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# Median Stats

The point-buy system included in 3.5 creates characters that are underpowered and bland compared to characters generated with the 4d6 method. This variant rule allows starting stats as long it would be at least as likely to get an absolutely better configuration as it would be to get absolutely worse configuration when rolling 4d6. That is, we accept starting stats below this generalized median.

## Derivation

Result of 4d6, drop lowest | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Symbol | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |

P(X) | Relative Frequency (Probability * 1296) | 1 | 4 | 10 | 21 | 38 | 62 | 91 | 122 | 148 | 167 | 172 | 160 | 131 | 94 | 54 | 21 |

F(X) | Probability <= | 1/1296 | 5/1296 | 5/432 | 1/36 | 37/648 | 17/162 | 227/1296 | 349/1296 | 497/1296 | 83/162 | 209/324 | 83/108 | 1127/1296 | 407/432 | 425/432 | 1 |

G(X) | Probability >= | 1 | 1295/1296 | 1291/1296 | 427/432 | 35/36 | 611/648 | 145/162 | 1069/1296 | 947/1296 | 799/1296 | 79/162 | 115/324 | 25/108 | 169/1296 | 25/432 | 7/432 |

F(X)/G(X) | 1/1296 | 1/259 | 15/1291 | 12/427 | 37/630 | 68/611 | 227/1160 | 349/1069 | 497/947 | 664/799 | 209/158 | 249/115 | 1127/300 | 1221/169 | 17 | 432/7 |

For multiple independent, ordered experiments, the ratio of results absolutely better than an outcome to results absolutely worse (R) is the product of the probabilities of outcomes better or equivalent less the product of the probability of the specific outcomes divided by the product of the probabilities of outcomes worse or equivalent less the produce of the probability of the specific outcomes.

R(x0, x1, x2...) = (product(G(x0), G(x1), G(x2) ...) - product(P(x0), P(x1), P(x2) ...) / (product(F(x0), F(x1), F(x2) ...) - product(P(x0), P(x1), P(x2) ...)

In the specific case of R = 1, one part over for every part under, the products of the specific probabilities can be removed from the equation, and R(x0, x1, x2...) >= 1 will hold as long as:

product(F(x0)/G(x0), F(x1)/G(x1), F(x2)/G(x2), ...) <= 1

Therefore, one could determine exactly if a set of stats were below this median by multiplying the number from the fourth line of the table for each stat and checking to see if the product is less than or equal to 1.

## Simplification

The logarithm function eats multiplications and turns them into additions, which are easier to do in one's head. The median constraint will be met if and only if:

ln(F(x0)/G(x0)) + ln(F(x1)/G(x1)) + ln(F(x2)/G(x2) + ... <= 0

Result of 4d6, drop lowest | |||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Symbol | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |||||||||||||||||

F(X)/G(X) | 1/1296 | 1/259 | 15/1291 | 12/427 | 37/630 | 68/611 | 227/1160 | 349/1069 | 497/947 | 664/799 | 209/158 | 249/115 | 1127/300 | 1221/169 | 17 | 432/7 | |||||||||||||||||

ll(X) | ln(F(X)/G(X)) | -7.1670 | -5.5568 | -4.4551 | -3.5718 | -2.8348 | -2.1955 | -1.6312 | -1.1194 | -0.6447 | -0.1850 | 0.2797 | 0.7725 | 1.3235 | 1.9775 | 2.8332 | 4.1225 | ||||||||||||||||

Approximation 1 | |||||||||||||||||||||||||||||||||

points | -26 | -20 | -16 | -13 | -10 | -8 | -6 | -4 | -2 | 0 | 1 | 3 | 5 | 7 | 10 | 15 | |||||||||||||||||

difference | 6 | 4 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 5 | ||||||||||||||||||

Approximation 2 | |||||||||||||||||||||||||||||||||

points | -33 | -25 | -20 | -16 | -13 | -10 | -7 | -5 | -3 | -1 | 1 | 4 | 6 | 9 | 13 | 19 | |||||||||||||||||

difference | 8 | 5 | 4 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 4 | 6 | ||||||||||||||||||

Approximation 3 | |||||||||||||||||||||||||||||||||

points | -43 | -33 | -27 | -21 | -17 | -13 | -10 | -7 | -4 | -1 | 2 | 5 | 8 | 12 | 17 | 25 | |||||||||||||||||

difference | 10 | 6 | 6 | 4 | 4 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 5 | 8 | ||||||||||||||||||

Approximation 5 | |||||||||||||||||||||||||||||||||

points | -76 | -59 | -47 | -38 | -30 | -23 | -17 | -12 | -7 | -2 | 3 | 8 | 14 | 21 | 30 | 44 | |||||||||||||||||

difference | 17 | 12 | 9 | 8 | 7 | 6 | 5 | 5 | 5 | 5 | 5 | 6 | 7 | 9 | 14 |