[D&D] Rolling System Statistics

[The Hive Custodian]

With the help of my TI-84, I've calculated the average results for six die rolling systems:

3d6: Average is 2,268 / 6^3, or 10.5.
4d6, drop lowest: Average is 15,869 / 6^4, or about 12.245.
5d6, drop two lowest: Average is 104,433 / 6^5, or about 13.430.

2d6, double highest: Average is 413 / 6^2, or about 11.472.
3d6, double highest, drop lowest: Average is 2,898 / 6^3, or about 13.417.
4d6, double highest, drop two lowest: Average is 18,907 / 6^4, or about 14.589.

This concludes another Nearly Useless Calculation.

[nightinverse]

Maybe I should have set this game up on 4d6, double highest, drop two lowest...

[Doomsmile]

That would be so sweet!
And since no-one else has their characters genned anyway (stupid lazy grumble grumble grumble....), why not? As long as I don't get anything too wacky compared to what I have now, it shouldn't mess up Candis too much.

[nightinverse]

I'll allow it! New plan: 4d6, double highest, drop two lowest!

[The Hive Custodian]

Question: 7th roll, like in Greyscale, or no?

[nightinverse]

Why not?

[Doomsmile]

Oh, cool!

[nightinverse]

Yar.

[Doomsmile]

Yar-har.

[The Hive Custodian]

Holy sh!t...

...you're going to make us solo hill giants and wyverns at 7th level, aren't you?

[Doomsmile]

I won't be soloing. I'll have an animal companion.

[The Hive Custodian]

Well, down to the 7th roll... is Ersi about to become smarter?

[Doomsmile]

I didn't use the modified version. The old stats were better. With the 7th re-roll, I might get something better. I should roll it.
EDIT: I don't think of Ersi as being that much smarter than average, anyway. A little, but not a lot. She can be determined and clever, but I don't picture her as a fount of knowledge or anything.

[The Hive Custodian]

I asked whether Ersi was about to become smarter as an indirect way of asking, "Any luck with that re-roll?" You have your lowest score in Intelligence, so if you don't improve your scores with your re-roll, then it won't change. No luck with the re-roll, Ersi doesn't get any smarter. If, on the other hand, you do get an improvement with the re-roll, then either you replace your lowest score, Intelligence, with the re-roll, making Ersi smarter, or you would shuffle another score in, which also makes Ersi smarter. So, if you had luck with the re-roll, Ersi gets smarter. Thus, the question "Is Ersi about to get smarter?" should have the same answer as "Any luck with that re-roll?"

So, let me ask you again, more directly:

Any luck with that re-roll?

[Doomsmile]

I got an 18. However, my scores were not rolled with that system, so I'm rolling a new set as well.
These scores are awesome. I'm just wondering if there's a way I can redistribute points. I really don't want Ersi to have an int higher than 13.

[nightinverse]

Holy sh!t...

...you're going to make us solo hill giants and wyverns at 7th level, aren't you?

I'll make you solo barbarian tribes, wyrmlings and dire elk.

[Doomsmile]

These scores are awesome. I'm just wondering if there's a way I can redistribute points. I really don't want Ersi to have an int higher than 13. No, really. Ersi isn't stupid, but she isn't smart enough to warrent a 15 intelegence. Anything I can do about it besides arbitrarily dropping two points of int

[The Hive Custodian]

Distribution of Ability Scores of Organic Characters (3d6)

......Chance..Cul..............Cul...........
Roll..in.216..Chance...Pct.....Pct.....Pctl..
3.......1.......1.....0.46%....0.46%...0.23%.
4.......3.......4.....1.39%....1.85%...1.16%.
5.......6......10.....2.78%....4.62%...3.24%.
6......10......20.....4.62%....9.26%...6.94%.
7......15......35.....6.94%...16.20%..12.73%.
8......21......56.....9.72%...25.93%..21.06%.
9......25......81....11.57%...37.50%..31.71%.
10.....27.....108....12.50%...50.00%..43.75%.
11.....27.....135....12.50%...62.50%..56.25%.
12.....25.....160....11.57%...74.07%..68.29%.
13.....21.....181.....9.72%...83.80%..78.94%.
14.....15.....196.....6.94%...90.74%..87.27%.
15.....10.....206.....4.62%...95.37%..93.06%.
16......6.....212.....2.78%...98.15%..96.76%.
17......3.....215.....1.39%...99.54%..98.88%.
18......1.....216.....0.46%..100.00%..99.77%.

Chance in 216: This many out of 216 people have this ability score.
Cul Chance: This many out of 216 people have this ability score or less.
Pct: This percentage of people have this ability score.
Cul Pct: This percentage of people have this ability score or less.
Pctl: The percentile of the average person of this score. The average person with this score is better than this percentage of the population.

[nightinverse]

Yar.

[Doomsmile]

So I definatly don't want a 15 intelegence. What can I do about it?

[The Hive Custodian]

I'll try to think up a workable system.

[The Hive Custodian]

LAP Ratings

Like a set of racial traits, a set of ability scores can be rated by LAP. The LAP rating of a set of ability scores is as follows:

Your highest score is worth 3 LAP per point. There is no reduction for odd scores.
Your second and third-highest scores are worth 2 LAP per point. For an odd score, the last point is only worth 0.5 LAP.
The rest of the scores are worth 1 LAP per point. For an odd score, the last point is only worth 0.5 LAP.

