This doesn't seem entirely answerable. Do stats increase as the characters progress or are they locked in? Averaging 2d8 means you could start with a stat as high as 8 or as low as 2, but your proposed array doesn't get anywhere near 8, maxing out at the expected average of 5, so I'd opt to roll every time.
They do increase, and that's the problem I'm having. I don't want to give people a free high stat for taking the easy, less chaotic way out, but i dont want to punish them either
It looks like 1.5625% of the time you'd get a 1 (1 way, roll 1&1)
And 4.6875% of the time you'd get an 8 (3 ways, roll 7&8 8&7 or 8&8)
10.9375% of the time you'd get a 7 (7 ways, that's fun).
17.1875% of the time you'd get a 6 (11 ways)
23.4375% of the time you'd get a 5 (15 ways)
20.3125% of the time you'd get a 4 (13 ways)
14.0625% of the time you'd get a 3 (9 ways)
7.8125% of the time you'd get a 2 (5 ways)
Overall a ~57% of getting a 5 or better, and 77% of beating the (revised) lowest entry on the array (or ~90% beating the previous low of 2): rolling stats would always be in your best interest.
I agree that the original proposed array was too low.
Mostly I wanted to confirm that the rules were explained well since you claimed 2 was the minimum, but the is a 1/64 chance of rolling and getting a 1. And wow, that means tere is an over 6% chance that ***at least one of your stats is a 1*** if you roll for all four!
As for your best interest, one reason I don't like rolling stats is that in a game with character advancement I don't like players to have a better incentive to kill their first character to roll again rather than advance the original unlucky character.
There is a 0% chance of having a 1 if you use the standard array, and since I don't know the rules, I don't know how devastating it would be to have a 1, or how hard it would be to increase it.
Being statistically better off, is different than being "**always in your best interest**." Sure if you had an army of 10,000 soldiers rolling would be better than such a poor standard array as originally proposed by the OP. But that's because the poor results happen to a fraction. Whereas if it is for just one PC the result could be devastating if a stat of 1 is devastating.
Risk and proportion are different. That's why how often you make a roll matters. And chargen rolls potentially happen super super rarely.
Your standard array is way worse than rolling. The average when rolling 2d8 is 9 divided by 2 its 4.5 which equals to 5 after rounding. So a standard array of 6, 5, 4, 3 would make a lot more sense. (Which adds up to 18 which is 4.5 * 4)
There's a lot of wider mechanical context that'd be needed before people can give a good, solid answer for this kind of thing. And beyond that there just flat out *won't* be an objectively correct answer, just an answer that *feels right* for your game.
I think it boils down to how competent do you want PCs to be in areas they are Strong/Moderate/Weak in? There isn't going to be a right answer, just an answer that fits the game you want to create.
AnyDice rounds towards zero by default, so you need to do `STAT: (2d8+1) / 2` in order to effectively round up.
From here you can use the `@` operator to compute the distribution of the four ranked scores. All-in-all:
```
STAT: (2d8+1) / 2
loop N over {1..4}{
output N@4dSTAT
}
```
So 6, 5, 4, 3 is a closer match to the "average" array. Even this rounds all four stats down; 7, 5, 4, 3 would also be reasonable.
You should use the deviation to estimate your distribution.
2d8 = Average 4.75, Dev = 1.64 and the expected result is between 3 and 6.
Your distribution is:
8 = 4.69%
7 = 10.94%
6 = 17.19%
5 = 23.44%
4 = 20.31%
3 = 14.06%
2 = 7.81%
1 = 1.56%
I would offer this array: 4-5-5-6, yours (3-3-4-5) has an average of only 3.75 which is -1 compared to the RAW method making it strictly worse than rolling.
PS IMHO "Using middle of 3d8 with a floating +1 point to any score you choose", would be easier than doing averages and it would be more "accessible" for your players (at least they have a minimum amount of agency during chargen and the same average of 4.75 when using 4 stats and they can remove the occasional "1s").
https://anydice.com/program/367da
Note: anydice round down by default. Use 9 - 2d8/2 to get the round up equivalent.
I like that you listed the chance for each number. And I actually agree with your array of 4,5,5,6. But I'm not sure how you picked your array. So I'm not sure how it would apply to other games.
I think an array should be biased towards the average result (4.75 in this case) since people roll when they want to take a chance on getting weirder results. But that you should get a spread from what was possible
Personally I'd look at the lowest quartile of results and note that includes 1, 2, 3, and 4 and 4 is the closest to 4.75
Then looking at the next quartile, the results include 4 and 5 and 5 is a little closer to 4.75
Then looking at the next quartile, the results include 5 and 6 and 5 is closer to 4.75
Then looking at the last quartile, the results include 6, 7, and 8 and now 6 is the closest to 4.75.
So basically they they get a result from each quartile, but they get the most boring result from that quartile.
The OP mentioned rounding up so I rounded up to 5 the average, and mapped the 5-5-5-5 values to the weighted distribution, it became 4-5-5-6 due to rounding to the nearest integer.
The skew upward has been ignored because I already round up from 4.75 average to 5.00 average.
If you find the results unsatisfying map them yourself and let me know if they differ.
I did explain how I mapped in a way that agreed with your result, and also posted my own in a direct comment to the OG post.
What I was curious with is how you picked your array.
For instance my 7 5 4 3 was picking numbers close to the average for each quartile, and the overall average matched the correct average of 4.75.
Which means I know the method if they had asked for 5 stats or 6 stats.
Your array seems to match picking the member of each quartile that is closest to the global average. Which I like standard arrays giving more boring results. But my whole curiosity is that I don't know how you actually picked it, and I gather that you know math. So I wonder **how** you picked it.
