A dataset W consists of 25 distinct positive numbers with mean M, standard deviation s > 0… — GRE General Test

A dataset $W$ consists of 25 distinct positive numbers with mean $M$ , standard deviation $s > 0$ , and interquartile range $IQR > 0$ . Dataset $V$ is constructed by adding $2M$ to every number in $W$ that is strictly greater than the median of $W$ , and subtracting $2M$ from every number in $W$ that is strictly less than the median of $W$ . The value equal to the median of $W$ itself remains unchanged. Which of the following statements MUST be true regarding dataset $V$ compared to dataset $W$ ?

Both the standard deviation and the interquartile range of dataset $V$ are greater than those of dataset $W$ .Cevap
B
The standard deviation of dataset $V$ is greater than $s$ , but the interquartile range of dataset $V$ is equal to $IQR$ .
C
The standard deviation of dataset $V$ is equal to $s$ , but the interquartile range of dataset $V$ is greater than $IQR$ .
D
Both the standard deviation and the interquartile range of dataset $V$ are equal to those of dataset $W$ .
E
The standard deviation of dataset $V$ is less than $s$ , while the interquartile range of dataset $V$ is greater than $IQR$ .

Cevap

Both the standard deviation and the interquartile range of dataset

V

are greater than those of dataset

W

The correct response identifies that both standard deviation and interquartile range increase. In dataset

W

, the 12 elements strictly below the median are reduced by

2M

, and the 12 elements strictly above the median are increased by

2M

. This leaves the mean

M

unchanged but increases the distance of every non-median element from

M

, thereby strictly increasing the standard deviation. Furthermore, the first quartile

Q_1

shifts down by

2M

and the third quartile

Q_3

shifts up by

2M

, expanding the interquartile range from

IQR

IQR + 4M

Adım Adım Çözüm

Analyze the impact of the transformation on the first and third quartiles (

Q_1

and

Q_3

In an ordered dataset of 25 distinct values, the median is the 13th element. The first quartile

Q_1

is in the lower half (below the median) and the third quartile

Q_3

is in the upper half (above the median).

Understanding where

Q_1

and

Q_3

fall relative to the median determines how their values change.

Calculate the new Interquartile Range (

IQR_V

Since

Q_1

is in the lower half, its new value is

Q_1 - 2M

. Since

Q_3

is in the upper half, its new value is

Q_3 + 2M

. Thus,

IQR_V = (Q_3 + 2M) - (Q_1 - 2M) = (Q_3 - Q_1) + 4M = IQR + 4M > IQR

because

M > 0

Evaluating the difference between the transformed 75th and 25th percentiles.

Analyze the impact on the mean and standard deviation.

Equal numbers of values (12 values) are shifted down by

2M

and shifted up by

2M

, so the mean of dataset

V

remains equal to

M

. For every shifted point

x_i

, its squared distance from the mean

(x_i - M)^2

strictly increases. Therefore, the variance and standard deviation strictly increase (

s_V > s

Standard deviation measures dispersion relative to the mean; pushing points further from the mean increases standard deviation.

Anahtar Kavram

Effect of non-uniform linear shifts on measures of dispersion (Standard Deviation and IQR)

Tahmini Süre:2m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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