Dice equivalence

UPDATE: I have made a website of the simulator (dnd.matteoferla.com).
SEEN ALSO: Description of battle simulator and some analysis, the GitHub code for the encounter simulator, analysis of dice equivalence

As explored previously, dice with the same mean can safely be replaced with each, a principle nicknamed dice equivalence in a previous post.

Utility: Changing a boring 1d8 for a crazier dice combination spices things up.
Sidedness: The number of sides of a die. Not sure if there is a word to describe the number of sides of a polyhedron. Hence this made up word.
Mean: This is half its sidedness plus 0.5. The 0.5 is because a die starts from 1, otherwise one would make a fencepost error. As a result a 2d4 is actually equivalent of a 1d9, not a 1d8. A bonus of +1 is equivalent to increasing the sidedness by 2.
Variance:

• The variance greatly increases with sidedness.
• The variance of multiple dice is the sum of each variance (Bienaymé formula).
• Bonuses do not affect variance and a fixed damage has therefore zero variance —damage dealt as when visiting the plane of Mechanus.
• Variance decreases the more dice are used and the distribution becomes closer to a bell curve —flipping coins would give a bell curve.

Here is a table of the equivalences:

Mean	Singles	Variance of single	Doubles	Plus-one	Minus-one	Coin-flip	Mixed
1.5	1d2	0.25	—	—	1d4-1	1d2	—
2	1d3	0.666666667	—	—	1d5-1	2d2-1	—
2.5	1d4	1.25	—	1d2+1	1d6-1	1d2+1	—
3	1d5	2	2d2	1d3+1	1d7-1	2d2	—
3.5	1d6	2.916666667	—	1d4+1	1d8-1	1d2+2	1d2+1d3
4	1d7	4	2d3	1d5+1	1d9-1	2d2+1	1d2+1d4
4.5	1d8	5.25	—	1d6+1	1d10-1	3d2	1d3+1d4
5	1d9	6.666666667	2d4	1d7+1	1d11-1	2d2+2	1d3+1d5
5.5	1d10	8.25	—	1d8+1	1d12-1	3d2+1	1d4+1d5
6	1d11	10	2d5	1d9+1	1d13-1	4d2	1d4+1d6
6.5	1d12	11.91666667	—	1d10+1	1d14-1	3d2+2	1d5+1d6
7	1d13	14	2d6	1d11+1	1d15-1	4d2+1	1d5+1d7
7.5	1d14	16.25	—	1d12+1	1d16-1	5d2	1d6+1d7
8	1d15	18.66666667	2d7	1d13+1	1d17-1	4d2+2	1d6+1d8
8.5	1d16	21.25	—	1d14+1	1d18-1	5d2+1	1d7+1d8
9	1d17	24	2d8	1d15+1	1d19-1	6d2	1d7+1d9
9.5	1d18	26.91666667	—	1d16+1	1d20-1	5d2+2	1d8+1d9
10	1d19	30	2d9	1d17+1	1d21-1	6d2+1	1d8+1d10
10.5	1d20	33.25	—	1d18+1	1d22-1	7d2	1d9+1d10

Obviously, there are more options. However, one issue is if a subtraction is present the maximum value of a subtrahend must be equal or less than the lowest value of the minuend. That is a 1d100-48 has the same mean as a 1d4, but the roll can give negative numbers which count as zero, so it is equivalent of rolling a 1d52.

All-or-nothing morningstar

When a die is subtracted from a bonus it is obviously the same as the die plus the difference of the bonus minus the sidedness. The same applies for multiple dice. Therefore if one wanted a weapon that gave tail values —i.e. nearly max or min damage— one cannot use dice, but has to use a Skipbo deck with the middle cards thinned or use a larger sided die with a conversion table. Despite that, this weapon, the all-or-nothing morningstar, is really interesting as it is quite unpredictable...

Appendix: Simulated proof

The cases where variance makes a difference would be when a single good hit makes the difference between life and death. A rare extreme case. Therefore a commoner was chosen with a 1d4 or a 1d6 weapon for various fights (5,000 times).
Duel: commoner vs. commoner (1d4 club).

Hard duel: commoner vs. rat

Team: two commoners vs. rat

Hard team: two commoners vs. two rats

In these extreme cases, there is a there is a slight advantage to higher variance (d6-1 or d8-1), but we are talking about a percentage point or two.