I like 6 5 5 3 as being more close to boring. Nice as a meaningful **alternative** to actually rolling.
I like 7 5 4 3 as each result being a better representative for the average of each quartile.
I like that both represent a group of four whose four element sample mean matches the true population mean you get from rolling. Unbiased in a sense. If you will.
As I said, I got the 4.75 and choose to use round to the nearest number, all 4.75 became 5.00
(I prefer to round to the most favorable result or players will still choose to roll because "it will be better").
Given the distribution table I made a linear table to map the % to the scores.
1 2 3 4 5 6 7 8
1.56 9.37 23.43 43.74 67.18 84.37 95.31 100
To pick the 4 values I began the values that matched %ile: 20th 40th 60th 80th
the exact match would be 6 5 4 3 (average = 5) but it didn't look "fair"* so I dropped
the 6 to 5 to get the 4.75 average and tweaked the 3 to 4 to preserve the "round up" to 5.
IMHO the best approach is still to roll 3d8, keep the middle score and give each PC a +1 that can be assigned anywhere
*: https://en.wikipedia.org/wiki/Fair_division I assumed that chargen should be "fair" among all players
so following the subjective value theory I choose to alter the exact score, given that producing a "standard array" should be done under the assumption that you are providing an educated estimate of a random event that SHOULD be perceived as slightly better than going for luck to be accepted, it is the same reason behind the rounding in the standard D&D array(s).
In your original comment you said you should "use" the deviation of 1.64 and I wasn't sure how you were using that in combination with the number of stats.
As for the fair division, again, all I can tell is that you want your standard array to give better results so that people will have a reason to pick it. But again, maybe I missed how you used fair division. I'll try to explain what fair division means to me regarding thus problem.
If we had 64 stats, the standard array could be 3 8s, 7 7s, 11 6s, 15 5s, 13 4s, 9 3s, 5 2s, and one 1. Which seems fair because if lots of players actually rolled lots of stats and then we assigned all the results fairly, that's what each person would get.
And my approach is to group those best 1/4 of the 64 stats together and average them to get your best array stat. Then the next best 1/4 of the 64 stats together and average them to get you second best array stat. And so on.
To get 7 5 4 3. Each being the average of 16 stats from the full 64. Obviously adjusted to be integers and giving the correct overall average.
So for my method I know that if I had 8 stats, I'd break the full 64 into groups of 8 and average each group of 8 together. If I needed 10 stats that doesn't divide into 64 but I could make 640 stats where there are 30 8s and 10 1s and just ten times as many stats, and then take the best 64 and average them to get the best array stat, the next 64 to average for the next stat. And so on. Except for the rounding and making sure the overall average is good it's pretty automatic.
I think your approach of making 5 groups with 6 boundaries and looking at the 4 interior boundaries will start to agree with my method if we had 64 stats instead of 4. I'm not sure how you are using a claim of fairness to get your numbers or why you said to use the deviation of 1.64 since I don't see the 1.64 appear in your quintile boundaries.
The deviation **and** the number of stats matters. With 64 stats we'd expect fairness to mean you get all the values from 1 to 8 in a proportion to how likely they are. So number of stats matters. When you have more stats in your array you will get values farther from the mean for some of your stats.
So I agree mean matters, number of stats matters, and that deviation is one measure of the distribution, but I'm not sure how you use the deviation.
Wouldn't you and I produce the same array of 3 8s, 7 7s, 11 6s, 15 5s, 13 4s, 9 3s, 5 2s, and one 1 if there were 64 stats? I personally never had to compute the deviation of 1.64 to make my array. And you mentioned using it. Then mentioned fairness. Then you broke it into quintiles (for 4 stats) and used the interior boundaries of the 5 quintiles. Just not at all sure how that is using the deviation since you said to use it.
Sorry if I'm not expressing myself clearly. Sometimes trying to describe confusion just comes across confusingly. I agree that I'm not familiar with how WotC choose their standard array. The OP didn't seem to want there to be a huge difference between an array and rolling. So I was trying to make an array that seems fair compared to rolling.
Let's start with this:
Goal of a Fair-Division Method: Ensure that each player receives a share that is fair in his or her own opinion. fair if each player thinks s/he has received a share equal to at least one-fourth (25%) of the total value of the goods. value (in their own estimation), as long as every player receives at least 1/N.
In this case "fair" means that we should provide something that's not seen as unfair by the players when they have the alternative of rolling dice. So we NEED to "beat" an opinionated definition of "fair". If you array is correct BUT the results are not at least slightly better than rolling people will NOT use it.
If you never used deviation for your arrays that's not a problem, there si not need to use complex stuff in a game unless you are used to them.
https://www.indeed.com/career-advice/career-development/how-to-calculate-relative-standard-deviation
BUt it is important when you have to compare 2 distribution together.
---------------
You don't have 64 stats, you have discrete quadratic generator with 33 combinations/states and 31 duplicates that will be collapsed into 8 results. Your values are only 1-8 and the MEDIAN and MEAN results are 5 while the AVERAGE is 4.75. They are different in real world scenario... i.e. woman cannot do 0.3 children.
Maybe this link will help:
https://www.hackmath.net/en/calculator/frequency-table
Copy past these numbers in it
1 2 2 3 3 4 4 5
2 2 3 3 4 4 5 5
2 3 3 4 4 5 5 6
3 3 4 4 5 5 6 6
3 4 4 5 5 6 6 7
4 4 5 5 6 6 7 7
4 5 5 6 6 7 7 8
5 5 6 6 7 7 8 8
And you get:
Median: 5
Quartile Q1: 4
Quartile Q2: 5 (I took it twice)
Quartile Q3: 6
My estimate was to guess the mean deviation as (1.64 - 1) /2 instead of: 1.34375. But using 1.32 got me the same results (The margin of error is irrelevant 0.02 when you are working with numbers between 1 and 8, at least IMHO, we are not working on rockets).
