Thanks in advance for the help.
I'm exploring a possible solution to a problem. In order to explore it, however, I need to find the maximum variance of a set of numbers on a given interval. For example, suppose I draw at random 5 continuous values from the interval [0,100]. Given these five values what is the maximum variance that I can expect?
What is the procedure to answering questions of this type?
note: I'm not sure what tags I should place on this under. I have picked statistics since the problem I wish to ultimately tackle is related to statistics. I have also added number theory for similar reasons. If there are any better tags please let me know.
Variance measures how spread out they are, so with five numbers, put two of them at the one endpoint and three at the other. Thus: $0,0,0,100,100$. With an even number of numbers, put have of them at one endpoint and half at the other.
For $0,0,0,100,100$, the variance is $\left(\frac 3 5\cdot\frac 2 5\right)\cdot100^2$.
(Here I am using $\frac 1 n \sum_{k=1}^n (x_k-\bar x)^2$ where $\bar x=\frac 1 n \sum_{k=1}^n x_k$, and not $\frac 1 {n-1} \sum_{k=1}^n (x_k-\bar x)^2$, which should be used only in doing estimates based on a random sample. But either way, the answer above will be the same.)