Dataset S consists of 4n distinct real numbers arranged in ascending order, where n \ge 5. The interquartile range of Dataset S is IQR_S, and its standard deviation is \sigma_S. A new dataset, T, is formed by transforming each value x in Dataset S into a corresponding value y as follows: y = \begin{

Dataset S consists of 4n distinct real numbers arranged in ascending order, where n \ge 5.… — GRE General Test

Dataset

S

consists of

4n

distinct real numbers arranged in ascending order, where

n \ge 5

. The interquartile range of Dataset

S

IQR_S

, and its standard deviation is

\sigma_S

. A new dataset,

T

, is formed by transforming each value

x

in Dataset

S

into a corresponding value

y

as follows:

y = \begin{cases} x + k & \text{if } x \le Q_3 \\ x + 2k & \text{if } x > Q_3 \end{cases}

where

Q_3

is the third quartile (75th percentile) of Dataset

S

, and

k

is a positive constant. Which of the following statements must be true regarding the interquartile range

IQR_T

and standard deviation

\sigma_T

of Dataset

T

relative to Dataset

S

$IQR_T = IQR_S$ and $\sigma_T > \sigma_S$ Cevap
B
$IQR_T > IQR_S$ and $\sigma_T > \sigma_S$
C
$IQR_T = IQR_S$ and $\sigma_T = \sigma_S$
D
$IQR_T > IQR_S$ and $\sigma_T = \sigma_S$
E
$IQR_T < IQR_S$ and $\sigma_T > \sigma_S$

Cevap

The interquartile range remains unchanged (

IQR_T = IQR_S

) while the standard deviation strictly increases (

\sigma_T > \sigma_S

The statement asserting that

IQR_T = IQR_S

and

\sigma_T > \sigma_S

is correct. Both

Q_1

and

Q_3

belong to the condition

x \le Q_3

, meaning both quartile values increase by exactly

k

. Thus,

IQR_T = (Q_3 + k) - (Q_1 + k) = Q_3 - Q_1 = IQR_S

. Meanwhile, the highest 25% of data values are shifted by an additional distance of

k

, increasing the overall spread of values around the mean, which strictly increases the standard deviation.

Adım Adım Çözüm

Analyze the impact of the transformation on the first quartile (

Q_1

) and third quartile (

Q_3

Since

Q_1 < Q_3

, the value corresponding to

Q_1

is less than or equal to

Q_3

, so it is shifted to

Q_1 + k

. The value corresponding to

Q_3

satisfies

x \le Q_3

, so it is also shifted to

Q_3 + k

The definition of the piecewise rule adds

k

to all values less than or equal to

Q_3

Calculate the new interquartile range

IQR_T

IQR_T = (Q_3 + k) - (Q_1 + k) = Q_3 - Q_1 = IQR_S

The constant shift

k

cancels out when taking the difference between the upper and lower quartiles.

Analyze the impact on standard deviation

\sigma_T

The lower 75% of elements are shifted by

+k

, while the upper 25% of elements are shifted further by

+2k

. This increases the relative distance between upper-tail data points and the rest of the dataset.

A non-uniform shift that spreads the upper tail farther from the rest of the distribution increases total variation around the mean, resulting in

\sigma_T > \sigma_S

Anahtar Kavram

Effect of non-linear piecewise transformations on dispersion measures (IQR invariance under equal quartile shifts vs. standard deviation sensitivity to upper-tail displacement)

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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