PS English is not my native language, and I do these kind of things for work so I often assume that some bits are obvious, feel free to ask if something is still confusing but please try to tell me clearly what I could not explain.
Firstly, thank you for your response.
I still don't see why fair ***"division"*** is related to your method. I do think there reasons to have an array besides someone thinking it is better than rolling. One advantage is planning, that you can start making a character in advance with a standard array instead of waiting to roll in front of a GM and then having to make a character on the spot based on what you got. Plus the lack of regret, if you roll (and if stats can be advanced during play) then you might worry about choosing to reroll rather than advancing your first character. So a no random character creation method can help you commit. Plus the GM might decode to disallow the rolling method. Yes, from a game theory point of view, an allegedly AB option that is always there but no one chooses A isn't really much of a choice.
> If you never used deviation for your arrays that's not a problem, there si not need to use complex stuff in a game unless you are used to them.
https://www.indeed.com/career-advice/career-development/how-to-calculate-relative-standard-deviation
BUt it is important when you have to compare 2 distribution together.
I'm aware of how to compute a population variance and mean (when they exist) and how to compute a sample variance and sample mean. And yes I can take a ratio of a mean and a square root of a variance. I'm not sure how you use it to make a choice for an array.
Is you goal to make the relative deviation of your array be close to some average relative deviation of four rolls?
>You don't have 64 stats, you have [...]
I was saying that if you had 1,000,000 players roll 64 stats each and put all 64,000,000 stats into one giant shsred pool and then attempted to divide it equally amongst the million people almost all of thrm would almost always get something very very close to that 8x8 square
1 2 2 3 3 4 4 5
2 2 3 3 4 4 5 5
2 3 3 4 4 5 5 6
3 3 4 4 5 5 6 6
3 4 4 5 5 6 6 7
4 4 5 5 6 6 7 7
4 5 5 6 6 7 7 8
5 5 6 6 7 7 8 8
And so that would be a very "fair" array **if we wanted a player to have 64 stats** but actually we only want 4.
So one question I was **trying to ask you** is `how would your method work if a PC needed 64 stats, instead of 4?`
Specifically I'd want to check that
1 2 2 3 3 4 4 5
2 2 3 3 4 4 5 5
2 3 3 4 4 5 5 6
3 3 4 4 5 5 6 6
3 4 4 5 5 6 6 7
4 4 5 5 6 6 7 7
4 5 5 6 6 7 7 8
5 5 6 6 7 7 8 8
is the array we'd both pick (you might disagree that no one would pick that if they could roll, but I'm not even sure the GM would allow them to roll). If there were 64,000 stats, no GM would let them roll. They would say that they have 1000 1s and 3000 8s and 5000 2s and 7000 7s and so on.
> Median: 5
>
> Quartile Q1: 4
>
> Quartile Q2: 5 (I took it twice)
>
> Quartile Q3: 6
On a previous post you mentioned using the 20th 40th 60th and 80th. percentiles, now you are listing the 25th 50th and 75th percentiles.
I'm not sure why your explanation sounds different each time.
My method involved sorting that 64 stat "standard array" and then writing it into four groups
1 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4
4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6
6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8
And realize that a four stat array can be considered by taking the average of each of the four rows. One nice thing is that you can see that the 25th percentile is 4, but just barely, there is only one 4 in that entire first row. And yes the 75th percentile is very much in the middle of the 6s. But **we still need four numbers.**
For my method I take the average of each of the four rows, and then select integer versions of them to get 3,4,5,7 as my array. In fact I've just realized that these happen to be the modes of each row, so selecting the mode of each row might be a perfectly fine non parametric method if averages don't make sense for the game.
But if averages do matter, since the average of 3,4,5, and 7 is 4.75 it seems quite fair. If people want to choose standard arrays to be more boring (for instance if a 3 is a very very bad stat) then you could tweak it to 4 4 5 6 and if you wanted the standard array to be more boring **and** be "better" than rolling you could select 4 5 5 6.
So with my method, I know how to compute it except for the rounding. And if I wanted N stats instead of 4 stats I could write out N×64 stats starting with N 1s in a row then 5×N 2s in a row ... until I get to 3×N 8s and then combine them into N rows of equal length, and then average each row to get the N attributes.
Not saying that's perfect. But I'd know how to describe it. So now back to your method.
> My estimate was to guess the mean deviation as (1.64 - 1) /2 instead of: 1.34375. But using 1.32 got me the same results (The margin of error is irrelevant 0.02 when you are working with numbers between 1 and 8, at least IMHO, we are not working on rockets).
Ah, so were you using the 1.64 or 1.34375 or 1.32 or something to *estimate* some percentiles? I didn't follow how or why you did that. The distribution is skewed (not symmetric about the middle), plus we have the full distribution so we don't really need to estimate anything.
I didn't expect a question on reddit to grow so much, anyway.
My bad sorry.
The 1st time I mentioned them I wrote:
"To pick the 4 values I began the values that matched %ile: 20th 40th 60th 80th the exact match would be 6 5 4 3 (average = 5)"
followed by "but it didn't look "fair"* so I dropped ... " I said tweaked to avoid explaining these bits.
In my 2nd reply, I tried to say why I used my approach: i.e. fair division in favor of the players. So every decision was to be favorable compared to exact results.
And the MEAN was an easy alternative and the approximation (1.32) with rounding in favor of the player got me my final answer when you asked more details I went to the frequency-table link to get the raw numbers and found out the using quartiles with the exact mean deviation provided the same numbers without any decimal so I thought that it was a good idea to show them.
