[...] Applying (Pearson's chi-squared test) for that dice should be plenty enough to tell you if the dice is likely to be biased. :-)
One of the more interesting "goodness-of-fit" applications of the chi-square test is to examine issues of "fairness" and "cheating" in games of chance, such as cards, dice, and roulette. ;-)
So we need to look to see if the 'observed values' are close to the 'expected values', which I crudely mentioned in the prior post.
[...]
Robust Statistical Analysis using (Chi-squared test)
Total amount of Observations (dice rolls): 1274
Expected Frequency per any d100 single face occurrence (1274 × 1/100): 12.74
Null Hypothesis (H0): Dice is fair (unbiased)
Alternative Hypothesis (H1): Dice is unfair (biased)
Level of Significance, i.e. area in the tail: (alpha) 0.01
Degrees of Freedom (DF); Sample size (total rolls, i.e. distinct possible categories, we used 100 rows to record), minus one (N – 1): 100 – 1 = (DF): 99.
For finding the Critical value (tail of graph and rejection region area). We'll just use 99 for DF, as I don't have specialised statistics software and I'm not calculating it longhand (DF 99 is solid enough) for this dice.
Looking on an official Chi-square table using (DF 99) and (alpha 0.01) Critical value, i.e.
P-value (probability): 134.642 rounded 134.64.
Therefore if our dice exceeds the (above)
P-value [134.642] "Daily Dice" is likely loaded.
Chi square: X2 = sum [(Observed - Expected)2/Expected]
For example using [
Post #10] data: Roll 1; Observed value 12, minus Expected value: 12.74 (squared) divisible by Expected 12.74, equals: 0.042983... Then rinse and repeat, i.e. total all the X
2 for 1274 rolls.
Our Test Statistic equals:
108.4176
We can easily see:
108.4176 is far less than
P-value: [134.642].
Therefore we fail to reject: Null Hypothesis (H0): (we accept) Dice is fair (unbiased)
The roll results and calculations can be found within: [
Post #42].
Conclusion
The "Daily dice" DO NOT appear to be loaded. In fact the statistical model shows there is an excellent chance of ~
99% percent the dice aren't loaded. (That's what the alpha 0.01 value we tested against means basically).