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:


  1. how weird, I was just reading that article yesterday

  2. I happened across this article while looking at other stuff and thought I'd stop by to comment. When I was twelve and my dad was writing that article, I was the guinea pig, rolling a million dice and either tabulating them on scratch pads or entering them on the computer (Atari 400 at that time) for him. This is the second time in all these years that I've run across a reference to that article and it gives me a smile.

    My personal take-away from this learning is that a die has to pretty out-of-whack to fail the chi-square (there were a lot of shitty, mis-molded dice back then and most of them passed). :-)

    1. Wow, Christopher, that's fascinating, thanks so much for sharing! Unfortunately it's true that the chi-square test is one of fairly lower power unless you're rolling hundreds of times for each die (link).

    2. ... and you're probably in the top 1% of being intimate with that fact. :-)