Dataset S consists of 80 distinct numerical scores arranged in ascending order, with an in… — GRE General Test

Dataset $S$ consists of $80$ distinct numerical scores arranged in ascending order, with an interquartile range denoted by $\text{IQR}_S$ . A new dataset, $T$ , is created by adding four scores to Dataset $S$ : two scores that are strictly smaller than the minimum value of Dataset $S$ , and two scores that are strictly greater than the maximum value of Dataset $S$ . If $\text{IQR}_T$ represents the interquartile range of Dataset $T$ , which of the following statements must be true?

$\text{IQR}_T > \text{IQR}_S$ Cevap
B
$\text{IQR}_T = \text{IQR}_S$
C
$\text{IQR}_T < \text{IQR}_S$
D
$\text{IQR}_T = \text{IQR}_S + 4$
E
The relationship between $\text{IQR}_T$ and $\text{IQR}_S$ cannot be determined from the given information.

Cevap

The interquartile range of Dataset

T

is strictly greater than the interquartile range of Dataset

S

(

\text{IQR}_T > \text{IQR}_S

The statement that

\text{IQR}_T > \text{IQR}_S

is correct. Inserting two items below the minimum value shifts the rank position of the first quartile to lower-indexed elements of the original dataset, reducing

Q_1

. Inserting two items above the maximum value shifts the rank position of the third quartile to higher-indexed elements of the original dataset, increasing

Q_3

. Since the original numbers are strictly increasing,

Q_3

increases while

Q_1

decreases, resulting in a larger interquartile range for Dataset

T

Adım Adım Çözüm

Determine the quartile positions for the original dataset

S

Dataset

S

has

n = 80

distinct elements

x_1 < x_2 < \dots < x_{80}

. Its first quartile

Q_{1,S}

is the average of

x_{20}

and

x_{21}

, and its third quartile

Q_{3,S}

is the average of

x_{60}

and

x_{61}

. Thus,

\text{IQR}_S = Q_{3,S} - Q_{1,S} = \frac{x_{60} + x_{61}}{2} - \frac{x_{20} + x_{21}}{2}

Quartiles partition sorted data into four equal parts based on element indices.

Determine the new element positions for dataset

T

Dataset

T

has

n = 84

elements ordered as

a, b, x_1, x_2, \dots, x_{80}, c, d

, where

a, b < x_1

and

c, d > x_{80}

. The elements of

S

are now shifted right by 2 index positions in

T

Inserting two values at the bottom of the ordered list increases the rank index of all original elements by 2.

Calculate the first and third quartiles for dataset

T

In Dataset

T

, the first quartile

Q_{1,T}

is the average of the 21st and 22nd elements of

T

, which correspond to

x_{19}

and

x_{20}

S

. The third quartile

Q_{3,T}

is the average of the 63rd and 64th elements of

T

, which correspond to

x_{61}

and

x_{62}

S

The 25th and 75th percentile positions in an 84-element dataset correspond to indices 21/22 and 63/64.

Compare

\text{IQR}_T

with

\text{IQR}_S

Since

S

consists of distinct values in strictly ascending order (

x_{19} < x_{20} < x_{21}

and

x_{60} < x_{61} < x_{62}

Q_{1,T} = \frac{x_{19} + x_{20}}{2} < \frac{x_{20} + x_{21}}{2} = Q_{1,S}

and

Q_{3,T} = \frac{x_{61} + x_{62}}{2} > \frac{x_{60} + x_{61}}{2} = Q_{3,S}

. Therefore,

\text{IQR}_T = Q_{3,T} - Q_{1,T} > Q_{3,S} - Q_{1,S} = \text{IQR}_S

Subtracting a smaller number from a larger number increases the overall difference.

Anahtar Kavram

Interquartile Range and Position Indices under Data Insertion

Soru

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