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<blockquote data-quote="Christian" data-source="post: 1926199" data-attributes="member: 381">1) Different versions of Excel have different statistical analysis packages. It's a useful tool even if it doesn't have any statistical functions in your version, because chi-squares are calcluation intensive operations, which is something spreadsheets are well-equipped to deal with ... To calculate a chi-square of a distribution, you just add up the squares of the differences between the expected number of rolls of each number and the actual number of rolls of that number, divided by the total number of rolls. Eg. for the two-hundred roll stage of my test (when the expected value is ten rolls of each number on the d20), I had numbers that looked like this:Value&nbsp;&nbsp; Rolls&nbsp; Square of Difference1&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 11&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 12&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 10&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 03&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 15&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 25...18&nbsp; &nbsp; &nbsp;&nbsp; 16&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 3619&nbsp; &nbsp; &nbsp;&nbsp; 10&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; 020&nbsp; &nbsp; &nbsp;&nbsp; 9&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1___________________TOTAL&nbsp; 200&nbsp; &nbsp; &nbsp;&nbsp; 4200Chi-square value: 4200/200 = 21.0Then, you can look that up in a chi-square table in your handy-dandy statistics textbook &amp; see that for nineteen degrees of freedom, this is in the 30-35% range of randomness. (You don't have a statistics textbook handy? Cretin!) =&gt; There is a significant non-random component to the results from this die.Actually, my version of Excel includes some chi-square statistical functions, that make life a bit easier. CHITEST(results table, expectations table) gives you the randomness percentage from a table of actual results and a matching table of expected results ... With dice, you're hoping that the CHITEST() output is very close to 1.2) How many rolls you need depends on how small a bias you want to be able to detect, and how certain you want to be of your conclusion, as with any statistical test. Rule of thumb with chi-square is, at an absolute minimum, an expected value of 5 or more for every possible value in your raw data. This means 100 rolls for a d20. In practical terms, this means you start with 100 rolls, and continue in 100 roll batches until you feel happy. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile&nbsp; &nbsp; :)"&nbsp; data-smilie="1"data-shortname=":)" /> With my first die, my first 100 rolls were in the 95%+ random region. That left me feeling confident that the die was probably fair, or that any bias was at least pretty small. My first 100 rolls with the black die have me a randomness percentage of less than 50%, which inspired me to continue doing tests; each subsequent 100 roll set, while sometimes showing a higher randomness percentage individually, resulted in an increase in the overall total chi-square; this made it rapidly clear that the initially perceived non-randomness was real and not a statistical artifact. 500 - 1000 rolls should be plenty for anybody, unless you're hoping to open a casino using those dice ... A bias small enough to pass through that level of testing probably is not significantly impacting your gaming.</blockquote>

[QUOTE="Christian, post: 1926199, member: 381"] 1) Different versions of Excel have different statistical analysis packages. It's a useful tool even if it doesn't have any statistical functions in your version, because chi-squares are calcluation intensive operations, which is something spreadsheets are well-equipped to deal with ... To calculate a chi-square of a distribution, you just add up the squares of the differences between the expected number of rolls of each number and the actual number of rolls of that number, divided by the total number of rolls. Eg. for the two-hundred roll stage of my test (when the expected value is ten rolls of each number on the d20), I had numbers that looked like this: Value Rolls Square of Difference 1 11 1 2 10 0 3 15 25 ... 18 16 36 19 10 0 20 9 1 ___________________ TOTAL 200 4200 Chi-square value: 4200/200 = 21.0 Then, you can look that up in a chi-square table in your handy-dandy statistics textbook & see that for nineteen degrees of freedom, this is in the 30-35% range of randomness. (You don't have a statistics textbook handy? Cretin!) => There is a significant non-random component to the results from this die. Actually, my version of Excel includes some chi-square statistical functions, that make life a bit easier. CHITEST(results table, expectations table) gives you the randomness percentage from a table of actual results and a matching table of expected results ... With dice, you're hoping that the CHITEST() output is very close to 1. 2) How many rolls you need depends on how small a bias you want to be able to detect, and how certain you want to be of your conclusion, as with any statistical test. Rule of thumb with chi-square is, at an absolute minimum, an expected value of 5 or more for every possible value in your raw data. This means 100 rolls for a d20. In practical terms, this means you start with 100 rolls, and continue in 100 roll batches until you feel happy. :) With my first die, my first 100 rolls were in the 95%+ random region. That left me feeling confident that the die was probably fair, or that any bias was at least pretty small. My first 100 rolls with the black die have me a randomness percentage of less than 50%, which inspired me to continue doing tests; each subsequent 100 roll set, while sometimes showing a higher randomness percentage individually, resulted in an increase in the overall total chi-square; this made it rapidly clear that the initially perceived non-randomness was real and not a statistical artifact. 500 - 1000 rolls should be plenty for anybody, unless you're hoping to open a casino using those dice ... A bias small enough to pass through that level of testing probably is not significantly impacting your gaming. [/QUOTE]

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