Dataset D consists of 40 distinct real numbers arranged in strictly increasing order: x_1 … — GRE General Test

Dataset $D$ consists of $40$ distinct real numbers arranged in strictly increasing order: $x_1 < x_2 < \dots < x_{40}$ . Let $R_D$ , $Q_D$ , and $\sigma_D$ denote the range, interquartile range, and standard deviation of Dataset $D$ , respectively.

A new dataset, Dataset $E$ , is created by replacing each of the $10$ largest numbers ( $x_{31}, x_{32}, \dots, x_{40}$ ) with the value of $x_{30}$ , which represents the $75\text{th}$ percentile of Dataset $D$ . Let $R_E$ , $Q_E$ , and $\sigma_E$ denote the range, interquartile range, and standard deviation of Dataset $E$ , respectively.

Which of the following statements MUST be true? Select all such statements.

The range of Dataset $E$ is strictly less than the range of Dataset $D$ ( $R_E < R_D$ ).Answer
The interquartile range of Dataset $E$ is equal to the interquartile range of Dataset $D$ ( $Q_E = Q_D$ ).Answer
The standard deviation of Dataset $E$ is strictly less than the standard deviation of Dataset $D$ ( $\sigma_E < \sigma_D$ ).Answer
D
The median of Dataset $E$ is strictly less than the median of Dataset $D$ .
E
The interquartile range of Dataset $E$ is strictly less than the interquartile range of Dataset $D$ ( $Q_E < Q_D$ ).

Answer

The statements asserting that the range of Dataset E is strictly less than the range of Dataset D (

R_E < R_D

), the interquartile range of Dataset E is equal to the interquartile range of Dataset D (

Q_E = Q_D

), and the standard deviation of Dataset E is strictly less than the standard deviation of Dataset D (

\sigma_E < \sigma_D

) are all correct.

The range of Dataset E decreases because its maximum value is reduced from

x_{40}

x_{30}

while its minimum value

x_1

remains the same. The interquartile range remains unchanged because the boundary values defining the

25\text{th}

percentile (

x_{10}

) and

75\text{th}

percentile (

x_{30}

) are not altered. The standard deviation strictly decreases because reducing the

10

largest values brings them closer to the rest of the distribution, decreasing total variation.

Step-by-Step Solution

Analyze the change in the maximum and minimum values to evaluate the Range.

In Dataset

D

\text{Range } R_D = x_{40} - x_1

. In Dataset

E

, the maximum element becomes

x_{30}

while the minimum remains

x_1

. Since all numbers are distinct and ordered,

x_{30} < x_{40}

, so

R_E = x_{30} - x_1 < R_D

. Thus,

R_E < R_D

is true.

Range is defined as

\text{Maximum} - \text{Minimum}

Analyze the

25\text{th}

percentile (

Q_1

) and

75\text{th}

percentile (

Q_3

) to evaluate the Interquartile Range.

For

N=40

Q_1 = x_{10}

and

Q_3 = x_{30}

. Dataset

E

modifies only the values strictly above

x_{30}

(

x_{31}

through

x_{40}

). Thus

x_{10}

and

x_{30}

retain their exact values in Dataset

E

. Therefore,

Q_E = x_{30} - x_{10} = Q_D

Interquartile range depends solely on

Q_3 - Q_1

, which relies on position cutoffs that were unaffected.

Analyze the effect of pulling extreme upper values inward on the Standard Deviation.

Standard deviation measures overall spread around the mean and can be calculated via pairwise differences

\text{Var}(X) = \frac{1}{2N^2} \sum_{i,j} (x_i - x_j)^2

. Replacing

x_{31}, \dots, x_{40}

with

x_{30}

strictly reduces the distances between these upper points and all other points, ensuring

\sigma_E < \sigma_D

Reducing the distance of tail observations from the rest of the dataset strictly decreases overall variance and standard deviation.

Key Concept

Impact of Data Transformations on Measures of Dispersion and Position

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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