Chapter 1.3, Problem 121E

(a)

To determine

To find: the four numbers that have the smallest possible standard deviation.

Answer to Problem 121E

2, 2, 2, 2

Explanation of Solution

The smallest possible standard deviation is 0; the reason is that the standard deviation is nonnegative.

The standard deviation would be zero, if all data values are equal.

A possible data list is then: 2, 2, 2, and 2

But could also pick, for example 10, 10, 10, 10

(b)

To determine

To find: the four numbers that have the largest possible standard deviation.

Answer to Problem 121E

0, 0, 10, 10

Explanation of Solution

The largest possible standard deviation is got if the values are as far from the mean as possible.

Although the numbers are between 0 and 10, this can the best be done if two of the values are 0 and two of the value are 10 the reason is this that the mean of 5 is exactly in the middle).

0, 0, 10, 10

(c)

To determine

To find: that the more on option possible in either part (a) or (b) explain.

Answer to Problem 121E

1. More than one option possible
2. Unique

Explanation of Solution

In part (a) more than one option is possible; the reason only if the sample is having one unique number would be the standard deviation is 0. Moreover, in part (a) it also give two possible options,

Therefore more than one option is possible.

In part (b) the answer is unique, because of changing only one of the values will change the mean from 5 and then the total deviation from the mean would decrease. Although the standard deviation depends on the total deviation, the standard deviation would decrease too.