In some role-playing games, the impacts of different types of weapons or spells are specified by designations such as 3d6 (the total of the roll of three six-sided dice, each one numbered from 1 to 6), 2d12, 4d8, and so on and so forth.
One would have a certain probability to hit or miss, modified by one's dexterity score, and if one had strength above that needed merely to wield one's weapon, one might get a damage bonus when one hits.
This involves a number of limitations.
Combinations need to be chosen from the available dice, and one would need multiple sets of dice to use several copies of the same die. Weapons even doing d8+d10+d12 damage are not common in RPGs. What the combinations are really attempting to approximate, at least in many cases, is a distribution with a given mean and standard deviation.
Adding something to damage as an adjustment is common enough, but multiplying damage by a factor is too awkward to be considered.
A much larger table could easily be made, for example based on the fact that the 53rd root of 2 allows good approximations to factors of 3 and 5 in integral powers thereof, (one might, in practice, choose to use the 176th root of 10) and one could use d100 or even d1000 to supply the input randomization, but something more practical, say based on the available values of resistors and capacitors with 5% tolerance, and perhaps using a d20 as input, ought to be sufficient.
A d20, however, has an even number of faces; what would really be liked is something with an odd number of faces, so that it would be possible to roll the exact average amount of damage. A further simplification, with the ability to extend further out to the edges of the distribution, without cluttering up the table with many close-together values, would be to use, say, 2d6 as input.
Thus, a table like this could be made the basis of a very powerful and flexible RPG system based on the ordinary 2d6, used for advancing one's token in humble and ordinary board games:
| Roll (7 is always 0)
Standard | 6 5 4 3 2 (- negative)
Deviation | 8 9 10 11 12 (+ positive)
----------+----------------------------------
100 | 39 76 115 159 220
112.2018 | 44 86 129 179 247
125,8925 | 49 96 145 201 277
141,2538 | 55 108 162 225 311
158.4893 | 62 121 182 253 349
177.8279 | 70 136 205 283 391
199.5262 | 78 153 230 318 439
223.8721 | 88 171 258 357 493
251.1886 | 99 192 289 400 553
281.8383 | 111 216 324 449 620
316.2278 | 124 242 364 504 696
354.8134 | 139 271 408 565 781
398.1072 | 156 304 458 634 876
446.6836 | 175 342 514 712 983
501.1872 | 197 383 577 799 1103
562.3413 | 221 430 647 896 1237
630.9573 | 248 483 726 1005 1388
707.9458 | 278 541 814 1128 1558
794.3282 | 312 607 914 1266 1748
891.2510 | 350 682 1025 1420 1961
1000 | 393 765 1150 1593 2201
So, if one has a sword which causes damage with a mean of 40 (actually 39.8107 rounded) and standard deviation of 10, but the damage is reduced by a multiplier of 0.8 (actually 0.7943282 rounded) we just need to move two steps down in the table for both items: 40 becomes 32 (31.62278 rounded), 10 becomes 8 (7.943282 rounded) and the table of damage for that sword with that multiplier contains entries derived as shown below:
Roll Damage Calculation Damage 2 31.62278 - 17.48 14 3 31.62278 - 12.66 19 4 31.62278 - 9.14 22 5 31.62278 - 6.07 26 6 31.62278 - 3.12 29 7 31.62278 32 8 31.62278 + 3.12 35 9 31.62278 + 6.07 38 10 31.62278 + 9.14 41 11 31.62278 + 12.66 44 12 31.62278 + 17.48 49
Although it looks complicated, the intent is that adjustments which involve multiplication as well as addition can be obtained through the simple operations of moving up or down in the table, adding or subtracting, and moving the decimal point left or right.
