Can someone who understands Statistics help me?

gabrion · Oct 29, 2005

I'm good at some math, but stats have never been a strong point of mine.

Using a 5d6, keep the highest three, system to roll stats (rolling six times altogether), what is the average roll one can expect?

More importantly, since there will no doubt be deviation in the results, I'm trying to figure out what this system equates to on average as far as point buy goes. Any help would be appreciated.

FEADIN · Oct 29, 2005

The average is the same, each die has an average roll of 3.5, but the more dice you roll the more chance that you get a 6.
With 3 dice it's 1/216 to have three 6 together, with 4 dice it's six times less because you only want three 6 on four dice and with 5 it's again six less so one in six to have three 6 together.
Remember that you can end with five 1 as easily

I hope that my answer is correct :heh:

For the point buy equivalent I would say that's it's equal at least to the highest possible, usually you end up with scores between 12 and 18 in all characs.

Thanee · Oct 29, 2005

Guessing, I'd say the average is about ~14 and PB equivalent would be around ~31.

Bye
Thanee

wuyanei · Oct 29, 2005

I ran the numbers in excel, the result:

Result Percentage
3 0.01
4 0.06
5 0.19
6 0.53
7 1.16
8 2.19
9 3.81
10 6.04
11 8.55
12 11.33
13 13.57
14 14.85
15 14.29
16 12.02
17 7.84
18 3.55
total 100

Average: 13.429

My method was to list all the possible number sets for 5 6-sided dice (ex. {2,2,2,1,1}), then calculate the number of permutations within each number set. The result is the Weight (Wt.) for this set. I then summed the weight for all set with the same number total (ex. set {2,2,2,1,1} has a total of 8), to obtain a Total Weight (TotWt) for each number between 3 and 18. This TotWt divided by 7776 (ie. 6^5 = 7767), gives us the percentage weight of each value. Calculating the average is then a straightfoward operation. The median lies within the value 14 block. The mode is also 14.

As for point buy equivalent, if the characters have 13 for every score, we need 30 pt point buy. If the characters have 14 for every score, we need 36 pt point buy. Scaling the difference linearly for 13.429, we need a point buy of:

30 + (0.429) * (6 x 6 - 6 x 5) = 32.574 or about 32 pt point-buy. QED

I have attached a graph mapping the percentage (y-axis) versus the total dice value (x-axis). The breakdown of my calculations are enclosed in the spoiler block below.

[sblock]

Code:

