Suppose I roll a 20-sided die 1000 times and count the number of times a particular value comes up. This gives an array of 20 counts, and the expected value of each is 1000/20 = 50.
I'd like to find an estimate for the maximum variation around this number. If I subtract 50 from each count and take the absolute value, what is the maximum I'm likely to find? This would be N where 50% of trials will have all counts within N of the expected value, and 50% will have one or more count beyond.
(A specific example for clarity, I'm really looking for a general answer about uniform distributions.)