In practice the quantities would be rounded before being added, so the results would actually be symmetrical around the rounded center point; and, since very large numbers would not normally be needed, in practice the table used would be more likely to look like this:
| Roll (7 is always 0)
Standard | 6 5 4 3 2 (- negative)
Deviation | 8 9 10 11 12 (+ positive)
----------+----------------------------------
10 | 4 8 12 16 22
11 | 4 9 13 18 25
13 | 5 10 14 20 28
14 | 6 11 16 23 31
16 | 6 12 18 25 35
18 | 7 14 20 28 39
20 | 8 15 23 32 44
22 | 9 17 26 36 49
25 | 10 19 29 40 55
28 | 11 22 32 45 62
32 | 12 24 36 50 70
35 | 14 27 41 57 78
40 | 16 30 46 63 88
45 | 18 34 51 71 98
50 | 20 38 58 80 110
56 | 22 43 65 90 124
63 | 25 48 73 101 139
71 | 28 54 81 113 156
79 | 31 61 91 127 175
89 | 35 68 103 142 196
100 | 39 76 115 159 220
And the table could be replaced by other tables, converting to the same set of steps in standard deviation from rolls of another chosen set of dice instead of the totals of 2d6, but this one table could be used to calculate all the various types of adjustments used in an RPG.
If one wishes to use a d20 instead of 2d6, the table, which, being now based on a dice roll with a linear scale, may illustrate the principle at work more clearly, becomes:
| Roll
Standard | 10 9 8 7 6 5 4 3 2 1 (- negative)
Deviation | 11 12 13 14 15 16 17 18 19 20 (+ positive)
----------+----------------------------------------------------------------
100 | 6 19 32 45 60 76 93 115 144 196
112.2018 | 7 21 36 51 67 85 105 129 162 220
125.8925 | 8 24 40 57 75 95 118 145 181 247
141.2538 | 9 27 45 64 84 107 132 162 203 277
158.4893 | 10 30 51 72 95 120 148 182 228 311
177.8279 | 11 34 57 81 106 134 166 205 256 349
199.5262 | 13 38 64 91 119 151 186 230 287 391
223.8721 | 14 42 71 102 134 169 209 258 322 439
251.1886 | 16 48 80 114 150 190 235 289 362 492
281.8383 | 18 53 90 128 168 213 263 324 406 552
316.2278 | 20 60 101 143 189 239 296 364 455 620
354.8134 | 22 67 113 161 212 268 332 408 511 695
398.1072 | 25 75 127 181 238 301 372 458 573 780
446.6836 | 28 84 142 203 267 337 417 514 643 876
501.1872 | 31 95 160 227 300 379 468 577 721 982
562.3413 | 35 106 179 255 336 425 526 647 810 1102
630.9573 | 40 119 201 286 377 477 590 726 908 1237
707.9458 | 44 134 226 321 423 535 662 814 1019 1388
794.3282 | 50 150 253 360 475 600 742 914 1143 1557
891.2510 | 56 169 284 404 533 673 833 1025 1283 1747
1000 | 63 189 319 454 598 755 935 1150 1440 1960
And, if you really want to get fancy, you can double the number of multiplicative steps, and roll 4d6, which leads to the following table:
| Roll (14 is always 0)
Standard | 13 12 11 10 9 8 7 6 5 4 (- negative)
Deviation | 15 16 17 18 19 20 21 22 23 24 (+ positive)
----------+----------------------------------------------------------------
100 | 28 56 84 114 144 174 207 242 283 337
105.9254 | 30 59 90 120 152 185 219 257 300 357
112.2018 | 31 63 95 127 161 196 232 272 318 378
118.8502 | 33 67 100 135 171 207 246 288 337 400
125.8925 | 35 71 106 143 181 220 260 305 357 424
133.3521 | 37 75 113 151 191 233 276 323 378 449
141.2537 | 40 79 119 160 203 246 292 342 400 476
149.6236 | 42 84 126 170 215 261 310 362 424 504
158.4893 | 44 89 134 180 227 277 328 384 449 534
167.8804 | 47 94 142 191 241 293 347 407 476 565
177.8279 | 50 100 150 202 255 310 368 431 504 599
188.3649 | 53 106 159 214 270 329 390 456 534 634
