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.

8, 16, 14, 8, 16

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

(b) Add 8 to each data value to get the new data set 16, 24, 22, 16, 24. Compute s. (Enter your answer to four decimal places.)

(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

3.6661

3.6661

A, Adding a constant does nothing to the standard deviation

Explanation:

I'm gonna assume s=standard deviation

The standard deviation is just the square root of the second moment minus the first moment squared

Because we were not told otherwise I think it's pretty safe to assume that all events are equally likely

Let's start by calculating the first moment (AKA The mean)

1/5(8+16+14+8+16)= 12.4

Let's then find the second moment

1/5(8²+16²+14²+8²+16²)= 167.2

√(167.2-12.4²)=3.6661

b.

While I could just tell you that adding something to the standard deviation (and the variane as well) doesn't do anything let's calculate it for fun

same process

.2(16+24+22+16+24)= 20.4

.2(16²+24²+22²+16²+24²)=429.6

√(429.6-20.4²)= 3.6661