2011-05-25

Balanced Dice in Dragon Magazine

One of the most-viewed posts on this blog (currently #3 in all-time number of views) is the article on "Testing a Balanced Die", with a simple presentation of the well-known Pearson's Chi-Square Test. What I just recently found out is that this was anticipated by D.G. Weeks in a Dragon magazine article more than 25 years before: "Be thy die ill-wrought? Only those that pass the chi-square test can play" (Dragon #78, Oct-1983, p. 62-65). Somehow this escaped my attention all these years (I don't think that I had a physical copy of this issue back in the day).

Not much of a surprise that Weeks' procedure was equivalent to the one I presented (it's very much a standardized process from circa 1900, at the advent of modern statistics). One thing he did differently: Whereas I presented chi-square values at the 5% significance level, which is sort of customary (only 5% chance that an unbiased die accidentally exceeds the given values), he gave two numbers, at the 10% and 1% significance levels (saying that if it exceeds the first, then the die is maybe-possibly biased, and if it exceeds the second, then there is overwhelming evidence that it's biased). He also presented a listing of BASIC program code you could type in (at the time, it was fairly common in Dragon magazine) that would do the tabulations and calculations for you. In addition, it could categorize the data in case you were testing a hypothesis that a specific known face or group of faces was coming up more frequently.

See below for Weeks' presentation of testing a d8 for fairness (80 total rolls, per face E=10, SSE = 76; concludes that the die is unbiased; same conclusion as from my presentation: SSE < X*E = 14.067*10 = 140.67). Compare to my scratch paper included at the bottom here.

Weeks' table of critical values for X at the 10% and 1% significance levels: