What is the statistically superior character creation method, twelve 3d6 or six 4d6? A. N. Other October 26, 2010 Regarding the question posted on StackExchange, “What is the statistically superior character creation method, twelve 3d6 or six 4d6?”: When you roll 4d6k3, each of your 6 ability scores follows the exact same probabilit probability y distribution. distribution. In statistics statistics lingo, your 6 ability ability scores are i.i.d.— independen independentt and identically identically distributed—ra distributed—random ndom variables variables.. Call one of these i.i.d. random random variables variables Y. The mean of Y is E[Y]=12.2445 E[Y]=12.244598765 987654321 4321,, and its standard deviation is σY = 2.8468444453115. Additionally, Additionally, this distribution is skewed to the left; its skewness is -0.283507652977282. By comparison, when you roll 3d6 once, you get a random variable X, with E[X]=10.5, E[X]=10.5, and standard standard deviation deviation σX = 2.95803 9580398915 989155. 5. Additionally Additionally,, it is symmetric, so its skewness is 0. However, when you roll 3d6 12 times and keep the highest 6, you get 6 different random variables (not i.i.d.), called the 7th through 12th order statistics, . E.g., X (12) is the maximum of the 12 rolls. Each order denoted X (7) , . . . , X(12) statistic statistic has its own mean, standard deviation, deviation, and skewness: skewness: Orde Orderr stat statis isti ticc mean ean stan stand dard ard devi deviaation tion skew skewne ness ss 10.8184 1.1411 -0.00564534 X (7) (7) X (8) 1 1 . 4 6 6 3 1 . 14865 -0.00978342 (8) X (9) 12.1517 1.1693 0.00712255 (9) X (10) 12.919 1.2154 0.0435863 (10) X (11) 1 3 . 8 5 9 8 1 . 3 0 4 5 5 0 .0503448 (11) X (12) 15.2263 1.44603 -0.125062 (12) Of course, you can easilyfind the average of the means of the 7th through 12 =7 E [X( ) ] 12th order statistics: µ = = 12.7403, so µ > E [Y ] by about a 6 1 point. But note note that that E [X (7) ] < E [X (8) ] < E [X (9) ] < E [Y ], meaning the 2 point. expected value of each of the 6 ability scores generated with 4d6k3 is greater than what you can expect from half the ability scores generated by the largest 6 of 12x(3d6). So the answer isn’t so simple. i
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