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.

10, 16, 15, 4, 6

(a) Compute s. (Enter your answer to one decimal place.)

(b) Add 4 to each data value to get the new data set 14, 20, 19, 8, 10. 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

(a)

The formula for s (the sample standard deviation) is given by:

where

n is the no. of observations;

is the sample mean; and

is the sum of square of the observations.

Now, the given dataset is:

10, 16, 15, 4, 6

Thus,

n = 5

Thus, the sample standard deviation is given by:

(b)

The new data set is 14, 20, 19, 8, 10

For the new dataset:

n = 5

Thus, the sample standard deviation is given by:

(c)

From part (a) and (b) we observe that the value of s remains unchanged if we add 4 to all the observations. [ANSWER]

The above conclusion is just a demonstration of the fact that adding a constant term c to all the observations doesn't change its sample standard deviation as was observed from part (a) and (b)

Thus, the correct option is:

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

The other three options are incorrect because the standard deviation does not change with adding a constant to all the observations. Moreover, we can conclusive say that it will remain unchanged and the statement "There is no distinct pattern when the constant is added to each data value" is incorrect.

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