An original dataset X consists of 20 distinct positive numerical values with standard devi… — GRE General Test

An original dataset $X$ consists of $20$ distinct positive numerical values with standard deviation $\sigma$ , range $R$ , and interquartile range $IQR$ . A new dataset $Y$ is created by transforming each data value $x_i$ in $X$ according to the rule $y_i = -3x_i + 5$ . Which of the following statements about the dispersion metrics of dataset $Y$ compared to dataset $X$ must be true? Select all such statements.

The standard deviation of dataset $Y$ is $3\sigma$ .Answer
The interquartile range of dataset $Y$ is $3 \times IQR$ .Answer
C
The standard deviation of dataset $Y$ is $3\sigma + 5$ .
D
The range of dataset $Y$ is $-3R + 5$ .
E
The interquartile range of dataset $Y$ is equal to $-3 \times IQR$ .

Answer

The statements asserting that the standard deviation of dataset Y is 3 times the standard deviation of dataset X, and that the interquartile range of dataset Y is 3 times the interquartile range of dataset X, are both correct.

For any data set undergoing a linear transformation

y = ax + b

, all measures of dispersion (including standard deviation, range, and interquartile range) are scaled by the absolute value of the multiplier

|a|

and remain completely unaffected by the additive constant

b

. Since

a = -3

|a| = |-3| = 3

. Therefore, both the standard deviation and the interquartile range scale by a factor of 3.

Step-by-Step Solution

Recall the general rule for linear transformation of measures of dispersion.

For any linear transformation

y = ax + b

, measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by

b

Measures of dispersion evaluate the spread/distance between points, which shifts unchanged when a constant is added, and scales non-negatively when multiplied by a constant.

Identify the values of the multiplicative constant

a

and additive constant

b

a = -3

and

b = 5

, giving

|a| = |-3| = 3

The transformation equation is

y_i = -3x_i + 5

Apply the scaling factor

|a| = 3

to each measure of dispersion.

Standard deviation of

Y

3\sigma

, IQR of

Y

3 \times IQR

, and Range of

Y

3R

All measures of dispersion scale by a factor of 3 regardless of the negative sign of the multiplier or the addition of 5.

Key Concept

Linear Transformations on Measures of Dispersion

Question

Answer

Step-by-Step Solution

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