Fair Random Stat Generation

[Camelot]

Okay, point buy is nice, and I prefer it myself for real games, but what about when you want to make random characters? Well, you do what the rulebook says: roll 4d6 and add the 3 highest dice for each stat. This way, you get a number from 8 to 18.

But how come you always end up with a lower than average ability score set? Oh, you say to yourself, it must just be bad luck. Maybe next time. But then you look at the rules and it clearly says that when you roll scores, you will "on average come out a little worse than if you had used the standard array." I don't want the average to be a little worse, I want the average to be average!

The average score with stat generation is about 13. If you buy stats to make them all the same, you will get 5 being 13 and one being 14. So we'll say 13 is average.

If you roll 2d6, add them together, then add 6, you will get a number from 8 to 18. And the average? 13 on the nose. Perfect! *Hey, that's not average!* What are you talking about, of course it's average! *No, average is 13 + 1/6 because of that 14!* Well, true, so you're still a teeny-tiny bit less than average, but 13 is closer to 13.1666 than 11, which is about the average from rolling stats the old way.

Is my math about right? I'm going to give this a try and let you know how it works. Let me know what you think!

 

[Nytmare]

Technically, when you roll 4d6 and you drop the lowest die, your average scores are average. When you use the standard array your scores are above average.

I guess you could also roll '3d6+2' or '4d6+1 drop lowest' if you wanted your average to fall somewhere nearer to the 13 mark.

 

[Pyrex]

I have a couple questions:

1) Just how random do you want your stat generation to be?

2) Just how fair do you want your stat generation to be?
-->2.1)Does "fair" mean character stats are close to expected point-buy average?
-->2.2)Or does "fair" mean each character in the party is approximately balanced?

One moderately-random but extremely "fair" (for both definitions) example:
-)Start all stats at 8.
-)In whatever order the player chooses, roll 1d6 per stat and spend that many ability points improving that stat.
-)Once you've rolled for all 6 stats, choose a new order for all six stats to roll them again.
-)Once a stat has been improved to 18, stop rolling for that stat.
-)Once you've spent a total of 32 build points (becuse you're starting your stats at 8 instead of an 8 and five 10's) you're done.

 

[Camelot]

I realize that my math may be off a bit, but that's why I'm posting this here. =)

By fair, I mean that the average score set is the same as the average score set for point buy. Let's say that my math is right and the average score is 13. I also want every score the same distance from the average to be have the same chance of being rolled. So, you have the same chance of getting a 12 as you do a 14, the same chance of getting an 11 as you do a 15, all the way to the same chance of getting an 8 as you do an 18. Rolling two dice solves that, because that's how they work. I think three or four dice would do the same thing, but the problem with the current rolling method is that by dropping the lowest that somehow skews it but my math skills aren't good enough to figure out how it is skewed.

I personally have rolled up many characters with the 4d6 minus lowest die method, but more have come up as underpowered characters than overpowered or even average characters once they start playing. It just seems like the game is harder for them than it is for a team made with point buy.

Thanks for the input so far!

 

[Camelot]

Also, 3d6+2 and 4d6+1 drop lowest would not work, because the lowest result for the first is 5 and the lowest for the second is 4. I want the range to be from 8 to 18, just like point buy, which from my experience is average.

I want to add that I don't want this method to replace the current method, as some may have had different experiences with it. I just want an alternative for those who prefer point buy but want some randomness.

 

[Garthanos]

There is also variance/standard deviation to be considered.
for instance...
6 + 4d6/2
is a way less extreme distribution than
6+ 2d6

both give you a range of 8 to 18 centered on 13.
It could be fun to use this to represent genetic variability.

Fey soul'd in my game world were genetically engineered aka one of the createds and thus vary... very little where as humans varied quite a bit.

 

[Camelot]

Yeah, that's a pretty good idea! Depending on the variability of the race (humans compared to something like goliaths), they have a wider range of stats.

I was searching for the probabilities and I found a handy chart. These are the probabilities of rolling a number with the two methods.

