Chapter 1.3, Problem 103E

(a).

To determine

Numbers that have smallest possible standard deviation.

Answer to Problem 103E

The smallest possible standard deviation is $0$ .

Explanation of Solution

Given:

Numbers: $0,1,2,3,4,5,6,7,8,9,10$

In case one will choose all $4$ numbers same for e.g. $1,1,1,1$ then standard deviation will definitely be $0$ because mean of this $4$ numbers is $1$ itself and so every deviation from mean is $0$ , so standard deviation is $4$ .

Conclusion:

So, the smallest possible standard deviation is $0$ .

(b).

To determine

Numbers that have largest possible standard deviation.

Answer to Problem 103E

The largest possible standard deviation is $5$ .

Explanation of Solution

Given:

Numbers: $0,1,2,3,4,5,6,7,8,9,10$

In case one can take all $4$ numbers like below we get largest possible standard deviation

For e.g.: $0,1,9,10$

Because mean of this $4$ numbers is $5$ and all this $4$ numbers are far away from mean as one can check it manually.

Conclusion:

So, the largest possible standard deviation is $5$ .

(c).

To determine

Possibility of more than one choice in either with reason.

Answer to Problem 103E

It is possible that more than $1$ choice.

Explanation of Solution

Given:

Numbers: $0,1,2,3,4,5,6,7,8,9,10$

It is possible that more than $1$ choice, that is one can take all same observations as $2,2,2,2$ or $3,3,3,3$ or $4,4,4,4$ .

The above asked case is only one choice of largest standard deviation and all other possible $4$ observation have definitely lower standard deviation.

Conclusion:

Thus, It is possible that more than $1$ choice.