Question

Select all of the true statements about the standard deviation of a quantitative variable. 1. Standard deviation is resistive to unusual values. 2.The standard deviation of a set of values is equal to 0 if and only if all of the values are the same. 3. Standard deviation is never negative. 4.Changing the units of a set of values (e.g., converting from inches to feet) does not affect its standard deviation. 5.If a set of values has a mean of 0 and a standard deviation that is not 0, then adding a new data point with a value of 0 will have no effect on the standard deviation. 6.Standard deviation represents how far a group of values are from the mean of those values, on average.

Answer 1

Considering the first statement

Standard deviation is resistive to unusual values

This statement is false because standard deviation is the numeric measure of deviation of the each observation from the mean

Considering the second statement

The standard deviation of a set of values is equal to 0 if and only if all of the values are the same.

This statement is true because standard deviation is the numeric measure of deviation of the each observation from the mean.

Considering the third statement

Standard deviation is never negative.

This statement is true because standard deviation is the numeric measure of deviation of the each observation from the mean.

Considering the fourth statement

Changing the units of a set of values (e.g., converting from inches to feet) does not affect its standard deviation

This statement is false because standard deviation is the numeric measure of deviation of the each observation from the mean

Considering the fifth statement

If a set of values has a mean of 0 and a standard deviation that is not 0, then adding a new data point with a value of 0 will have no effect on the standard deviation.

This statement is false because , let take an example

x -4 - 3 0 3 4

Generally the mean is mathematically evaluated as

`\= x =  (\sum x_i )/(n)`

=>
`\= x =  (-4+ (- 3)+ 0 + 3 + 4 )/(5)`

=>
`\= x =  0`

Generally the standard deviation is mathematically evaluated as

`\sigma  =  \sqrt{(\sum (x_ i - \= x )^2)/(y) }`

`\sigma  =  \sqrt{( (-4  - 0 )^2 + (- 3 - 0 )^2 + (0 - 0 )^2 + (3 - 0)^2 + (4 - 0 )^2)/(5) }`

=>
`\sigma  = 3.16`

Now when zero is removed the standard deviation is

`\sigma_1  =  \sqrt{( (-4  - 0 )^2 + (- 3 - 0 )^2 + (3 - 0)^2 + (4 - 0 )^2)/(4) }`

=>
`\sigma_1  =  3.54`

Since
`\sigma  \\e  \sigma _1` the above statement is false

Considering the sixth statement

Standard deviation represents how far a group of values are from the mean of those values, on average.

This statement is true because standard deviation is the numeric measure of deviation of the each observation from the mean.

Explanation: