[Nerdity Warning] Alternate attribute generation methods and statistics of existing ones.

[Feuerrabe]

Hello,

I've always liked to toy a little with the magic of chance and numbers and chance, so I've been toying with some

And even though my statistics are a bit rusty and I would have to read terms like "median" "golden cut" and "standard deviation", which I vaguely remember but wouldn't know how to calculate from scratch.

A collague at my work who has studied statistics has been extremely helpful pointing me to the fact that I can calculate the probability of things like "If I roll six scores in this manner, how likely is it that I get two or more results of less than ten?" using the binomial distribution, for which a lot of online calculators across the internet exist. Which is a good thing, since I was already about to write a script to build a binary tree out of all possibilities and then traverse it, tracing the probaility and result of each branch.
A very helpful site is for example: http://stattrek.com/online-calculator/binomial.aspx

Anyway here are the first two alternate methods I've been considering. Everyone is more than welcome to comment on it, especially if they know more about statistics than I do.

All methods assume that we're doing six times the exact same thing. I've considered other possibilities, but it only makes it unnescessarily complicated, imho.

4d4+2:
Result range: 6 - 18
Expectancy Value: 12
Likelyhood of at least one score less than 8: 11.16%
Likelyhood of at least one 16 or higher: 30.39%
Likelyhood of at least two values lower than 10: 19.27%
Diagram: http://ubuntuone.com/59CZxx1doXcM2SG2iZaen4 (PNG)

5d4, delete lowest, +2:
Result range: 6 - 18
Expectancy Value: 13.23
Likelyhood of at least one score less than 8: 3.47%
Likelyhood of at least one 16 or higher: 65.40%
Likelyhood of at least two values lower than 10: 3.87%
Diagram: http://ubuntuone.com/1w6b7Jrkpy51LK0LVaASXI (PNG)

For comparison:
2d6+6:
I didn't calculate that out in detail, but the expectancy value is 13, the curve is symmetric since you don't delete a dice and more flat than with 4d4+2.

Maximum Average with Point Buy 25: 13.5 (14, 14, 14, 14, 13, 12) or (16, 13, 13, 13, 13, 13)

Maximum Average with Point Buy 20: 13.17 (14, 13, 13, 13, 13, 13)
* with one 16: 12.67 (eg. 16, 12, 12, 12, 12, 12)

Maximum Average with Point Buy 15: 12.5 (13, 13, 13, 12, 12, 12)
* with one 16: 11.83 (eg. 16, 12, 12, 11, 10, 10)

4d6, delete lowest:
Result Range: 3 - 18
Expectancy Value: 12.24
Likelyhood of at least one score less than 8: 29.72%
Likelyhood of at least one 16 or higher: 56.76%
Likelyhood of at least two values lower than 10: 28.38%
Diagram: http://ubuntuone.com/0c9HI1IvrwByFCL1hetL08 (PNG)

5d6, delete lowest 2:
Result Range: 3 - 18
Expectancy Value: 13.48
Likelyhood of at least one score less than 8: 11.17%
Likelyhood of at least one 16 or higher: 79.83%
Likelyhood of at least two values lower than 10: 7.64%
Diagram: http://ubuntuone.com/5YYSs2rFynRu0witsoQku4

Looking at the stats I think I like the dsitribution 5d4, delete lowest, add two a lot better than 5d5, delete lowest two, but it it will still yield epic range chars, even though the expectancy value is slightly lower. What disturbs me about the 4d4+2 is the probability to get a value lower than 8. Two d6+6 remains a quite fine method.

There are a few other possibilties that I've been considering. Not add up the dice of a role, but use the average of two or three d10 or d12. Or use the geometric_middle(4d4) + 8 (which would make an 18 impossible, since four rolled 4s would yield sqrt(4² + 4² + 4² + 4²) = 8.

That would of course be a bit tough to explain to the players. They will feel cheated of they don't understand how you get from the rolls to the scores. But a DM who likes these stuff could formulate it somewhat mystical, making a kind of horoscope. (Eg. Your strength is determined by a star the four dimensions of your dice roll show. The first dimension is that of your parents... and make diagram of the first three dimensions with a stars background with bars in glowing colours. The fourth dimension might be the shifting chaos of the abyss which can never be visually represented.)

What I am trying to reach is a symmetric curve that is flat for the middle four or five stats and steep at the ends. And try to avoid values lower than 8, while making an 8 a realistic possibility. Unfortunately, I will have to dig a lot deeper into statistics, to reach a good, simple mode to that end.

[MechE_]

If you want a flatter curve with a lower realistic minimum (8, for example) roll more dice. Rolling 6d3 (d6/2) would yield a maximum of 18, a minimum of 6 (both of which are highly unlikely) with an average of 12. The middle would be pretty flat because of the increased sample size while the ends, specifically the possibility of a 6, 7, 17, or 18, would be quite low.

Run your numbers on that and see what you come up with. =)

Alternatively, you could d 5d3 + 3 if you REALLY wanted to avoid the possibility of a 6 or 7 occuring, but I think you'll find the probability of those low enough to be satisfactory. Also, 5d3 + 3 gives an average of 13 instead of 12, which I'm not a fan of, but ymmv.

[Feuerrabe]

Ooops, wrong thread. Well.

I think I am gonna stick with the 5d4, delete lowest, +2 mode.

Maybe I am gonna optionally let the players choose Point Buy 22 or so as well, but if they roll, they gotta use it. I am gonna run the numbers on the d3 method you suggest as well. I wonder if I can make actual d3s out of pencils with three edges and mark their cornes with colors...

[Feuerrabe]

hrm. Alright. It gets down to a matter of taste.

The 6d3 method (yielding standard range characters) has a standard deviation of 2.00.

The 5d3+3 method (yielding high fantasy - epic range characters) has a standard deviation of 1.83.

The 5d4, discard lowest, +2 method I stuggested earlier (yielding epic range characters) has a standard deviation of 2.24.

I must say, for a standard range campaign there is a lot that speaks for the simple, straightforward 6d3. "2" sounds like a good standard deviation to me.

PS: Does anyone know whether there are actual d3 for sale? I've seen an offer in a fantasy shape, but I'd prefer a normal cube that has the each of the markings twice.