using the numbers only 0 through 10 only whole numbers with repeats allowed

Question

pick four numbers that would have the smallest standard deviation

I said four zeros

pick four numbers that would have the largest standard deviation

I said 0 0 0 10

is there more than one correct answer for either of these questions

Answer 1

Standard deviation σ is given by:
σ&sup2 = Σ (x-x&#772)²/n
So if you can find the smallest difference between x and x&#772, you have found the smallest value.
(0,0,0,0) would work as well as any other number such as (5,5,5,5).

For the maximum value, we would see the biggest spread possible. For four numbers, I would try (0,0,10,10). Try out other combinations (such as 0,0,0,10 and 0,10,10,10 to convince yourself that you have got the maximum.

Answer 2

Any four numbers that are the same would have a standard deviation of zero.

The set of numbers 0,0,10,10 would have the largest average deviation from the mean (5). I suspect that would also have the largest standard deviation (also 5). Try others and see