I was trying to think of a way to show how "normal" a set of dice rolls are in a game like catan. In Catan, each turn you roll 2d6 and the sum of those 2d6 dictate game outcomes. A game can last anywhere from 20 - 50+ turns.
For examples sake, let's say we roll 36 times in a game. For a perfectly "normal" game the dice sum 2 would roll exactly once, as would 12. The dice sum 7 would roll 6 times. This forms a triangular distribution. In the distribution of all possible sets of dice results, this would be exactly in the middle of the distribution.
One way to do this might be to look at the total sum of all the dice sums (2d6) throughout the game. At the lower extrema someone could roll 2 36 times (so 72) and at the highest extrema someone could roll 12 36 times (432). The average then would be 7 * 36 = 252. So you'd have a distribution of sum results with around 252 with a min of 72 and max of 252. Is the distribution also triangular? I think so.
However, this isn't a very good score of how "normal" a game might be, because there is a big difference between rolling specific numbers. For example if no 7s were rolled the entire game this would be very abnormal and greatly affect gameplay.
So looking at the sum isn't a very good score.
I then thought about trying to compute the Z score for each result and somehow combine them. We know the expected frequencies of each 2d6 result in the totals number of rolls. We can then take the actual frequency and divide by the standard deviation for each roll to get this.
(E(2d6r) - O(2d6r)) / Std(2d6r) where
2d6r is a specific result on 2d6 (like 12) and
E is expected,
O is observed and
Std is standard deviation.
At this point we could sum up the absolute values of these z scores and get a total score. If the distribution matched exactly this score would be 0. In the most extreme case of rolling only 2s or 12s this number would be the largest it could be. How is this result distributed? Obviously it's going to be right skewed but it is normally distributed where I can say x% of results are less than some positive X value?
Is this z score calculation the most statistically appropriate for determining the "normal"-ness of a specific set of results given that we care about the frequencies of each results and not just the sum?