D1	D2	D3	D4	D5	Value	Wt.	TotWt	%
1	1	1	1	1	3	1	1	0.01
1	1	1	1	2	4	5	5	0.06
1	1	1	1	3	5	5	15	0.19
1	1	1	2	2	5	10		
1	1	1	1	4	6	5	41	0.53
1	1	1	2	3	6	20		
1	1	2	2	2	6	10		
1	2	2	2	2	6	5		
2	2	2	2	2	6	1		
1	1	1	1	5	7	5	90	1.16
1	1	1	2	4	7	20		
1	1	1	3	3	7	10		
1	1	2	2	3	7	30		
1	2	2	2	3	7	20		
2	2	2	2	3	7	5		
1	1	1	1	6	8	5	170	2.19
1	1	1	2	5	8	20		
1	1	1	3	4	8	20		
1	1	2	2	4	8	30		
1	1	2	3	3	8	30		
1	2	2	2	4	8	20		
1	2	2	3	3	8	30		
2	2	2	2	4	8	5		
2	2	2	3	3	8	10		
1	1	1	2	6	9	20	296	3.81
1	1	1	3	5	9	20		
1	1	1	4	4	9	10		
1	1	2	2	5	9	30		
1	1	2	3	4	9	60		
1	1	3	3	3	9	10		
1	2	2	2	5	9	20		
1	2	2	3	4	9	60		
1	2	3	3	3	9	20		
1	3	3	3	3	9	5		
2	2	2	2	5	9	5		
2	2	2	3	4	9	20		
2	2	3	3	3	9	10		
2	3	3	3	3	9	5		
3	3	3	3	3	9	1		
1	1	1	3	6	10	20	470	6.04
1	1	1	4	5	10	20		
1	1	2	2	6	10	30		
1	1	2	3	5	10	60		
1	1	2	4	4	10	30		
1	1	3	3	4	10	30		
1	2	2	2	6	10	20		
1	2	2	3	5	10	60		
1	2	2	4	4	10	30		
1	2	3	3	4	10	60		
1	3	3	3	4	10	20		
2	2	2	2	6	10	5		
2	2	2	3	5	10	20		
2	2	2	4	4	10	10		
2	2	3	3	4	10	30		
2	3	3	3	4	10	20		
3	3	3	3	4	10	5		
1	1	1	4	6	11	20	665	8.55
1	1	1	5	5	11	10		
1	1	2	3	6	11	60		
1	1	2	4	5	11	60		
1	1	3	3	5	11	30		
1	1	3	4	4	11	30		
1	2	2	3	6	11	60		
1	2	2	4	5	11	60		
1	2	3	3	5	11	60		
1	2	3	4	4	11	60		
1	3	3	3	5	11	20		
1	3	3	4	4	11	30		
2	2	2	3	6	11	20		
2	2	2	4	5	11	20		
2	2	3	3	5	11	30		
2	2	3	4	4	11	30		
2	3	3	3	5	11	20		
2	3	3	4	4	11	30		
3	3	3	3	5	11	5		
3	3	3	4	4	11	10		
1	1	1	5	6	12	20	881	11.33
1	1	2	4	6	12	60		
1	1	2	5	5	12	30		
1	1	3	3	6	12	30		
1	1	3	4	5	12	60
1	1	4	4	4	12	10
1	2	2	4	6	12	60
1	2	2	5	5	12	30
1	2	3	3	6	12	60
1	2	3	4	5	12	120
1	2	4	4	4	12	20
1	3	3	3	6	12	20
1	3	3	4	5	12	60
1	3	4	4	4	12	20
1	4	4	4	4	12	5
2	2	2	4	6	12	20
2	2	2	5	5	12	10
2	2	3	3	6	12	30
2	2	3	4	5	12	60
2	2	4	4	4	12	10
2	3	3	3	6	12	20		
2	3	3	4	5	12	60		
2	3	4	4	4	12	20		
2	4	4	4	4	12	5		
3	3	3	3	6	12	5		
3	3	3	4	5	12	20		
3	3	4	4	4	12	10		
3	4	4	4	4	12	5		
4	4	4	4	4	12	1		
1	1	1	6	6	13	10	1055	13.57
1	1	2	5	6	13	60		
1	1	3	4	6	13	60		
1	1	3	5	5	13	30		
1	1	4	4	5	13	30		
1	2	2	5	6	13	60		
1	2	3	4	6	13	120		
1	2	3	5	5	13	60
1	2	4	4	5	13	60
1	3	3	4	6	13	60
1	3	3	5	5	13	30
1	3	4	4	5	13	60
1	4	4	4	5	13	20
2	2	2	5	6	13	20
2	2	3	4	6	13	60
2	2	3	5	5	13	30
2	2	4	4	5	13	30
2	3	3	4	6	13	60
2	3	3	5	5	13	30
2	3	4	4	5	13	60
2	4	4	4	5	13	20
3	3	3	4	6	13	20
3	3	3	5	5	13	10
3	3	4	4	5	13	30		
3	4	4	4	5	13	20		
4	4	4	4	5	13	5		
1	1	2	6	6	14	30	1155	14.85
1	1	3	5	6	14	60		
1	1	4	4	6	14	30		
1	1	4	5	5	14	30		
1	2	2	6	6	14	30		
1	2	3	5	6	14	120		
1	2	4	4	6	14	60		
1	2	4	5	5	14	60		
1	3	3	5	6	14	60		
1	3	4	4	6	14	60		
1	3	4	5	5	14	60		
1	4	4	4	6	14	20		
1	4	4	5	5	14	30		
2	2	2	6	6	14	10
2	2	3	5	6	14	60
2	2	4	4	6	14	30
2	2	4	5	5	14	30
2	3	3	5	6	14	60
2	3	4	4	6	14	60
2	3	4	5	5	14	60
2	4	4	4	6	14	20
2	4	4	5	5	14	30
3	3	3	5	6	14	20
3	3	4	4	6	14	30
3	3	4	5	5	14	30
3	4	4	4	6	14	20
3	4	4	5	5	14	30
4	4	4	4	6	14	5
4	4	4	5	5	14	10
1	1	3	6	6	15	30	1111	14.29
1	1	4	5	6	15	60		
1	1	5	5	5	15	10		
1	2	3	6	6	15	60		
1	2	4	5	6	15	120		
1	2	5	5	5	15	20		
1	3	3	6	6	15	30		
1	3	4	5	6	15	120		
1	3	5	5	5	15	20		
1	4	4	5	6	15	60		
1	4	5	5	5	15	20		
1	5	5	5	5	15	5		
2	2	3	6	6	15	30		
2	2	4	5	6	15	60		
2	2	5	5	5	15	10		
2	3	3	6	6	15	30		
2	3	4	5	6	15	120		
2	3	5	5	5	15	20		
2	4	4	5	6	15	60		
2	4	5	5	5	15	20		
2	5	5	5	5	15	5		
3	3	3	6	6	15	10		
3	3	4	5	6	15	60		
3	3	5	5	5	15	10		
3	4	4	5	6	15	60		
3	4	5	5	5	15	20		
3	5	5	5	5	15	5		
4	4	4	5	6	15	20		
4	4	5	5	5	15	10		
4	5	5	5	5	15	5		
5	5	5	5	5	15	1		
1	1	4	6	6	16	30	935	12.02
1	1	5	5	6	16	30
1	2	4	6	6	16	60
1	2	5	5	6	16	60
1	3	4	6	6	16	60
1	3	5	5	6	16	60
1	4	4	6	6	16	30
1	4	5	5	6	16	60
1	5	5	5	6	16	20
2	2	4	6	6	16	30
2	2	5	5	6	16	30
2	3	4	6	6	16	60
2	3	5	5	6	16	60
2	4	4	6	6	16	30
2	4	5	5	6	16	60
2	5	5	5	6	16	20
3	3	4	6	6	16	30
3	3	5	5	6	16	30		
3	4	4	6	6	16	30		
3	4	5	5	6	16	60		
3	5	5	5	6	16	20		
4	4	4	6	6	16	10		
4	4	5	5	6	16	30		
4	5	5	5	6	16	20		
5	5	5	5	6	16	5		
1	1	5	6	6	17	30	610	7.84
1	2	5	6	6	17	60		
1	3	5	6	6	17	60		
1	4	5	6	6	17	60		
1	5	5	6	6	17	30		
2	2	5	6	6	17	30		
2	3	5	6	6	17	60		
2	4	5	6	6	17	60		
2	5	5	6	6	17	30		
3	3	5	6	6	17	30		
3	4	5	6	6	17	60		
3	5	5	6	6	17	30		
4	4	5	6	6	17	30		
4	5	5	6	6	17	30		
5	5	5	6	6	17	10		
1	1	6	6	6	18	10	276	3.55
1	2	6	6	6	18	20		
1	3	6	6	6	18	20		
1	4	6	6	6	18	20		
1	5	6	6	6	18	20		
1	6	6	6	6	18	5		
2	2	6	6	6	18	10		
2	3	6	6	6	18	20		
2	4	6	6	6	18	20		
2	5	6	6	6	18	20		
2	6	6	6	6	18	5		
3	3	6	6	6	18	10		
3	4	6	6	6	18	20		
3	5	6	6	6	18	20		
3	6	6	6	6	18	5		
4	4	6	6	6	18	10		
4	5	6	6	6	18	20		
4	6	6	6	6	18	5		
5	5	6	6	6	18	10		
5	6	6	6	6	18	5		
6	6	6	6	6	18	1		
					13	7776	7776	100

