find that having an unexpected weakness can really help me give some personality to a character.

[D+1]

Want some more ways to roll stats? Try this on:
http://home.earthlink.net/~duanevp/d...generation.htm
I sorta stopped adding to it because there really isn't much limit to the variations and I think people's eyes just start glazing over after the first half dozen or so.

[Eloel]

Let me get something straight. You can code. You see 4d6, and your first response is to try 100 times, multiple sets, instead of simply exhaustively going through the whole 1296 of them?

[Meepo_]

Let me get something straight. You can code. You see 4d6, and your first response is to try 100 times, multiple sets, instead of simply exhaustively going through the whole 1296 of them?

Umm... yeah... it was for science...?

[Eloel]

Umm... yeah... it was for science...?

Science doesn't experiment when results can be theoretically proven with no room for error. I think.

[Eldan]

I did 4d6b3 exhaustively once, years ago. By hand, though. With averages.

[Solaris]

I don't like a lot of the higher-rolling methods. I guess I don't see why PCs have to be above average in everything and even one-in-a-thousand in one or two things. I even less see the absolute panic some players seem to have over every stat under 10, much less stats under 8. Remember: a 7 is as "bad" as a 14 is "good". I've played quite viable characters with a 5 in unexpected places, it can work. (There was the wizard with 6 dex, and the druid with 5 charisma...).
Now, I see why too much randomness can be bad and ruin a build, but on the other side, I just find that having an unexpected weakness can really help me give some personality to a character.

'Cause being average doesn't play well into the power fantasy that is D&D. We can do mediocre in real life, why do it in a game?

[Meepo_]

Science doesn't experiment when results can be theoretically proven with no room for error. I think.

Guess I overthought that one. But I'll never have to manually roll stats again! (I still will, it's too satisfying)

[Meepo_]

I did 4d6b3 exhaustively once, years ago. By hand, though. With averages.

Looks like you did what Eloel suggested. At least we got similar results.

[Aetol]

Let me get something straight. You can code. You see 4d6, and your first response is to try 100 times, multiple sets, instead of simply exhaustively going through the whole 1296 of them?

Damn, I was going to say that.

Besides, coding is the lazy way out. A real man does the math by hand. And not just for 4d6b3, but for XdYbZ.

[LadyFoxfire]

Most of my DMs use the 4d6 drop lowest method. I hate rolling my stats in order though, I like to pick my class myself rather than letting the dice do it for me.

[Aetol]

I heard the argument that rolling your stats is only worthwhile if you do it in order. Basically, rolling then changing the order is the worst of both world : it has neither the wacky randomness of rolling in order, nor the even power level of point buy.

[YossarianLives]

'Cause being average doesn't play well into the power fantasy that is D&D. We can do mediocre in real life, why do it in a game?

Actually I've always wanted to play a character that is average or slightly above average. In many ways, I think a story about average folks overcoming terrifying beasts and horrifying sorcery through their own luck and wits is far more interesting than a story about larger-than-life heroes who battle gods and save the cosmos.

That said, one of the best campaigns I've ever played in was quite high-powered and involved the PCs performing ridiculous and implausible stunts.

[Arbane]

'Cause being average doesn't play well into the power fantasy that is D&D. We can do mediocre in real life, why do it in a game?

Being forced to play an inept wimp due to bad die-rolls is good for ROLEPLAYING.

I prefer point-buy, personally.

[Aetol]

Being forced to play an inept wimp due to bad die-rolls is good for ROLEPLAYING.

Oh right, that old "powergaming and roleplaying are mutually incompatible" notion.

I get why some would find the "I don't get to chose what character I play" kind of game. That's basically how you tell the game "surprise me". But that's not for me, either.

[Draconium]

Oh right, that old "powergaming and roleplaying are mutually incompatible" notion.

Erm, I think he was being sarcastic... that's what another color (usually blue) tends to mean. Unless he's using purple to be evil, like Red Fel does. Either way, I'm pretty sure he wasn't serious.

[Remedy]

Erm, I think he was being sarcastic... that's what another color (usually blue) tends to mean. Unless he's using purple to be evil, like Red Fel does. Either way, I'm pretty sure he wasn't serious.

Even when we don't look the part, Evil is always serious.

... I mean, uh, even when they don't look the part. Yeah.

[Pex]

Here I go again.

27-25-23

Step 1: Roll 4d6, best 3, three times. These are your first three scores. If a roll is less than 7 count it as 7.

Step 2: Take one score and subtract from 27 for your 4th score. Ideally force a max 18, so no 27 - 7 = 20 for a score.

Step 3: Take another score and subtract from 25 for your 5th score. Note that 25 - 7 = 18.

Step 4: The third score is subtracted from 23 for the 6th score.

Step 5: Add +2 to any one score. Ideally force a max 18.

Step 6: Arrange the 6 scores as desired and apply racial modifiers.

Example, wanting to play a human paladin in Pathfinder

Step 1: 16, 13, 8

Step 2: 27 - 16 = 11

Step 3: 25 - 8 = 17

Step 4: 23 - 13 = 10

Step 5: 16 + 2 = 18

Step 6: Apply racial modifier +2 to Con where the 13 will go.

ST: 18 DX: 11 CO 15: IN: 8 WI: 10 CH: 17

Plans: Level 4 increase CH to 18. Level 8 increase CO to 16.

5E variant human paladin, same scores:

ST: 18 DX: 12 CO: 14 IN: 8 WI: 10 CH: 17

Plans: Take Heavy Armor Master to get ST to 19. Level 4 increase ST to 20 and CH to 18. Definitely more powerful than 5E Point Buy, but that to me is a feature because I loathe 5E Point Buy. However, that's a different topic.