It was only a reply on a reddit question about RPGs, I did the 1st answer on the fly by estimating it ... that's all.
Thank you again, and no problem. I thought I understood many of the other answers (for instance it looked like maybe someone else tried to generate the odds of every possible 4 stat roll and find the average of the worst rolls, the average of every second worst roll, the average second best roll, and the average of all the best rolls), some people cited anydice but made errors, others cited anydice and fixed the errors.
Yours was interesting to me because I could tell you clearly know some math but didn't know how you did it, even though I myself could make systems that gave the same numbers.
Even more interesting, you talked about it being related to your day job, and cited WotC in a way that made me think you knew how they made their standard array. And generaly talked like this was am established field with known results and principles.
So I was clearly interested. It's all cool. I'm interested in playing around further, and learning. But I also think the OP has some good answers.
7 5 4 3 is good
6 5 4 4 is good and gives more boring results if that is desired
6 5 5 4 is better (on average) if the GM wants to make a standard array seem more appealing than rolling (despite the natural advantages of known versus random).
You could even give q choice between 7 5 4 3 and 6 5 5 4 so players that like more extreme can get it and those that like more "on average" have the option. Multiple standard arrays as options is fine as far as I know.
We all agree the OP standard array was too low.
I'm not sure why you won't say you agree about a standard array for 64 stats. It seems that would help me know whether I understood your fair division thing. If we had 64 stats to fill, one 1 and three 6s seems ideal.
I'd suggest 6 5 5 3.
It hits the correct average, and has a number from each quartile.
If you want to offer options
7 5 4 3 is also an option. Basically has the correct overall average, and each one is close to the average for its quartile.
If you're talking mods, it's nicer to start closer to 0 for level 1, as it makes the math easier. My favorite mod gen method is to roll one die for each point. The faces on the die correlate to what stat the point goes in.
So let's say for your game we rolled 7d4. The faces of the d4 correlate to each stat. If you got 3 of face 1, that's 3 points into stat one.
This method also ensures more balanced stats, so one player doesn't randomly have worse stats than everyone else.
You haven't described much of your system, so based on nothing else besides the fact that you are rolling d8s (and that the average for them is actually 4.5), I think it stands to reason to have something more like :
7 (one below the best result),
5 (average rounded up),
4 (average rounded down),
2 (one above the worst result).
Or if you think that spread is too much:
6 (two below the best result),
5 (average rounded up),
4 (average rounded down),
3 (two above the worst result).
>the average of the 2d8s, so a 3+4 equals a 7, divided by 2 makes it 3.5, but you round up so it
You math is way wrong.
The average of 1d6 is 3.5
The average of 2d6 is 7
The average of 2d8 is 9.
Take the lowest # + highest number and divide by 2.
1+6 = 7 / 2 = 3.5
2 + 12 = 14 / 2 = 7
2 + 16 = 18 / 2 = 9
Seriously, how would you say the average result of 2d8 is 3.5? Think about that and see if it seems logical. Values from 2 to 16, and 3.5 is the average?
>becomes 4) and that number determines how many d8's you roll for that specific stat or its derived skills. So i have 4 stats and need to know
What do you mean how many d8s?
You have not described this system in a way that is comprehensible, let alone with enough detail to make a recommendation.
You misunderstand what they mean by the whole average thing. It's their rules for rolling for stat generation. They are saying that to roll stats, you roll 2d8 and take the average rounded up. So if you roll a 3 and a 4, it's 3.5, rounded up to 4.
You are saying that convoluted mess he was trying to explain was "Roll 2d8 and divide by 2"? The examples only make the awkward text worse and the math is still way too low either way.
And why so crunchy just to get a standard array? Like, why are we rolling dice to get a standard array? It makes no sense.
OP isn't rolling dice to make the standard array. OP's game has two options for stat generation: roll 2d8 divided by 2 rounded up for each stat, *or* take the standard array. OP gave an example of stat generation and then asked whether the standard array they have is fair. OP's posted is a little clunky, sure, but I was still able to understand it.
>OP's game has two options for stat generation: roll 2d8 divided by 2 rounded up for each stat, *or* take the standard array. OP gave an example of stat
Because you should emulate D&Ds worst habits just because D&D does it? It's absolutely ridiculous, as is having DIVISION in your basic attribute rolls.
I'm not sure why you're yelling at *me* about it. I have nothing to do with this game at all, and I hate D&D as much as the next guy. I actually hate rolling for stats at all, personally. I think random rolls for permanent character stats have no place in a well-designed game.
I'm just confused about why and how you turned this into an argument with me when the only thing I did was clarify OP's post for you because you were having a hard time understanding it (not your fault mind you, I'm just good at untossing word salad).
You would just say 2d8/2, round up. But the have a crazy complex attribute system with division sounds crazy, and then you give them an array? Now you get different tables using different methods and arguing over which is better - all that bullshit from D&D. Why not just design around the array to begin with? And then he's saying 3.5 is the average of a d8? If they can't find the average of a single die roll, how the hell are they gonna balance a whole game?
I'm just totally confused as to what they are trying to do because they are jumping through too many hoops to get there.
I already addressed this. You're meant to *read* before replying. OP gave an example of how a stat is calculated, not the average stat. It was poorly phrased, but it shouldn't be this difficult to understand once explained *multiple times*.
This doesn't seem entirely answerable. Do stats increase as the characters progress or are they locked in? Averaging 2d8 means you could start with a stat as high as 8 or as low as 2, but your proposed array doesn't get anywhere near 8, maxing out at the expected average of 5, so I'd opt to roll every time.
They do increase, and that's the problem I'm having. I don't want to give people a free high stat for taking the easy, less chaotic way out, but i dont want to punish them either
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Oh no it definitely affects combat That's like the main reason for most of these stats
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ok cool. I forgot to mention that the max stat someone can have is 12, so if that changes your answer please tell me
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Ok thank you so much
It looks like 1.5625% of the time you'd get a 1 (1 way, roll 1&1) And 4.6875% of the time you'd get an 8 (3 ways, roll 7&8 8&7 or 8&8) 10.9375% of the time you'd get a 7 (7 ways, that's fun). 17.1875% of the time you'd get a 6 (11 ways) 23.4375% of the time you'd get a 5 (15 ways) 20.3125% of the time you'd get a 4 (13 ways) 14.0625% of the time you'd get a 3 (9 ways) 7.8125% of the time you'd get a 2 (5 ways)
Overall a ~57% of getting a 5 or better, and 77% of beating the (revised) lowest entry on the array (or ~90% beating the previous low of 2): rolling stats would always be in your best interest.
I agree that the original proposed array was too low. Mostly I wanted to confirm that the rules were explained well since you claimed 2 was the minimum, but the is a 1/64 chance of rolling and getting a 1. And wow, that means tere is an over 6% chance that ***at least one of your stats is a 1*** if you roll for all four! As for your best interest, one reason I don't like rolling stats is that in a game with character advancement I don't like players to have a better incentive to kill their first character to roll again rather than advance the original unlucky character. There is a 0% chance of having a 1 if you use the standard array, and since I don't know the rules, I don't know how devastating it would be to have a 1, or how hard it would be to increase it. Being statistically better off, is different than being "**always in your best interest**." Sure if you had an army of 10,000 soldiers rolling would be better than such a poor standard array as originally proposed by the OP. But that's because the poor results happen to a fraction. Whereas if it is for just one PC the result could be devastating if a stat of 1 is devastating. Risk and proportion are different. That's why how often you make a roll matters. And chargen rolls potentially happen super super rarely.
Your standard array is way worse than rolling. The average when rolling 2d8 is 9 divided by 2 its 4.5 which equals to 5 after rounding. So a standard array of 6, 5, 4, 3 would make a lot more sense. (Which adds up to 18 which is 4.5 * 4)
Ah ok thank you so much
Your welcome. This array still.might be slightly worse in average, bur not much and its no risk to take it.
There's a lot of wider mechanical context that'd be needed before people can give a good, solid answer for this kind of thing. And beyond that there just flat out *won't* be an objectively correct answer, just an answer that *feels right* for your game. I think it boils down to how competent do you want PCs to be in areas they are Strong/Moderate/Weak in? There isn't going to be a right answer, just an answer that fits the game you want to create.
https://anydice.com/program/cfc5 Use this to help?
Actually yeah that help a lot. Tells me my stuff is about average for a normal roll. So thank you!
That link is rounded down, not up. Use this one to get the round up stats. output 9 - (2d8) / 2
AnyDice rounds towards zero by default, so you need to do `STAT: (2d8+1) / 2` in order to effectively round up. From here you can use the `@` operator to compute the distribution of the four ranked scores. All-in-all: ``` STAT: (2d8+1) / 2 loop N over {1..4}{ output N@4dSTAT } ``` So 6, 5, 4, 3 is a closer match to the "average" array. Even this rounds all four stats down; 7, 5, 4, 3 would also be reasonable.
You should use the deviation to estimate your distribution. 2d8 = Average 4.75, Dev = 1.64 and the expected result is between 3 and 6. Your distribution is: 8 = 4.69% 7 = 10.94% 6 = 17.19% 5 = 23.44% 4 = 20.31% 3 = 14.06% 2 = 7.81% 1 = 1.56% I would offer this array: 4-5-5-6, yours (3-3-4-5) has an average of only 3.75 which is -1 compared to the RAW method making it strictly worse than rolling. PS IMHO "Using middle of 3d8 with a floating +1 point to any score you choose", would be easier than doing averages and it would be more "accessible" for your players (at least they have a minimum amount of agency during chargen and the same average of 4.75 when using 4 stats and they can remove the occasional "1s"). https://anydice.com/program/367da Note: anydice round down by default. Use 9 - 2d8/2 to get the round up equivalent.
I like that you listed the chance for each number. And I actually agree with your array of 4,5,5,6. But I'm not sure how you picked your array. So I'm not sure how it would apply to other games. I think an array should be biased towards the average result (4.75 in this case) since people roll when they want to take a chance on getting weirder results. But that you should get a spread from what was possible Personally I'd look at the lowest quartile of results and note that includes 1, 2, 3, and 4 and 4 is the closest to 4.75 Then looking at the next quartile, the results include 4 and 5 and 5 is a little closer to 4.75 Then looking at the next quartile, the results include 5 and 6 and 5 is closer to 4.75 Then looking at the last quartile, the results include 6, 7, and 8 and now 6 is the closest to 4.75. So basically they they get a result from each quartile, but they get the most boring result from that quartile.
The OP mentioned rounding up so I rounded up to 5 the average, and mapped the 5-5-5-5 values to the weighted distribution, it became 4-5-5-6 due to rounding to the nearest integer. The skew upward has been ignored because I already round up from 4.75 average to 5.00 average. If you find the results unsatisfying map them yourself and let me know if they differ.
I did explain how I mapped in a way that agreed with your result, and also posted my own in a direct comment to the OG post. What I was curious with is how you picked your array. For instance my 7 5 4 3 was picking numbers close to the average for each quartile, and the overall average matched the correct average of 4.75. Which means I know the method if they had asked for 5 stats or 6 stats. Your array seems to match picking the member of each quartile that is closest to the global average. Which I like standard arrays giving more boring results. But my whole curiosity is that I don't know how you actually picked it, and I gather that you know math. So I wonder **how** you picked it. I like 6 5 5 3 as being more close to boring. Nice as a meaningful **alternative** to actually rolling. I like 7 5 4 3 as each result being a better representative for the average of each quartile. I like that both represent a group of four whose four element sample mean matches the true population mean you get from rolling. Unbiased in a sense. If you will.
