How do I calculate the extreme possible values of a population given mean and standard deviation?

Question

If I know that the mean value of some random variable in a population is 11.52 and the standard deviation is 18.55182, can I estimate the value of some of the most extreme data points?

My question is inspired by the following article: https://www.rcmalternatives.com/2012/03/if-bill-gates-were-as-tall-as-his-money/

If Bill Gates were as tall as his money, he would be a "300 mile tall man."

Answer 1

If you do not assume a particular distribution then you can not say very much about extreme values beyond something like:

• either at least one value in the population is $\mu+\sigma$ or more, or at least one value is $\mu-\sigma$ or below: in your example, you cannot have all the values strictly between $-7.03182$ and $30.07182$
• if all the values are non-negative, then the largest value must be at least $\mu+\frac{\sigma^2}{\mu}$: in your example the largest value would then have to be more than $41.39587$