Maximum Scattering

(a) When there are $2$ numbers, their total is $2$. The possibilities for the values of the two numbers can be represented as a straight line $x+y=2$ on a graph. The mean is the point $(1,1)$ in the centre of the line. The standard deviation ($\sigma$) is greatest when the distance between the values and the mean is largest. The endpoints are furthest from the centre so $\sigma$ is largest at the point $(2,0)$ when it is equal to $1$.

(b) When there are $3$ values, their total is $3$. This is represented by a plane. As the numbers are non-negative the only values for the numbers are in a triangle with the corners at $(0,0,3)$, $(0,3,0)$ and $(3,0,0)$. These are the points furthest away from the mean which is at $(1,1,1)$ so are where $\sigma$ is greatest and it is $\sqrt{2}$.

(c) When there are $n$ values, the total is $n$. The region where this is true and they are non-negative is a $n-1$-dimensional shape with $n$ corners which have coordinates $(0,0,...,0,n)$ each with the $n$ in a different position and $n-1$ $0$s. These are furthest from the mean (which is at $(1,1,...,1,1)$) so are where the value of $\sigma$ is greatest. $\sigma$ is the square root of the difference between the square of mean and the mean of the squares. The mean is $1$ so the mean squared is $1$. The squares are $n^{2}$, 0, 0, ... so the total of the squares is $n^{2}$ and their mean is $n$. This makes $\sigma= \sqrt{n-1}$.