Stat 4d6 drop lowest 2d6 + 6
3 .1% 0%
4 .3% 0%
5 .8% 0%
6 1.6% 0%
7 2.9% 0%
8 4.8% 2.8%
9 7% 5.6%
10 9.4% 8.3%
11 11.4% 11.1%
12 12.9% 13.9%
13 13.3% 16.7%
14 12.3% 13.9%
15 10.1% 11.1%
16 7.3% 8.3%
17 4.2% 5.6%
18 1.6% 2.8%

Clearly, 2d6+6 will get you higher stats, no questions asked. And the average of 4d6 drop lowest is 12.25, whereas 2d6+6 brings it right to 13. I think the slight raise in stats is nice for player characters, so the players don't feel like their characters got the short end of the stick when they rolled numbers lower than 8. The site I found this on discusses other D&D probabilities (from 3rd Edition, however) if you're interested. D&D Statistics

In my search I came across a site that'll roll character stats for you, and it gave you the option of multiple methods. One of those methods was 2d6+6, so I guess I wasn't the first to think of it (but of course I wasn't, it seems so obvious).

The site I linked to also proposes this method:
1. Roll 4d6 and remove the lowest die. Do this for each stat.
2. If none of the stats are higher than 13, start over at step 1.
3. If the total modifiers added up are 0 or less, start over at step 1.

You end up getting an average total modifier of about +6, which means that the average score is 12 or 13. So around the same area. The problem is, this seems much more time consuming and I wouldn't want to have to reroll the scores for one character even more than once.

Some other score rolling ideas:

3d6 added together. Pretty brutal, though.
3d6 three times, use highest result.
4d6 drop lowest, but reroll 1s.

 

[Paul Strack]

It all depends on what you mean by "fair". Comparing 2d6+6 to a point-buy, you would need to consider a weight distribution, because the point-cost of an ability score increase dramatically as it gets closer to 18.

The weighted attribute value of traits would be:

8: -2 x (1 / 36)
9: -1 x (2 / 36)
10: 0 x (3 / 36)
11: 1 x (4 / 36)
12: 2 x (5 / 36)
13: 3 x (6 / 36)
14: 5 x (5 / 36)
15: 7 x (4 / 36)
16: 9 x (3 / 36)
17: 12 x (2 / 36)
18: 16 x (1 / 36)

Weighted Average: 4.11 * 6 = 24.67, + 2 to account for the single point-buy stat beginning at 8. So, the weight average of your curve is 26.27 point-value, compared to the 22 points in the "official" point-buy. That's considerably better.

I will do the math on 4d6 drop 1, to see how it compares to point-buy.

 

[Paul Strack]

Here is the same math for 4d6 drop 1 die. Since is not possible to "buy" a stat below 8, I've extrapolated the negative values for lower scores. The odds of a getting a score below 7 is less than 6%, so those scores don't have a big impact, anyhow.

3: -7 x (1 / 1296)
4: -6 x (4 / 1296)
5: -5 x (10 / 1296)
6: -4 x (21 / 1296)
7: -3 x (38 / 1296)
8: -2 x (62 / 1296)
9: -1 x (91 / 1296)
10: 0 x (122 / 1296)
11: 1 x (148 / 1296)
12: 2 x (167 / 1296)
13: 3 x (172 / 1296)
14: 5 x (160 / 1296)
15: 7 x (131 / 1296)
16: 9 x (94 / 1296)
17: 12 x (54 / 1296)
18: 16 x (21 / 1296)

Weighted Average: 3.34 * 6 = 20.05, + 2 to account for the single point-buy stat beginning at 8. The weighted average is 22.05 point-value, almost identical to the 22 points in the "official" point-buy.

So, statistically the 4d6 drop 1d6 is very close to the point buy system, in the sense the "average" point-value for your rolls will be almost identical to what you would get through the point buy system.

However, it isn't really comparable, because you will be getting random values, and are not guaranteed the 16 primary score that you need to have an good character (there is about a 50% chance of getting at least one 16+ roll). So you can make an argument that the random averages should be better than point-buy to balance not being able to optimize your character.

 

[Nytmare]

Also, 3d6+2 and 4d6+1 drop lowest would not work, because the lowest result for the first is 5 and the lowest for the second is 4.

Man. Allergy medicine is a hell of a drug...