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Is there a maximum number of standard deviations between the mean of set and the set's maximum value? From a few basic examples I found this, but is this always true? For set $p$ of length $n$ : $$\frac {p_n - \overline {p} }{\sigma p} < \sqrt {n-1}$$

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Hint:

Take the set of observations $\{0, 1, 1, 1, 1, \dots, 1, 2\}$ where there are $n$ observations of $1$. What's the standard deviation of this observation? What is the mean and maximum element? What happens to the ratio between standard deviation and difference between mean and max as $n$ becomes large?

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