[Drynwyn]

For 3.5, I'm okay with rolling and point buy. For 5e, though, I'm honestly flabbergasted that they would suggest die rolls as the "default" method- stats are simply too important in 5th edition, due to the smaller non-stat bonuses, to be left to chance- you can end up with characters of DRAMATICALLY different power levels.

[goto124]

So, why roll stats? Why not just go point buy or stat array?

Especially when I wanted to play a STR-based character, and I end up rolling a 5 on STR? If I can change my rolls to suit my chars, why waste time rolling and just go straight for point buy? Why not have all the numbers in place before coming to the table, instead of having to rush through character generation and finish in 5 min to give enough time for the actual game?

[Mrs Kat]

You should post standard deviation as well as mean. Otherwise your results aren't really meaningful.

A histogram or probability plot of each method would be interesting, too. I imagine 3d6 is a bell curve, and 4d6 drop lowest is a really skewed bell curve.

[tgva8889]

If I am playing a game where stats are a relevant part of a large number of game mechanics (like D&D), I much prefer standardized stats. That is, no randomness between players, everyone has the capability to produce identical stats. I've played in games where we roll for stats and let me tell you, rolling the worst on stats makes it really frustrating to play because the game expects you to rise to the same level as everyone else.

In 5e the effect is super-magnified, so I'd just always suggest any sort of standardized array.

[Fri]

So, why roll stats? Why not just go point buy or stat array?

Especially when I wanted to play a STR-based character, and I end up rolling a 5 on STR? If I can change my rolls to suit my chars, why waste time rolling and just go straight for point buy? Why not have all the numbers in place before coming to the table, instead of having to rush through character generation and finish in 5 min to give enough time for the actual game?

The thing that I found recently and I can't believe I never thought of it before, is simply having one stat freely as "bonus" to be assigned.

If your game want randomly rolled stat instead of point buy, just, roll your stats with low stat array or whatever, but have one "free" above average stat as bonus to be freely assigned (and maybe make a ruling that if you have 3 above average stat you can't have this free above average stat or something).

This means you can have your randomly stated character, but you can also still have most of character idea you're thinking.

[Aetol]

The thing that I found recently and I can't believe I never thought of it before, is simply having one stat freely as "bonus" to be assigned.

If your game want randomly rolled stat instead of point buy, just, roll your stats with low stat array or whatever, but have one "free" above average stat as bonus to be freely assigned (and maybe make a ruling that if you have 3 above average stat you can't have this free above average stat or something).

This means you can have your randomly stated character, but you can also still have most of character idea you're thinking.

But that only really works for SAD characters. If I want to play a MAD character and roll poorly in more than one stat you need, your fix won't do much good.

[Anonymouswizard]

Want some more ways to roll stats? Try this on:
http://home.earthlink.net/~duanevp/d...generation.htm
I sorta stopped adding to it because there really isn't much limit to the variations and I think people's eyes just start glazing over after the first half dozen or so.

I disagree with most of the reasoning here, but mainly the one used for why the 3.5 standard array is bad. The idea was to just let you give the character an average roll (I think I once saw the claim that it was a statistical average of what set you'd get if you rolled 4d6b3, but I think it sacrifices the most likely for a bit of variety). Otherwise, kinda interesting, but not my cup of tea (that would be tea).

'Cause being average doesn't play well into the power fantasy that is D&D. We can do mediocre in real life, why do it in a game?

I can see a handful of reasons, but I've come to the view that a system that forces someone to start at the same level as the rest of the party is bad. In GURPS, if I (for some reason) want to play Sidekick Man then I don't have to spend all my Character Points, and if I do get bored of being weaker I can spend a session or two boosting my stats and powers to get to the party's level,

So, why roll stats? Why not just go point buy or stat array?

Especially when I wanted to play a STR-based character, and I end up rolling a 5 on STR? If I can change my rolls to suit my chars, why waste time rolling and just go straight for point buy? Why not have all the numbers in place before coming to the table, instead of having to rush through character generation and finish in 5 min to give enough time for the actual game?

I agree with this, but then I also have a loathing for strongly classed systems as well (which is why I prefer Dark Heresy to D&D, the classes are looser). Either stop whinging and generate every last scrap of your character, or actually give the players the ability to play Bob Sir Knight if they don't want to play Wizard McFireball but rolled 18 Int and 4 Str.

[goto124]

Either stop whinging and generate every last scrap of your character, or actually give the players the ability to play Bob Sir Knight if they don't want to play Wizard McFireball but rolled 18 Int and 4 Str.

Er... this may be a problem if stats have great impact on making your character remotely competent at doing anything (consider the amount of effort to make a 4 Str Warrior that can contribute decently). Or if the random rolling results in big gaps in a set of roles (there're no healers/social skillers/etc), forcing the GM to redesign the entire adventure at the last minute.

System-dependant, I suspect.

I personally don't see the purpose of randomness in character generation. Especially when I end up trying to adjust a lot of things that will affect me for the rest of the campaign*... 3 minutes before the game starts. At best a day, and my ideas of 'I get to play Bob Sir Knight who does this and that which requires 15 Str' are shattered.

*Or the rest of the character's life. Wait... must I roll for my second character as well?

[Anonymouswizard]

Er... this may be a problem if stats have great impact on making your character remotely competent at doing anything (consider the amount of effort to make a 4 Str Warrior that can contribute decently). Or if the random rolling results in big gaps in a set of roles (there're no healers/social skillers/etc), forcing the GM to redesign the entire adventure at the last minute.

System-dependant, I suspect.

Sorry, the intention was to imply that they'd be able to change to the stats they want, because I find rolling stupid (I've only ever pulled it out to stop people from playing the same character for the umpteenth time).