199.5262 | 56 112 169 227 286 348 413 483 565 672
211.3489 | 59 119 179 240 303 369 437 512 599 711
223.8721 | 63 126 189 254 321 391 463 542 634 754
237.1373 | 66 133 200 269 340 414 491 574 672 798
251.1886 | 70 141 212 285 361 438 520 608 711 846
266.0724 | 75 149 225 302 382 464 550 645 754 896
281.8382 | 79 158 238 320 404 492 583 683 798 949
298.5381 | 84 167 252 339 428 521 618 723 846 1005
316.2276 | 89 177 267 359 454 552 654 766 896 1065
334.9652 | 94 188 283 380 481 584 693 811 949 1128
354.8132 | 99 199 300 403 509 619 734 859 1005 1194
375.8372 | 105 211 318 427 539 656 778 910 1065 1265
398.1069 | 112 223 336 452 571 695 824 964 1128 1340
421.6962 | 118 237 356 479 605 736 872 1021 1194 1420
446.6833 | 125 251 377 507 641 779 924 1082 1265 1504
473.1509 | 133 265 400 537 679 826 979 1146 1340 1593
501.1868 | 140 281 423 569 719 874 1037 1214 1420 1687
530.8840 | 149 298 449 603 762 926 1098 1286 1504 1787
562.3408 | 158 315 475 639 807 981 1163 1362 1593 1893
595.6616 | 167 334 503 676 855 1039 1232 1443 1687 2005
630.9567 | 177 354 533 716 906 1101 1305 1528 1787 2124
668.3433 | 187 375 565 759 959 1166 1383 1619 1893 2250
707.9451 | 198 397 598 804 1016 1235 1465 1715 2005 2383
749.8934 | 210 421 634 852 1076 1308 1551 1816 2124 2524
794.3274 | 223 446 671 902 1140 1386 1643 1924 2250 2674
841.3942 | 236 472 711 955 1208 1468 1741 2038 2383 2832
891.2499 | 250 500 753 1012 1279 1555 1844 2159 2524 3000
944.0598 | 265 530 798 1072 1355 1647 1953 2287 2674 3178
1000 | 280 561 845 1136 1435 1745 2069 2422 2832 3366
Or, scaled down, and more practical, as above, but still using 4d6, the table could be:
| Roll (14 is always 0)
Standard | 13 12 11 10 9 8 7 6 5 4 (- negative)
Deviation | 15 16 17 18 19 20 21 22 23 24 (+ positive)
----------+----------------------------------------------------------------
1 | 0 1 1 1 1 2 2 2 3 3
1.1 1 | 0 1 1 1 2 2 2 3 3 4
1.3 1 | 0 1 1 1 2 2 3 3 4 4
1.4 1 | 0 1 1 2 2 2 3 3 4 5
1.6 2 | 0 1 1 2 2 3 3 4 4 5
1.8 2 | 0 1 2 2 3 3 4 4 5 6
2 2 | 1 1 2 2 3 3 4 5 6 7
2.2 2 | 1 1 2 3 3 4 5 5 6 8
2.5 3 | 1 1 2 3 4 4 5 6 7 8
2.8 3 | 1 2 2 3 4 5 6 7 8 9
3.2 3 | 1 2 3 4 5 6 7 8 9 11
3.5 4 | 1 2 3 4 5 6 7 9 10 12
4 4 | 1 2 3 5 6 7 8 10 11 13
4.5 4 | 1 3 4 5 6 8 9 11 13 15
5 5 | 1 3 4 6 7 9 10 12 14 17
5.6 6 | 2 3 5 6 8 10 12 14 16 19
6.3 6 | 2 4 5 7 9 11 13 15 18 21
7.1 7 | 2 4 6 8 10 12 15 17 20 24
7.9 8 | 2 4 7 9 11 14 16 19 22 27
8.9 9 | 2 5 8 10 13 16 18 22 25 30
10 | 3 6 8 11 14 17 21 24 28 34
11 | 3 6 9 13 16 20 23 27 32 38
13 | 4 7 11 14 18 22 26 30 36 42
14 | 4 8 12 16 20 25 29 34 40 48
16 | 4 9 13 18 23 28 33 38 45 53
18 | 5 10 15 20 26 31 37 43 50 60
20 | 6 11 17 23 29 35 41 48 57 67
22 | 6 13 19 25 32 39 46 54 63 75
25 | 7 14 21 29 36 44 52 61 71 85
28 | 8 16 24 32 40 49 58 68 80 95
32 | 9 18 27 36 45 55 65 77 90 106
35 | 10 20 30 40 51 62 73 86 100 119
40 | 11 22 34 45 57 69 82 96 113 134
45 | 13 25 38 51 64 78 92 108 127 150
50 | 14 28 42 57 72 87 104 121 142 169
56 | 16 32 48 64 81 98 116 136 159 189
63 | 18 35 53 72 91 110 131 153 179 212
71 | 20 40 60 80 102 124 146 171 201 238
79 | 22 45 67 90 114 139 164 192 225 267
89 | 25 50 75 101 128 156 184 216 252 300
100 | 28 56 84 114 144 174 207 242 283 337
Even with this table, often one would be dividing by 10 and rounding to use it for calculations, so this time while the steps are as in the earlier tables, it is longer than the others because it has been extended downwards.