---

Using this system:

I presume your new set is 18, 17, 16, 15, 15, 15? That would be 162 LAP. Giving up 2 points of Intelligence would give you 2 LAP to work with. You could raise one of the 15s to 16 (1.5 LAP), and the 16 to 17 (0.5 LAP). Your scores would then be 18, 17, 17, 16, 15, 13. You could give up another point of Intelligence (0.5 LAP) to increase the 16 by one, giving 18, 17, 17, 17, 15, 12. (While this may look like a large increase, the net change in actual ability modifiers is only +1 to the fourth-highest and -1 to the last.)

[Doomsmile]

NI, would this be acceptable? I wouldn't be plugging any points into my 18. I just don't think Ersi is smarter than 95% of people. Probably not even 80%, but I want the bonus language and skill. This is the problem of using a pre-concieved character, I guess...
(I would probably be putting a point into the 17 con and dumping the rest because it's not part of the template and I don't want to turn this into a modified point-buy.)

[nightinverse]

It's acceptable, for now.

[Doomsmile]

Cool!

[The Hive Custodian]

I've added a new column to the table, which should give a clearer measure of how good a certain score is. The last column, Pctl, gives the percentile of the average person with that score a la STAR testing results.

[The Hive Custodian]

Tonight's installment of Yet More Useless Statistics is brought to you by the following:

Me.
My TI-84.
$10.60.
4 hours of my time.
BART.
Caltrain.
And one unfortunate life, who, from what I hear, neglected to look both ways before crossing the train tracks.

So, what was bought at such a large cost? I managed to calculate the probability of rolling a certain total of ability scores on 6 x 3d6. I won't list every probability (since there's nearly a hundred possible totals), but here are some key points:




The standard elite array (15, 14, 13, 12, 10, 8) totals 72 points. 90.44% of people have a total of 72 points or below.
The average total score on 6 x 4d6 drop lowest is about 73 points. On average, this is the 91.52nd percentile.
The average total score on 6 x 4d6 drop two lowest, double highest is about 88 points. On average, this is the 99.98062nd percentile.
To have a higher ability score total than 75% of the general populaton, you need a total of at least 68 points.
To have a higher ability score total than 90% of the general populaton, you need a total of at least 73 points.
To have a higher ability score total than 95% of the general populaton, you need a total of at least 76 points.
To have a higher ability score total than 99% of the general populaton, you need a total of at least 81 points.
To have a higher ability score total than 99.9% of the general populaton, you need a total of at least 86 points.
To have a higher ability score total than 99.99% of the general populaton, you need a total of at least 90 points.
The character in Greyscale with the lowest ability score total is Orion Mure. His ability scores total 72 points. On average, this puts him at the 89.15th percentile.
The character in Greyscale with the highest ability score total is Jens Baecker. His ability scores total 96 points. On average, this puts him at the 99.99994188th percentile. This is literally one in a million; about 1 in 1,720,448 to be more exact.
Ersi is only one point below Jens. This is still good enough for the 99.9998613rd percentile, or about 1 in 721,117. Candis is one point behind Ersi; she's at the 99.9996845th percentile, or about 1 in 316,906.
Assuming the current world's population is distributed according to the calculated probabilities, about 60 people have triple-digit totals. However, it is likely that nobody has a total of above 103.
Lastly, the probability of getting 6 18s on 6 x 3d6 is 1 in 1.0156 x 10^14, a number on the magnitude of the net worth of the world in US dollars.

[nightinverse]

And we wonder how long it will take before I roll up an all 18 character? There we go.

Now, does it matter that Greyscale World-Earth has around 9.5 billion souls?

[The Hive Custodian]

It makes it more likely that someone has a 103. 104 is far less likely, though; less than 1 in 7 chances of that happening with 9.5 billion.

[Doomsmile]

With Ersi's scores, I can lie and atribute more to the template that the teplate actually garnered (13 points of strength and 6 points of con probably came from the template. The 13 dex is probably about what she had before, as the metal bits didn't slow her reaction time. She is, though, uncharacteristicly clumsy for a 13 dex character- even after all these years.)

I think I'm going to go gen Paladin, Necro, and Katherine to see how amazingly gifted they are, too. Y'know, just for yucks. And in case NI doens't feel like trying to guess how I picture these people and make sheets for them.

[The Hive Custodian]

Standard Array

Dividing the probability table for 4d6 into 6 equally likely sections, and taking the median of each, I found that the standard array should really be 16, 14, 13, 12, 10, 8 instead of 15, 14, 13, 12, 10, 8.

[nightinverse]

Standard Array

Dividing the probability table for 4d6 into 6 equally likely sections, and taking the median of each, I found that the standard array should really be 16, 14, 13, 12, 11, 10, 8 instead of 15, 14, 13, 12, 10, 8.

That's seven values...

[The Hive Custodian]

Eck! Fixed it. Somehow I put an extra 11 in there...

[nightinverse]

We all make mistakes, indeed we do.

[The Hive Custodian]

Standard Arrays, Continued

I've now also calculated standard arrays for 3d6 and 4d6, drop 2 lowest, double highest:

3d6:::::::::::::: 15,13,11,10,8 ,6
4d6, DropL::::::: 16,14,13,12,10,8
4d6, Drop2L, 2xH: 18,17,16,14,13,10

So it appears that the average person, is, in fact, fairly good at something, assuming 3d6 is a good indicator of the general population.