Dataset S consists of 10 distinct numerical values. Dataset T is created by replacing the … — GRE General Test

Dataset $S$ consists of 10 distinct numerical values. Dataset $T$ is created by replacing the maximum value of Dataset $S$ with a number that is strictly greater than that maximum value, while all other 9 values remain unchanged. Which of the following statements comparing Dataset $S$ and Dataset $T$ must be true? Select all that apply.

The range of Dataset $T$ is greater than the range of Dataset $S$ .Answer
The interquartile range (IQR) of Dataset $T$ is equal to the interquartile range of Dataset $S$ .Answer
C
The standard deviation of Dataset $T$ is equal to the standard deviation of Dataset $S$ .
D
The median of Dataset $T$ is strictly greater than the median of Dataset $S$ .
E
The 75th percentile of Dataset $T$ is strictly greater than the 75th percentile of Dataset $S$ .

Answer

The statements asserting that the range of Dataset

T

is greater than the range of Dataset

S

and that the interquartile range (IQR) of Dataset

T

is equal to the interquartile range of Dataset

S

are correct.

The statement regarding the range is correct because increasing the maximum value increases the difference between the maximum and minimum values. The statement regarding the interquartile range is correct because the 25th percentile (

Q_1

) and 75th percentile (

Q_3

) depend only on the first 8 ordered elements of a 10-element dataset, so changing the 10th element leaves

Q_1

and

Q_3

identical.

Step-by-Step Solution

Analyze the impact on Range

\text{Range}(T) > \text{Range}(S)

Range is defined as

\text{Maximum} - \text{Minimum}

. Since the minimum is unchanged and the maximum increases, the range must increase.

Analyze the impact on Interquartile Range (IQR)

\text{IQR}(T) = \text{IQR}(S)

IQR is

Q_3 - Q_1

. For 10 ordered elements,

Q_1

and

Q_3

depend on the positions of the bottom 75% of the data. Modifying only the 10th (largest) element leaves

Q_1

and

Q_3

unchanged.

Analyze the impact on Median and Standard Deviation

Median is unchanged; Standard deviation increases.

The median depends on the 5th and 6th values, which are unaffected. Moving an extreme value further out increases distance from the mean, strictly increasing standard deviation.

Key Concept

Effect of outlier modification on measures of position (

Q_1, Q_3

, Median) versus measures of dispersion (Range, Standard Deviation, IQR).

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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