Question

Consider the following.

21, 11, 5, 26, 7, 4

Compute the population standard deviation of the numbers. (Round your answer to two decimal place.)

_____________

(a) Double each of your original numbers and compute the standard deviation of this new population. (Round your answer to two decimal place.)

___________


(b) Use the results of part (a) and inductive reasoning to state what happens to the standard deviation of a population when each data item is multiplied by a positive constant k.

1.The standard deviation is multiplied by -k.

2. The standard deviation is divided by k.    

3. The standard deviation is unchanged.

4. The standard deviation is multiplied by k.

5. The standard deviation is k.

Answer 1

Given data : 21, 11, 5, 26, 7, 4

Population standard deviation is given as,

where, is obtained as,

i
1	21	8.67	75.1689
2	11	-1.33	1.7689
3	5	-7.33	53.7289
4	26	13.67	186.8689
5	7	-5.33	28.4089
6	4	-8.33	69.3889
Sum	12.33333		415.3334

Thus, population standard deviation is

(a) Doubling each of the original data, we get

42, 22, 10, 52, 14, 8

Here,

i
1	42	17.33	300.3289
2	22	-2.67	7.1289
3	10	-14.67	215.2089
4	52	27.33	746.9289
5	14	-10.67	113.8489
6	8	-16.67	277.8889
Sum	148		1661.333

Hence, the standard deviation of this new population will be,

(* Note : Here, 16.64 = 2*(8.32), where 8.32 is the standard deviation of the original data points.)

(b)

Using the result of part (a) we see that if the data is doubled, i.e., when the data is multiplied by 2 (> 0), we ultimately get the new standard deviation as double of standard deviation of original data.

Hence, if we multiply the data with any positive constant k (i.e, k > 0) , then the standard deviation of population is multiplied by k,

Thus, option 4. is the correct option.