Mathematical tests for fair dice

My previous post on fair dice focused on some intuitive ideas about die fairness, and the beginnings of a mathematical approach. Now I’d like to describe the tests my die roller does. These are what I’ve settled on so far as a way to get some numbers to describe and compare the fairness of different dice.

The chi-squared test

First off, there’s the chi-squared goodness-of-fit test. This is a test that looks at the deviations in the histogram to get a total number (the chi-squared statistic) that characterizes how far the histogram deviates from ideal. You can also compare the result statistic to a mathematically-determined threshold that will give you a confidence value for the test; a 95% confidence is often chosen.

So hey, that’s great! We can reduce the whole set of results and its histogram to a single number for a given die, and then find out whether that die is fair or not with 95% confidence! Super! Right?

What it means to disprove a null hypothesis

The chi-squared test is useful in answering a very specific question, and this sometimes gets overlooked outside of statistics class. Officially, you use a chi-squared test (and most statistical tests) to “disprove the ;dip;hypothes.o Q Inis oucaseff, tto ̶e ;dip;hypotheso Q thio ̶, tta given 1-stbumatiocouholribispreduled baon fair e,o Q (ans to disprovt it means that that yot havreswnat thaeihether eat die iunn faay, ot yot havs bell&nbsunluckyen.sectrg>n.

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