25 or 32...what's the REAL "standard" point buy?

[Creamsteak]

There is a certain degree of error with my numbers. I'll expand on it. Basically I turned the 6^24 possibilities into a table. Though, honestly, since we don't have a linear increase I don't think examining JUST one stat is the way to go (though apparently it came up with the same number as me). The computer generated information above looks right to me, though.

What I did to get the number of equations down to around 1 million was first to generate a table of all values based on how often they occur in the 1296 rolls of 4d6-L. So 3 occurs 1/1296, 8 occurs 72/1296 iirc, and such. Then run a loop from 3 to 18 multiplying the fractions (so 3, 3, 3, 3, 3, 3 occurs 1/1296^6 times). I was using doubles, so theres a certain degree of imprecision right there. There are six nexted loops going through 3 to 18, so we get to see how often each ability array occurs. This loop also has a second array which keeps track of the points value * the percentage that it occurs in. The sum of all these numbers came out to be 1.0000000005971 or such, so I had a number that was slightly greater than 100%.

So basically the program tracks 1) How often each ability occurs out of all abilities, 2) How many abilities exist given the criteria, 3) The average points value of all accepted totals. Then the program divides the total points by the number of accepted criteria, and gives an output of that number.

Theres problems with the way I wrote it though... I screwed up at one point and was rounding something off before it should have been.

 

[Creamsteak]

I re-wrote the code into visual basic, so I made a handy little button presser. You'll need visual basic 6.0 to run this, I think.

Here is the relevant portion of the code:

For a = 3 To 18
    For b = 3 To 18
        For c = 3 To 18
            For d = 3 To 18
                For e = 3 To 18
                    For f = 3 To 18
                        If (Check1.Value = 0) Or ((Check1.Value = 1) And ((a Or b Or c Or d Or e Or f) > 14)) Then
                            If (Check2.Value = 0) Or ((Check2.Value = 1) And ((Modifier(a) + Modifier(b) + Modifier(c) + Modifier(d) + Modifier(e) + Modifier(f)) > 0)) Then
                                If (Check3.Value = 0) Or ((Check3.Value = 1) And ((a And b And c And d And e And f) > 7)) Then
                                    Sum = Sum + ((Points(a) + Points(b) + Points(c) + Points(d) + Points(e) + Points(f)) * Dice(a) * Dice(b) * Dice(c) * Dice(d) * Dice(e) * Dice(f))
                                    Track = Track + (Dice(a) * Dice(b) * Dice(c) * Dice(d) * Dice(e) * Dice(f))
                                End If
                            End If
                        End If
                    Next f
                Next e
            Next d
        Next c
    Next b
Next a
List1.AddItem (Sum / Track & " is the average points value of")
List1.AddItem (Track * 100 & "% of the dice.")
End Sub

And if anyone knows how to turn the data into an exe, I'd like to know.

 

[dvvega]

I went back to the proverbial drawing board and rethought my statistics.

What I did was create the sample space from every possible combination available, filtering for the 2 rules of character generation. I added a third rule, that no ability could be less than 8 just to see if it made much difference. It was negligible.

Why is this a valid space? Because if I gave you every possible combination out there, and you added "just one more" character, they would average with the other copy of themselves to become themselves.

Then I went through and performed the averaging of abilities versus the averaging of points.

Same situation arose with the averages: abilities around the 12.5 mark, point score average around the 30 mark.

Speaking to two seperate professors I received two seperate answers. On the on hand, one categorically stated that you average the independent results, the other professor stated you average the ability points.

The first opinion is based on the fact that in the average space your average set of abilities tends to 12.5s.

The second opinion is based on the fact that the points table does not have results that are fractional therefore your average of 12.5 is "lost in limbo" unless you apply a rounding rule.

Now since d20 requires rounding it causes issues of when to round. Anyways, I'm pretty "over" this affair. If I can't get a straight answer out of statistics professors then I'm giving up.

 

[Creamsteak]

I think that the all abilities must be 8 or better is somewhat important, though. For every set you eliminate (3, 4, 5, 6, and 7s) your removing more "high" numbers than "low" numbers with respect to points value, so you should get a lower average.

 

[dvvega]

Yes I agree the average should be lower and it is around 12.25 or something similar since most of the 8s and lower combinations were removed by applying the first 2 rules.

For example ... there are 16^6 combinations initially (6 abilities in the range of 3-18 each). All combinations with no ability 14+ removes 6^6 combinations from the "pool" (6 abilities in the range of 3-13).

