Give a set of 3 integers between 1 and 9 that would give a) the smallest value and b) the largest value of s (sample standard deviation) and find the value of s: i)if repeats are allowed and ii)if no repeats are allowed?
Is there an approach to this question or is it done by trial or error? Thanks
Let's take a stab at that portion of the problem this not answered in comments under the question. The three numbers are $1,x,9$. The mean is $$ \bar x = \frac {1+x+9} 3 $$ so the sum of squares of deviations from the mean is \begin{align} & (1-\bar x)^2 + (x- \bar x)^2 + (9-\bar x)^2 \\[10pt] = {} & \left( \frac{ -7-x } 3 \right)^2 + \left( \frac {2x - 10} 3 \right)^2 + \left( \frac {17-x} 3 \right)^2 \\[10pt] = {} & \frac{ 6x^2 - 60 x + 438 } 9 = \frac {6(x-5)^2 + 288} 9. \end{align}
This last expression is equal to $\dfrac{288} 9 = 32$ if $x=5$, but is bigger than that if $x\ne5$. (And $5$ is the average of $1$ and $9$.)