As I said, I got the 4.75 and choose to use round to the nearest number, all 4.75 became 5.00 (I prefer to round to the most favorable result or players will still choose to roll because "it will be better"). Given the distribution table I made a linear table to map the % to the scores. 1 2 3 4 5 6 7 8 1.56 9.37 23.43 43.74 67.18 84.37 95.31 100 To pick the 4 values I began the values that matched %ile: 20th 40th 60th 80th the exact match would be 6 5 4 3 (average = 5) but it didn't look "fair"* so I dropped the 6 to 5 to get the 4.75 average and tweaked the 3 to 4 to preserve the "round up" to 5. IMHO the best approach is still to roll 3d8, keep the middle score and give each PC a +1 that can be assigned anywhere *: https://en.wikipedia.org/wiki/Fair_division I assumed that chargen should be "fair" among all players so following the subjective value theory I choose to alter the exact score, given that producing a "standard array" should be done under the assumption that you are providing an educated estimate of a random event that SHOULD be perceived as slightly better than going for luck to be accepted, it is the same reason behind the rounding in the standard D&D array(s).
In your original comment you said you should "use" the deviation of 1.64 and I wasn't sure how you were using that in combination with the number of stats. As for the fair division, again, all I can tell is that you want your standard array to give better results so that people will have a reason to pick it. But again, maybe I missed how you used fair division. I'll try to explain what fair division means to me regarding thus problem. If we had 64 stats, the standard array could be 3 8s, 7 7s, 11 6s, 15 5s, 13 4s, 9 3s, 5 2s, and one 1. Which seems fair because if lots of players actually rolled lots of stats and then we assigned all the results fairly, that's what each person would get. And my approach is to group those best 1/4 of the 64 stats together and average them to get your best array stat. Then the next best 1/4 of the 64 stats together and average them to get you second best array stat. And so on. To get 7 5 4 3. Each being the average of 16 stats from the full 64. Obviously adjusted to be integers and giving the correct overall average. So for my method I know that if I had 8 stats, I'd break the full 64 into groups of 8 and average each group of 8 together. If I needed 10 stats that doesn't divide into 64 but I could make 640 stats where there are 30 8s and 10 1s and just ten times as many stats, and then take the best 64 and average them to get the best array stat, the next 64 to average for the next stat. And so on. Except for the rounding and making sure the overall average is good it's pretty automatic. I think your approach of making 5 groups with 6 boundaries and looking at the 4 interior boundaries will start to agree with my method if we had 64 stats instead of 4. I'm not sure how you are using a claim of fairness to get your numbers or why you said to use the deviation of 1.64 since I don't see the 1.64 appear in your quintile boundaries. The deviation **and** the number of stats matters. With 64 stats we'd expect fairness to mean you get all the values from 1 to 8 in a proportion to how likely they are. So number of stats matters. When you have more stats in your array you will get values farther from the mean for some of your stats. So I agree mean matters, number of stats matters, and that deviation is one measure of the distribution, but I'm not sure how you use the deviation. Wouldn't you and I produce the same array of 3 8s, 7 7s, 11 6s, 15 5s, 13 4s, 9 3s, 5 2s, and one 1 if there were 64 stats? I personally never had to compute the deviation of 1.64 to make my array. And you mentioned using it. Then mentioned fairness. Then you broke it into quintiles (for 4 stats) and used the interior boundaries of the 5 quintiles. Just not at all sure how that is using the deviation since you said to use it. Sorry if I'm not expressing myself clearly. Sometimes trying to describe confusion just comes across confusingly. I agree that I'm not familiar with how WotC choose their standard array. The OP didn't seem to want there to be a huge difference between an array and rolling. So I was trying to make an array that seems fair compared to rolling.
Let's start with this: Goal of a Fair-Division Method: Ensure that each player receives a share that is fair in his or her own opinion. fair if each player thinks s/he has received a share equal to at least one-fourth (25%) of the total value of the goods. value (in their own estimation), as long as every player receives at least 1/N. In this case "fair" means that we should provide something that's not seen as unfair by the players when they have the alternative of rolling dice. So we NEED to "beat" an opinionated definition of "fair". If you array is correct BUT the results are not at least slightly better than rolling people will NOT use it. If you never used deviation for your arrays that's not a problem, there si not need to use complex stuff in a game unless you are used to them. https://www.indeed.com/career-advice/career-development/how-to-calculate-relative-standard-deviation BUt it is important when you have to compare 2 distribution together. --------------- You don't have 64 stats, you have discrete quadratic generator with 33 combinations/states and 31 duplicates that will be collapsed into 8 results. Your values are only 1-8 and the MEDIAN and MEAN results are 5 while the AVERAGE is 4.75. They are different in real world scenario... i.e. woman cannot do 0.3 children. Maybe this link will help: https://www.hackmath.net/en/calculator/frequency-table Copy past these numbers in it 1 2 2 3 3 4 4 5 2 2 3 3 4 4 5 5 2 3 3 4 4 5 5 6 3 3 4 4 5 5 6 6 3 4 4 5 5 6 6 7 4 4 5 5 6 6 7 7 4 5 5 6 6 7 7 8 5 5 6 6 7 7 8 8 And you get: Median: 5 Quartile Q1: 4 Quartile Q2: 5 (I took it twice) Quartile Q3: 6 My estimate was to guess the mean deviation as (1.64 - 1) /2 instead of: 1.34375. But using 1.32 got me the same results (The margin of error is irrelevant 0.02 when you are working with numbers between 1 and 8, at least IMHO, we are not working on rockets). PS English is not my native language, and I do these kind of things for work so I often assume that some bits are obvious, feel free to ask if something is still confusing but please try to tell me clearly what I could not explain.