[/sblock]
Hope I've been helpful!

Happy gaming!
Yanei Wu

KarinsDad · Oct 29, 2005

Thanee said:
Guessing, I'd say the average is about ~14 and PB equivalent would be around ~31.

You are correct on the average (I think it is 13.5 or so, I do not quite remember exactly), but the average point buy is probably close to 40.

If you rolled 3 13's and 3 14's, you would have a point buy of 33.

The fact that every 15 to 18 you roll ups that by 2 to 10 whereas every 8 to 13 or lower only lowers it by 6 to 1 means that you usually will have a higher average than 33.

I used this system for my current campaign and the equivalent point buys ended up being:

70 (he rolled out his butt with 17, 17, 17, 18, 16, 13 and we all watched him do it)
51
40
42
43
44

Note: I also had two house rules that upped this slightly.

1) If you did not get at least one 16 stat, you could re-roll all of the stats from scratch. One player did a 25 point buy with a 15 as highest, so lower rolls do occur. This happened one time out of six here.

2) I allowed a single re-roll if you rolled real bad on one of your rolls, but you had to take the new roll, even if it was lower. This probably ups the average point buy by about 5 because you throw away a 9 (or sometimes 8 or 10) and get a 14 (on average).

Given this and dropping out the two extremes of 25 and 70 in our group, results in our group would have averaged about 39 (51 + 40 + 42 + 43 + 44 = 220 / 5 = 44 - 5 = 39) if we did not have the second house rule.

dcollins · Oct 29, 2005

This is a surprisingly hard math problem. I've spent days in the past picking out problems in people's alleged solutions to the general pick M out of N dice problem.

Anyway, just wrote a C++ simulator for this one, ran it a million times: average is 13.43. I figure that comes out just the same as a "high-powered campaign", 32 point buy (if all 13's, 5x6=30; count this as 5x6 + .43 x 1 = 32.58).

KarinsDad · Oct 29, 2005

wuyanei said:
As for point buy equivalent, if the characters have 13 for every score, we need 30 pt point buy. If the characters have 14 for every score, we need 36 pt point buy. Scaling the difference linearly for 13.429, we need a point buy of:

30 + (0.429) * (6 x 6 - 6 x 5) = 32.574 or about 32 pt point-buy. QED

You should add an equation and column to your spreadsheet to figure out the exact point buy (use -1 for 7s, -2 for 6s, etc.) because the point buy is much higher than 32 due to the drastic increase at 15 to 18.

Lonely Tylenol · Oct 29, 2005

Something to point out when die-roll methods vs. point buy systems comes up: hopeless characters. If you take the standard 4d6-drop-lowest method and roll a million times, the point buy average comes out around 28 points. But if you remember that many of those characters rolled will be "hopeless" and so excluded from the final calculation, it works out closer to 30 points. Removing the lowest end of the distribution moves the mean.

It would be the same in this case, too. However, with the 5d6-drop-2 system there will be fewer hopeless characters. I'd still ad-hoc one or two points of point buy onto the estimated average, at least until someone produces the inevitable program that simulates a billion die rolls, discarding hopeless characters, and averages the calculated point buy for each.

KarinsDad · Oct 29, 2005

I recalculated this with Excel and came up with the same 13.43 average as everyone else.

However, I also did a count of each result and did the point buy for that (with 3 being -5, 4 being -4, etc. for numbers below 8) and I came up with a point buy average of 39.6.

Also, less than 2% of the totals on the dice come out 7 or less, so it would be extremely rare to have a character with more than one terrible stat and very unusual to have a character with one of them. One character in nine will have one or more stats 7 or lower, typically one stat.

wuyanei · Oct 29, 2005

Hmm... KarinsDad, you have a point. We should use the 'Average point-buy value' instead of 'The point-buy value of a character with all average scores'.

In other words, we should calculate:

AvgPBv = Sum3 to 18[ PBv(N) x P(N) ];

where PBv(N) is the point buy value of score N, P(N) is the percentage chance of rolling N.

PBv(3) = -5; PBv(4) = -4; PBv(5) = -3; PBv(6) = -2; PBv(7) = -1; PBv(8) = 0; PBv(9) = 1; PBv(10) = 2; PBv(11) = 3; PBv(12) = 4; PBv(13) = 5; PBv(14) = 6; PBv(15) = 8; PBv(16) = 10; PBv(17) = 13; PBv(18) = 16;

Code:

N	P(N)	N x P(N)/100	PBv(N) x P(N)/100
3	0.01	0.0003		-0.0005
4	0.06	0.0024		-0.0024
5	0.19	0.0095		-0.0057
6	0.53	0.0318		-0.0106
7	1.16	0.0812		-0.0116
8	2.19	0.1752		0
9	3.81	0.3429		0.0381
10	6.04	0.604		0.1208
11	8.55	0.9405		0.2565
12	11.33	1.3596		0.4532
13	13.57	1.7641		0.6785
14	14.85	2.079		0.891
15	14.29	2.1435		1.1432
16	12.02	1.9232		1.202
17	7.84	1.3328		1.0192
18	3.55	0.639		0.568
total	100	13.429		6.3397

6.3397 x 6 = 38.0382 ...so a 38 pt point-buy might be a better representation of 5d6 drop two lowest.

BTW, I'm rather curious on how you came to the 39.6 pt number. Would you mind sharing your calculation method?

Thanks for replying!
Yanei Wu

Can someone who understands Statistics help me?

gabrion

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FEADIN

Explorer

Thanee

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wuyanei

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KarinsDad

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dcollins

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KarinsDad

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Lonely Tylenol

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KarinsDad

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wuyanei

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