Question

In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the following data set.

16, 12, 9, 4, 7

(a) Use the defining formula, the computation formula, or a calculator to compute s. (Enter your answer to one decimal place.)

(b) Add 8 to each data value to get the new data set 24, 20, 17, 12, 15. Compute s. (Enter your answer to one decimal place.)

(c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

Adding the same constant c to each data value results in the standard deviation remaining the same.

Adding the same constant c to each data value results in the standard deviation increasing by c units.    

Adding the same constant c to each data value results in the standard deviation decreasing by c units.

There is no distinct pattern when the same constant is added to each data value in a set.

Answer 1

Solution: Let us use the data set to find the standard deviation.

Standard deviation is calculated using formula

where  =Standard deviation, Xi=given data sets, =mean, N=size of data set=5

a)

Data set	xi-	(xi-)2
16	6.4	40.96
12	2.4	5.76
9	-0.6	0.36
4	-5.6	31.36
7	-2.6	6.76
=9.6		(xi-)2=85.2

So,

=85.2/5

=4.13

b) Let us calculate Standard deviation with the new data set.

Data set	xi-	(xi-)2
24	6.4	40.96
20	2.4	5.76
17	-0.6	0.36
12	-5.6	31.36
15	-2.6	6.76
=17.6		(xi-)2=85.2

So, =85.2/5 =4.13

c) Comparing the results of (a) and (b), we can say that there is no difference in standard deviation when a constant c(here its 8) added to each data value.

So option 1 is the answer.(Adding the same constant c to each data value results in the standard deviation remaining the same.)