Posted by statistics on Tuesday, July 28, 2009 at 11:46am.
Standard deviation σ is given by:
σ² = Σ (x-x̄)²/n
So if you can find the smallest difference between x and x̄, 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.
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