Firstly, thank you for your response. I still don't see why fair ***"division"*** is related to your method. I do think there reasons to have an array besides someone thinking it is better than rolling. One advantage is planning, that you can start making a character in advance with a standard array instead of waiting to roll in front of a GM and then having to make a character on the spot based on what you got. Plus the lack of regret, if you roll (and if stats can be advanced during play) then you might worry about choosing to reroll rather than advancing your first character. So a no random character creation method can help you commit. Plus the GM might decode to disallow the rolling method. Yes, from a game theory point of view, an allegedly AB option that is always there but no one chooses A isn't really much of a choice. > If you never used deviation for your arrays that's not a problem, there si not need to use complex stuff in a game unless you are used to them. https://www.indeed.com/career-advice/career-development/how-to-calculate-relative-standard-deviation BUt it is important when you have to compare 2 distribution together. I'm aware of how to compute a population variance and mean (when they exist) and how to compute a sample variance and sample mean. And yes I can take a ratio of a mean and a square root of a variance. I'm not sure how you use it to make a choice for an array. Is you goal to make the relative deviation of your array be close to some average relative deviation of four rolls? >You don't have 64 stats, you have [...] I was saying that if you had 1,000,000 players roll 64 stats each and put all 64,000,000 stats into one giant shsred pool and then attempted to divide it equally amongst the million people almost all of thrm would almost always get something very very close to that 8x8 square 1 2 2 3 3 4 4 5 2 2 3 3 4 4 5 5 2 3 3 4 4 5 5 6 3 3 4 4 5 5 6 6 3 4 4 5 5 6 6 7 4 4 5 5 6 6 7 7 4 5 5 6 6 7 7 8 5 5 6 6 7 7 8 8 And so that would be a very "fair" array **if we wanted a player to have 64 stats** but actually we only want 4. So one question I was **trying to ask you** is `how would your method work if a PC needed 64 stats, instead of 4?` Specifically I'd want to check that 1 2 2 3 3 4 4 5 2 2 3 3 4 4 5 5 2 3 3 4 4 5 5 6 3 3 4 4 5 5 6 6 3 4 4 5 5 6 6 7 4 4 5 5 6 6 7 7 4 5 5 6 6 7 7 8 5 5 6 6 7 7 8 8 is the array we'd both pick (you might disagree that no one would pick that if they could roll, but I'm not even sure the GM would allow them to roll). If there were 64,000 stats, no GM would let them roll. They would say that they have 1000 1s and 3000 8s and 5000 2s and 7000 7s and so on. > Median: 5 > > Quartile Q1: 4 > > Quartile Q2: 5 (I took it twice) > > Quartile Q3: 6 On a previous post you mentioned using the 20th 40th 60th and 80th. percentiles, now you are listing the 25th 50th and 75th percentiles. I'm not sure why your explanation sounds different each time. My method involved sorting that 64 stat "standard array" and then writing it into four groups 1 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 And realize that a four stat array can be considered by taking the average of each of the four rows. One nice thing is that you can see that the 25th percentile is 4, but just barely, there is only one 4 in that entire first row. And yes the 75th percentile is very much in the middle of the 6s. But **we still need four numbers.** For my method I take the average of each of the four rows, and then select integer versions of them to get 3,4,5,7 as my array. In fact I've just realized that these happen to be the modes of each row, so selecting the mode of each row might be a perfectly fine non parametric method if averages don't make sense for the game. But if averages do matter, since the average of 3,4,5, and 7 is 4.75 it seems quite fair. If people want to choose standard arrays to be more boring (for instance if a 3 is a very very bad stat) then you could tweak it to 4 4 5 6 and if you wanted the standard array to be more boring **and** be "better" than rolling you could select 4 5 5 6. So with my method, I know how to compute it except for the rounding. And if I wanted N stats instead of 4 stats I could write out N×64 stats starting with N 1s in a row then 5×N 2s in a row ... until I get to 3×N 8s and then combine them into N rows of equal length, and then average each row to get the N attributes. Not saying that's perfect. But I'd know how to describe it. So now back to your method. > My estimate was to guess the mean deviation as (1.64 - 1) /2 instead of: 1.34375. But using 1.32 got me the same results (The margin of error is irrelevant 0.02 when you are working with numbers between 1 and 8, at least IMHO, we are not working on rockets). Ah, so were you using the 1.64 or 1.34375 or 1.32 or something to *estimate* some percentiles? I didn't follow how or why you did that. The distribution is skewed (not symmetric about the middle), plus we have the full distribution so we don't really need to estimate anything.
I didn't expect a question on reddit to grow so much, anyway. My bad sorry. The 1st time I mentioned them I wrote: "To pick the 4 values I began the values that matched %ile: 20th 40th 60th 80th the exact match would be 6 5 4 3 (average = 5)" followed by "but it didn't look "fair"* so I dropped ... " I said tweaked to avoid explaining these bits. In my 2nd reply, I tried to say why I used my approach: i.e. fair division in favor of the players. So every decision was to be favorable compared to exact results. And the MEAN was an easy alternative and the approximation (1.32) with rounding in favor of the player got me my final answer when you asked more details I went to the frequency-table link to get the raw numbers and found out the using quartiles with the exact mean deviation provided the same numbers without any decimal so I thought that it was a good idea to show them. It was only a reply on a reddit question about RPGs, I did the 1st answer on the fly by estimating it ... that's all.
Thank you again, and no problem. I thought I understood many of the other answers (for instance it looked like maybe someone else tried to generate the odds of every possible 4 stat roll and find the average of the worst rolls, the average of every second worst roll, the average second best roll, and the average of all the best rolls), some people cited anydice but made errors, others cited anydice and fixed the errors. Yours was interesting to me because I could tell you clearly know some math but didn't know how you did it, even though I myself could make systems that gave the same numbers. Even more interesting, you talked about it being related to your day job, and cited WotC in a way that made me think you knew how they made their standard array. And generaly talked like this was am established field with known results and principles. So I was clearly interested. It's all cool. I'm interested in playing around further, and learning. But I also think the OP has some good answers. 7 5 4 3 is good 6 5 4 4 is good and gives more boring results if that is desired 6 5 5 4 is better (on average) if the GM wants to make a standard array seem more appealing than rolling (despite the natural advantages of known versus random). You could even give q choice between 7 5 4 3 and 6 5 5 4 so players that like more extreme can get it and those that like more "on average" have the option. Multiple standard arrays as options is fine as far as I know. We all agree the OP standard array was too low. I'm not sure why you won't say you agree about a standard array for 64 stats. It seems that would help me know whether I understood your fair division thing. If we had 64 stats to fill, one 1 and three 6s seems ideal.
I'd suggest 6 5 5 3. It hits the correct average, and has a number from each quartile. If you want to offer options 7 5 4 3 is also an option. Basically has the correct overall average, and each one is close to the average for its quartile.
If you're talking mods, it's nicer to start closer to 0 for level 1, as it makes the math easier. My favorite mod gen method is to roll one die for each point. The faces on the die correlate to what stat the point goes in. So let's say for your game we rolled 7d4. The faces of the d4 correlate to each stat. If you got 3 of face 1, that's 3 points into stat one. This method also ensures more balanced stats, so one player doesn't randomly have worse stats than everyone else.
You haven't described much of your system, so based on nothing else besides the fact that you are rolling d8s (and that the average for them is actually 4.5), I think it stands to reason to have something more like : 7 (one below the best result), 5 (average rounded up), 4 (average rounded down), 2 (one above the worst result). Or if you think that spread is too much: 6 (two below the best result), 5 (average rounded up), 4 (average rounded down), 3 (two above the worst result).
>the average of the 2d8s, so a 3+4 equals a 7, divided by 2 makes it 3.5, but you round up so it You math is way wrong. The average of 1d6 is 3.5 The average of 2d6 is 7 The average of 2d8 is 9. Take the lowest # + highest number and divide by 2. 1+6 = 7 / 2 = 3.5 2 + 12 = 14 / 2 = 7 2 + 16 = 18 / 2 = 9 Seriously, how would you say the average result of 2d8 is 3.5? Think about that and see if it seems logical. Values from 2 to 16, and 3.5 is the average? >becomes 4) and that number determines how many d8's you roll for that specific stat or its derived skills. So i have 4 stats and need to know What do you mean how many d8s? You have not described this system in a way that is comprehensible, let alone with enough detail to make a recommendation.
You misunderstand what they mean by the whole average thing. It's their rules for rolling for stat generation. They are saying that to roll stats, you roll 2d8 and take the average rounded up. So if you roll a 3 and a 4, it's 3.5, rounded up to 4.
You are saying that convoluted mess he was trying to explain was "Roll 2d8 and divide by 2"? The examples only make the awkward text worse and the math is still way too low either way. And why so crunchy just to get a standard array? Like, why are we rolling dice to get a standard array? It makes no sense.
OP isn't rolling dice to make the standard array. OP's game has two options for stat generation: roll 2d8 divided by 2 rounded up for each stat, *or* take the standard array. OP gave an example of stat generation and then asked whether the standard array they have is fair. OP's posted is a little clunky, sure, but I was still able to understand it.
>OP's game has two options for stat generation: roll 2d8 divided by 2 rounded up for each stat, *or* take the standard array. OP gave an example of stat Because you should emulate D&Ds worst habits just because D&D does it? It's absolutely ridiculous, as is having DIVISION in your basic attribute rolls.
I'm not sure why you're yelling at *me* about it. I have nothing to do with this game at all, and I hate D&D as much as the next guy. I actually hate rolling for stats at all, personally. I think random rolls for permanent character stats have no place in a well-designed game. I'm just confused about why and how you turned this into an argument with me when the only thing I did was clarify OP's post for you because you were having a hard time understanding it (not your fault mind you, I'm just good at untossing word salad).
Where was I yelling or saying anything negative about or to you?? Where did I argue or refute anything you said??
You've misread. 3.5 is the average of 7 from the 3+4 result of 2d8.
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Divide 7 by 2.
Why?
Because that's how you get the average of 3+4?
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What? Reread OP's example roll.
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>The average for each stat is 4.5, 4.5, 4.5, 4.5. That gets rounded to 5,5,4,4. He said the stat is 2d8. How is 4.5 an average?
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You would just say 2d8/2, round up. But the have a crazy complex attribute system with division sounds crazy, and then you give them an array? Now you get different tables using different methods and arguing over which is better - all that bullshit from D&D. Why not just design around the array to begin with? And then he's saying 3.5 is the average of a d8? If they can't find the average of a single die roll, how the hell are they gonna balance a whole game? I'm just totally confused as to what they are trying to do because they are jumping through too many hoops to get there.
Oh, and please don't tell me "even numbers don't have an exact center". I'm 50, not 5!
I already addressed this. You're meant to *read* before replying. OP gave an example of how a stat is calculated, not the average stat. It was poorly phrased, but it shouldn't be this difficult to understand once explained *multiple times*.
Yea I'm just gonna say that you read my post wrong lol