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## Research Topic: Normally Distributed Percentile Dice

By Ian Mander BSc, written 29 May 2016, posted 14 November 2018.

Question: What is the best way of getting a normally distributed percentile die roll? (In other words, using dice to make a number from 0-100, with a mean, median, and mode of 50, with the middle numbers having a greater probability of occurring, in a normally distributed bell curve.) If we include rolls that give a 1-100 result, or even narrower ranges, non-continuous ranges, are there any interesting (or convenient) rolls?

Answer: In a previous research topic Six Sided Dice Roll Distribution we see that a bell curve that approximates a normal distribution needs at least three dice. The more dice used the narrower the curve and the less likely a low or high result is. This implies the best rolls will likely use three or four dice.

I've used the AnyDice online dice tool to graph some of the best approximations. It allows others to be tested.

### Real dice, not full range

The first option, 3d30+4, doesn't have much going for it. There are options with a wider range. The dice can be found on eBay.

A 4d24 roll gives results from 4 to 96 with a single mode. It's very simple since no modification is needed and gives almost the full range of percentile results. 24 sided dice are available on eBay with only a small amount of searching, in two different forms.

The next option, 5d20-2, has a slightly wider range and uses more common 20-sided dice. Downsides are it's bimodal and it has a narrower peak than 4d24.

A very interesting option is 4d6*5-20 which gives a non-continuous distribution with possible totals every fifth number; it's a scaling of the 21 results of a 4d6 roll. The dice are very common and the maths is simple enough to do in one's head.

Roll 3d30+4 4d24 5d20-2 4d6*5-20
Range 7-94 4-96 3-98 non-continuous 0-100;
every 5th number
Mode 50, 51 50 50, 51 50
Spread/FWHM 40 (31~70) 35 (33~67) 32 (35-66) 41 (30~70)
Approx. SD
ranges
±1sd 35-66
±2sd 21-80
±3sd 12-88
±1sd 36-64
±2sd 23-77
±3sd 13-87
±1sd 38-63
±2sd 25-76
±3sd 15-86
±1sd
±2sd
±3sd
Availability of dice ***
Can be found on eBay.
***
Can be found on eBay with triangular faces (basically a gently stellated cube).
**
Harder to find with kite shaped faces.
****
Pretty commonly available.
*****
Many homes already have four d6.
Simplicity ****
Just a roll with an easy modification.
*****
Almost as easy as it gets – just roll 4 dice and add up.
****
Quite a few dice and a modification.
***
Somewhat complicated, but multiplying by 5 is pretty easy (multiply by 10 then halve it).
Representation **
Doesn't give full range.
***
Doesn't give full range.
***
Doesn't give full range. Very unlikely to give results ≤10 or ≥90.
**
Full range but only has every 5th number.
Name of solid Rhombic triacontahedron. Tetrakis hexahedron or deltoidal icositetrahedron. Icosahedron. Cube (hexahedron).

### Real dice, full range

If we are going to limit ourselves to using real dice, the simplest way of getting a 1-100 bimodal result is 3d34-2. It gives a range of 1-100 with a good chance of low and high results. Unfortunately, while they do exist, 34-sided dice are extremely hard to find. They are an unusual shape for dice, being bipyramidal.

The next simplest is 3d100/3. This is the average of three d100 rolls, each using a pair of reasonably commonly available 10-sided percentile dice (which themselves give a flat distribution). It gives a very similar 1-100 bimodal result to 3d34-2 but is more difficult to work out, and requires rounding.

Because 11d10-10 and 20d6-20 use so many dice they both have a very narrow peak.

Roll 3d34-2 3d100/3 11d10-10 20d6-20
Range 1-100 1-100 1-100 0-100
Mode 50, 51 50, 51 50, 51 50
Spread/FWHM 44 (29~72) 44 (29~72) 24 (39~62) 19 (41~59)
Approx. SD
ranges
±1sd: 33-67
±2sd: 17-83
±3sd: 7-93
±1sd: 33-67
±2sd: 17-83
±3sd: 7-93
±1sd: 41-60
±2sd: 32-69
±3sd: 23-78
±1sd: 43-57
±2sd: 35-65
±3sd: 28-72
Availability of dice *
Extremely hard to find.
****
Easy to find.
***
Reasonably easy to find, but you'll need quite a few.
****
Commonly available but maybe not in this quantity for a reasonable price.
Simplicity ****
Just a roll with an easy modification.
**
Somewhat tricky division and rounding.
***
***
That's a lot of dice to roll at once.
Representation *****
*****
**
Very unlikely to give low and high results; almost no chance of results <20 or >80.
**
Very close to a normal distribution (due to Central Limit Theorem) but very unlikely to give low and high results; almost no chance of results <25 or >75.
Name of solid 17th-order bipyramid. A pair of pentagonal trapezohedra. Pentagonal trapezohedron. Cube (hexahedron).

Note – d10 assumed to give a result 1-10, not 0-9.

### Virtual dice

A 0-100 single mode distribution is possible in a number of ways from "virtual" dice – dice which are easy to calculate on a computer with a random number generator, but which don't actually exist.

The simplest way is to take the number of dice in the roll as a factor of 100, the number of sides for each die one more than the factor pair of the first factor, and for a modifier subtract the number of dice. If x * y = 100, the roll is xd(y+1)–x. This effectively gives x dice each providing 0-y.

For example, 4 * 25 = 100. Take 4 as the number of dice, 25 + 1 = 26 sides per die, and subtract 4, the number of dice: 4d26-4.

5d21-5 has a slightly narrower spread.

10d11-10 is getting quite narrow. Interestingly, 10d11-10 gives a 0-100 result but 11d10-10 (listed above) gives a 1-100 result.

20d6-20 is an example of this sort of calculation using real dice; it's listed above.

25d5-25 is very narrow.

50d3-50 is extremely narrow.

100d2-100 is crazy narrow, very little chance of a result outside of 35-65. This roll is the equivalent of summing the tosses of 100 coins, each with the two sides labelled 0 and 1.

Roll 4d26-4 5d21-5 10d11-10 25d5-25 50d3-50 100d2-100
Range 0-100 0-100 0-100 0-100 0-100 0-100
Mode 50 50 50 50 50 50
Approx. SD
ranges
±1sd:
±2sd:
±3sd:
±1sd:
±2sd:
±3sd:
±1sd:
±2sd:
±3sd:
±1sd:
±2sd:
±3sd:
±1sd:
±2sd:
±3sd:
±1sd:
±2sd:
±3sd:
Representation *****
****
***
***
**
*

Sorry, some figures missing from that table still.

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