# Maximum number of standard deviations between set mean and max

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 $$n$$ : $$\frac {p_n - \overline {p} }{\sigma p} < \sqrt {n-1}$$

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?