Dataset A consists of n numerical values with a standard deviation of \sigma_A and an inte… — GRE General Test

Dataset $A$ consists of $n$ numerical values with a standard deviation of $\sigma_A$ and an interquartile range of $\text{IQR}_A$ . Dataset $B$ is constructed by multiplying each value in Dataset $A$ by $4$ and then subtracting $10$ from each resulting value. Which of the following correctly expresses the standard deviation $\sigma_B$ and the interquartile range $\text{IQR}_B$ of Dataset $B$ in terms of $\sigma_A$ and $\text{IQR}_A$ ?

A
$\sigma_B = 4\sigma_A - 10$ and $\text{IQR}_B = 4\text{IQR}_A - 10$
$\sigma_B = 4\sigma_A$ and $\text{IQR}_B = 4\text{IQR}_A$ Answer
C
$\sigma_B = 16\sigma_A$ and $\text{IQR}_B = 4\text{IQR}_A$
D
$\sigma_B = 4\sigma_A$ and $\text{IQR}_B = \text{IQR}_A$
E
$\sigma_B = 4\sigma_A - 10$ and $\text{IQR}_B = 4\text{IQR}_A$

Answer

\sigma_B = 4\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

When a dataset undergoes a linear transformation of the form

Y = cX + k

, any measure of dispersion

D

(such as standard deviation or interquartile range) transforms according to

D_Y = |c| \cdot D_X

. The constant shift

k

shifts all points equally and does not affect spread. Therefore, multiplying by

4

multiplies both

\sigma_A

and

\text{IQR}_A

4

, and subtracting

10

has no effect on either measure.

Step-by-Step Solution

Analyze the effect of multiplying each value in a dataset by a positive constant

c = 4

Multiplying all data values by

c

scales all measures of spread (including standard deviation, range, and interquartile range) by

|c| = 4

. Thus, initial scaled values give

4\sigma_A

and

4\text{IQR}_A

Measures of dispersion reflect distances between data points, which scale linearly when all values are multiplied by a constant.

Analyze the effect of subtracting a constant

k = 10

from each data point.

Subtracting a constant shifts the entire distribution along the number line without altering the relative distances between any data points.

Because measures of dispersion measure relative spread rather than absolute location, adding or subtracting a constant has zero effect on standard deviation or interquartile range.

Combine the transformation effects to state

\sigma_B

and

\text{IQR}_B

\sigma_B = 4\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

The multiplicative factor scales both metrics by 4, and the additive shift of

-10

leaves both metrics unchanged.

Key Concept

Linear Transformations on Measures of Dispersion

Estimated Time:1m 15s

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