In the long run there are 7.7 million sets of valid abilities. Due to the sheer number of them the average still tends to the gaussian distribution of 12-13. Removing the 8s and lower keeps them closer to 12, without removing them keeps them closer to 12.5/13.

However one thing that must be made clear is that removing combinations with <8 is NOT a valid thing to do in this hypothesis. What we started debating here was if 25 point buy was reasonably accurate compared to 4d6-Lowest. In essence which is fairer, or if as argued, 4d6-Lowest is closer to 30 points. 4d6-L allows for abilities from 3-18. Point buy allows them from 8-18. You cannot remove those sets with < 8 in them from the 4d6-L combinations since they're valid in that space. The only reason you would consider it is that you want a "perfect" match with point buy.

 

[Creamsteak]

My numbers put the average of all values at 28, and the average with the PHB definitions for all valid arrays at 30.

 

[stephenh]

Well I guess that we all more or less agree that 4d6 drop lowest is roughly equivalent to about 28-29 points and once you apply some sort of cut for minimum viable stats you get an average of about 30-31 ish points? I guess I originally wanted to just point out that it was higher than 25 points.

Only minor issues after this point...

Beyond this I guess we are mostly quibbling over minor details. I guess I should point out that in some of my earlier analysis I used a "non viable" definition that was a bit different to what the book suggests (I was trying to remember off the top of my head).

Using exactly the same definitions as Anax I get values that are all within 0.1 points of everything he quotes (using my random generation method).

Anax and Creamstreak have both done a "complete" statistical analysis essentially using the same method, although your values are actually a little bit different from one another. As I said earlier I use a totally different method and get values that agree more closely with Anax.

To Anax: Some of the differences you see between our results are because I also used a definition of at least one stat of better than 14 rather than one stat of at least 14, as you used when comparing with me. (again I used this because I was guessing what the non viable definition was).

To dvvega: I would say that it sounds to me like one of the professors you asked didn't fully understand the problem, because averaging ability scores is not the right way to do this.

To Creamsteak: Firstly On calculating average points per stat:- This is a valid thing to do if you are treating all stats in the same way. It is not valid if you are treating some stats differently. i.e. Applying a rule that no stat can be lower than 8 is ok becasue the same rule applys to all stats. In contrast, specifying that one stat (out of the 6) must be larger than 14, means that you cannot average the points for the stats because one stat is being treated differently to the others. This is why applying a cut of 8 to all stats must increase the average rolled.
In further evidence of this consider the average (mean) set of stats rolled after applying a cut of total modifiers +1 or better and at least one stat above 13. You get: 15.95, 14.45, 13.22, 12.02, 10.65, 8.72 (sorted into order).
Given this:
1) On average the lowest stat you roll is 8.72.
2) Applying the rule of "all stats above 7" can be simplified to "the lowest stat must be greater than 7" -- this rule is then totally independant of the values of any of the other stats, which will have averages of the values given earlier.
3) Given the average of 8.72, in the majority of cases the lowest stat is greater than 7 -- so applying a cut of all stats equal to 7 or lower should not make have significant effect on the results.
4) In the small number of cases where the lowest value is 7 or lower you are removing cases where the lowest stat is given a points value of 0. So your average must increase. Remember, irrespective of what the lowest value actually is, the average values of the higher stats are unchanged.

How about adjusting your code to calculate the average points you get when you only consider cases where at least one value is 7 or less. By the numbers you quoted earlier is must be in the high 30s at least (because when you removed them your average dropped from 30 ish to 26.5 ish). The averages stats I quoted above show that only a small fraction of cases will have any stats below 8. -- There are many more cases where the stats all all above 7 -- well over 50% of cases must already satisfy the condition of all stats above 8.

The math just doesn't hang together. Look at it in reverse. You are saying that when you include stats sets with very low stat values (which would have low points values), your average points value actually goes up!!???

 

[stephenh]

All values > 8	One value > 14	Total mod > 0	Average Points Value
Check		Check		Check		26.616669536546
		Check		Check		30.0628924089566 
						28.5277777775552

Ahh I see now why some of your values disagree with those calculated by myself and Anax. I just programed in your method and I notice that the 30.0628924089566 average actually corresponds to the case where the sum of modifiers is +1 and there is at least 1 stat *>11* rather than at least one stat >14 as you claim in your original post. Using 1 stat of at least 12 I reproduce this number exactly (well to 10 significant figures anyway, which is about the limit of float precision).