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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.)

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Suppose I roll a D20 /// ... What is a D20? A 20-sided die? A meeting of dollar-denominated countries? A donut? Or something you smoke? –  wolfies Jul 2 '13 at 16:06
    
There is no such thing as "maximum variance" of a given and parametrized distribution. The variance is a central moment of the distribution, hence a fixed number. The only way your question could make sense would be :"suppose we estimate the variance of a distribution. What would be the variance of the estimator of the distribution's variance?" - permitting you to construct a confidence interval. –  Alecos Papadopoulos Jul 30 '13 at 23:52

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