Dice Statistics: The Normal Distribution


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Recently I’ve had the benefit of attending a statistics course at MUN … or at least I had to take it for my program ;) . In the end it hasn’t been so bad, and we did discuss some dice mechanics. Given this fortunate overlap of my hobby and academic life, I figured I’d use it as an excuse to write an article. This article discusses the significance of using normally distributed dice mechanics for task resolution in roleplaying systems.

The normal distribution is a pattern that shows up in nature quite frequently. You might have heard it called the “Bell Curve”. If you really have no idea what I’m talking about wikipedia is here to help: wikipedia (normal distribution). Basically, the idea is that if you sample something with a normal distribution you will tend to get a result close to average. In the GURPS and the HERO systems, two pen-and-paper rpg systems, you roll 3d6 for success rolls. These rolls would be normally distributed, and therefore the result would tend to be close to average. For 3d6 is 10.5 is the average roll. Since we can’t have half a number 10/11 are the most common rolls.
If we were to roll just one dice, we would get a continuous uniform distribution. Once again here’s a link towikipedia (continuous uniform distribution). This is the case for the d20 system. There is an equal chance of rolling any number between 1-20 (5%).
So what difference does it make? Well in terms of probability quite a bit. For one it means that when you roll for success in the d20 system you are much more susceptible to “the whims of fate”. I’m sure we’ve all rolled those low rolls, the ones where your Fighter get completely pwned by an orc. And we’ve all had times were we get a high roll, bringing success where we should probably have failed miserably. It can be fun. It can be downright hilarious. It can add tension to the game. There are a lot of merits to this. But is it realistic?
In GURPS the aim is to be realistic, and as I mentioned earlier nature loves the normal distribution. So it only makes sense to use a normal distribution. This means a character will tend to do about as well on a task as their skill and the situation suggests. On occasion there will be extremely high or low rolls, but these will be far and few between. In GURPS/HERO you must roll equal to or below a target number, which generally equal to a characters attribute/skill. Due to the shape of a normal distribution, as you increase your level in attribute/skill you will get less and less return. Increasing from 10->11 will raise the probability of success by about 12%, raising it from 17-18 will increase the probability of success by only about 0.5%. However, raising your attribute/skill to a high level will still protect against higher negative modifiers. Basically, you are so skilled that you can work around a situation that isn’t exactly ideal. Additionally, if degree of success, the amount by which you exceeded the target number, is important you will tend to roll a certain level higher than the dice roll consistently. So a high level will consistently bring a high degree of success. A similar, but opposite, situation occurs for very low attribute scores. Here positive modifiers will have less effect on your overall success. Think about it this way: If you know nothing about chemistry, being in a fully equipped chemistry lab would do little to help you synthesize a compound. For degree of failure, the amount by which you did not meet the target number, you will tend to get a high degree of failure most of the time.
So there you have it. Do you want thrill of chance or consistency? If you do play GURPS/HERO and want more chance you could roll a d16+2 (or a d20 since they are easier to come by). This has a range of 3-18, just like 3d6, but a continuous uniform distribution. Maybe you see some situations as inherently more affected by chance than others. You could use d16+2 specifically for these situations. You could also use 3d6 for the d20 system, as explained here. If you wanted to be really particular you could use 2d6, a d10 then subtract 2. The rolls will be between 1-20, just like a d20. Basically, you shouldn’t feel constrained by the “rules” if you want to do something different.

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