Dataset W consists of 101 distinct real numbers arranged in strictly increasing order: x_1… — GRE General Test

Dataset $W$ consists of $101$ distinct real numbers arranged in strictly increasing order: $x_1 < x_2 < \dots < x_{101}$ . The interquartile range of $W$ is $Q$ , its range is $R$ , and its standard deviation is $\sigma$ . A new dataset $W'$ is constructed by adding a positive constant $k$ to each of the highest $25$ observations (that is, $x_{77}, x_{78}, \dots, x_{101}$ become $x_{77}+k, x_{78}+k, \dots, x_{101}+k$ ), while the first $76$ observations remain unchanged. Which of the following statements MUST be true regarding the statistics of $W'$ compared to dataset $W$ ?

The interquartile range of $W'$ is equal to $Q + \frac{k}{2}$ .Answer
B
The interquartile range of $W'$ is equal to $Q + k$ .
C
The interquartile range of $W'$ is equal to $Q$ .
D
The standard deviation of $W'$ is equal to $\sigma$ .
E
The median of $W'$ is equal to the median of $W$ plus $\frac{k}{2}$ .

Answer

The statement specifying that the interquartile range of

W'

is equal to

Q + \frac{k}{2}

must be true.

For an ordered dataset of 101 elements, the first quartile

Q_1

is the median of the lower 50 values (

\frac{x_{25} + x_{26}}{2}

) and the third quartile

Q_3

is the median of the upper 50 values (

\frac{x_{76} + x_{77}}{2}

). Since elements

x_1

through

x_{76}

are unchanged,

Q_1

remains identical. However,

x_{77}

increases by

k

, causing

Q_3

to increase by

\frac{k}{2}

. Consequently, the new interquartile range is

Q' = Q_3' - Q_1' = Q + \frac{k}{2}

Step-by-Step Solution

Determine the position of quartiles for a 101-element dataset.

For

101

sorted values, the median is

x_{51}

. The lower half consists of

50

values (

x_1

through

x_{50}

), whose median

Q_1

\frac{x_{25} + x_{26}}{2}

. The upper half consists of

50

values (

x_{52}

through

x_{101}

), whose median

Q_3

\frac{x_{76} + x_{77}}{2}

Establishing the exact index formulas for

Q_1

and

Q_3

is required to evaluate changes in IQR.

Analyze how

Q_1

changes in dataset

W'

x_{25}

and

x_{26}

are among the first

76

observations, so they are unchanged. Thus,

Q_1' = Q_1

Since the modified values start at index 77, the first quartile remains completely unaffected.

Analyze how

Q_3

changes in dataset

W'

W'

x_{76}' = x_{76}

and

x_{77}' = x_{77} + k

. Therefore,

Q_3' = \frac{x_{76} + (x_{77} + k)}{2} = \frac{x_{76} + x_{77}}{2} + \frac{k}{2} = Q_3 + \frac{k}{2}

One of the two values used to calculate

Q_3

is shifted by

k

, so their average increases by

\frac{k}{2}

Compute the new interquartile range

Q'

Q' = Q_3' - Q_1' = \left(Q_3 + \frac{k}{2}\right) - Q_1 = (Q_3 - Q_1) + \frac{k}{2} = Q + \frac{k}{2}

Subtracting

Q_1'

from

Q_3'

yields the new interquartile range directly.

Key Concept

Quartile Interpolation and Selective Data Transformations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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