In the lower extension, the standard deviation rounded to tenths has been shown to allow rows to be indicated unambiguously; it is also then shown rounded to units so that that scale can be used to multiply the means.
From the table, it is easy to see that a roll of 4d6 approximates a distribution with a mean of 14 and a standard deviation close to 3.2.
So far, we've seen how to use dice to approximate a distribution with a certain mean and a certain standard deviation.
If one is working with an existing FRP game, however, such things as the damage done by a given weapon are not expressed in those terms, they're expressed in types of dice rolls.
So we need another table telling us which table to use.
One die gives a uniform distribution, and two dice give us a triangular distribution; this is true with any identical dice, not just the d6.
It's only with three or more dice that an approximation to the bell curve of the normal distribution is obtained.
So this table shows the mean and standard deviation for several common dice combinations:
3d4 4d4 5d4 6d4 7d4 8d4
7.5 10 12.5 15 17.5 20
1.9365 2.2361 2.5 2.7386 2.958 3.1623
3d6 4d6 5d6 6d6 7d6 8d6
10.5 14 17.5 21 24.5 28
2.958 3.4156 3.8188 4.1833 4.5184 4.8305
3d8 4d8 5d8 6d8 7d8 8d8
13.5 18 22.5 27 31.5 36
3.9686 4.5826 5.1235 5.6125 6.0622 6.4807
3d10 4d10 5d10 6d10 7d10 8d10
16.5 22 27.5 33 38.5 44
4.9749 5.7446 6.4226 7.0356 7.5993 8.124
3d12 4d12 5d12 6d12 7d12 8d12
19.5 26 32.5 39 45.5 52
5.9791 6.9041 7.719 8.4558 9.1333 9.7639
3d20 4d20 5d20 6d20 7d20 8d20
31.5 42 52.5 63 73.5 84
9.9875 11.5326 12.8938 14.1244 15.2561 16.3095
In the case of the d10, since the faces are marked with the digits 0 to 9 instead of the numbers 1 to 10, the corrected means are shown, where 0 is read as 10; the uncorrected means would be the same as for the d8.
Dice with 34 sides have been made because the totals from three such dice vary from 3 to 102. Subtract 2, and the range is from 1 to 100; and so this makes a bell curve distribution that matches well with the linear numbers from percentile dice in much the same way that the totals from 3d6 work well with those from a d20.
But 9d12 give totals that vary from 9 to 108 - subtract 9, one gets 1 to 100; 11d10 give totals that vary from 11 to 110 - subtract 10, one gets 1 to 100. Why are exotic dice with 34 sides needed?
In the case of 3d34, the standard deviation is 16.9926; in the case of 9d12, it is 10.3562, and in the case of 11d10, it is 9.5263.
So the distributions are very different; with nine or eleven dice, as opposed to three, the range of possible totals includes extremely unlikely events very distant from the center of the bell curve, with the likely totals all closer to the mean. Three dice produce an approximate bell curve more appropriate for an FRP game.