Given a set with a known cardinality, least upper and greatest lower bound; how can I calculate the maximum possible standard deviation for any set of values within the set.


As an example:


Given a set {1,50}, with mean of 25.5, the std. dev. for both members at 34.65. 
This std. dev. is the possible for a set with a cardinality of **2**, greatest lower bound of **1**, and least upper bound of **50**.


Here are some examples I have calculated by hand:


F(**2**, **1**, **50**) = 34.65 (as above)


F(3, 1, 50) = 28.29


F(4, 1, 50) = 28.29


F(5, 1, 50) = 26.84


F(6, 1, 50) = 26.84


F(7, 1, 50) = 26.19


F(8, 1, 50) = 26.19



Calculate largest possible standard deviation for a set
The sample deviation is maximized when half of the observations are at each extreme. If you want a formula for the maximum standard deviation as a function of the interval endpoints and the number of samples, you will probably want to divide the formula up depending on whether the number of samples is even or odd.


