Dice Testing with the Running Chi-Square Distribution

Dice are not fair. Producing a geometrically precise, uniform die is not possible. Casino dice come as close as possible to perfectly random dice because they are machined to an accuracy of a few ten-thousandths of an inch and the putty used for the pips (dots) is the same density as the plastic die body. Polyhedral dice used in games like Dungeons and Dragons are far more difficult to manufacture to high tolerances. Some dice are fairer than others. Since 20-sided dice (D20s) are very difficult to manufacture precisely, they were the focus of this study. The running chi-square distribution (Campbell and Dolan 2019, 2020) was the tool of choice for these tests. The chi-square statistic has several interesting mathematical characteristics that are described. An unfair die will have a constant term and a linear trend added to the running chi-square statistic. The steeper the slope of the trend, the more unfair the die. 30 dice were rolled at least 3000 times each with an automated dice-rolling device, and some of the same dice were rolled 3000 times by hand in a dice tower. Results indicate that the method of rolling heavily influences the outcome