Dataset A consists of 100 distinct positive numbers. Dataset B is created by replacing eve… — GRE General Test

Dataset $A$ consists of 100 distinct positive numbers. Dataset $B$ is created by replacing every number in Dataset $A$ that is strictly greater than the 75th percentile of Dataset $A$ with the value of the 75th percentile of Dataset $A$ . All numbers less than or equal to the 75th percentile remain unchanged. Which of the following statements comparing Dataset $B$ to Dataset $A$ must be true? Select all such statements.

The 25th percentile of Dataset $B$ is equal to the 25th percentile of Dataset $A$ .Answer
The interquartile range of Dataset $B$ is equal to the interquartile range of Dataset $A$ .Answer
C
The standard deviation of Dataset $B$ is greater than the standard deviation of Dataset $A$ .
The range of Dataset $B$ is strictly less than the range of Dataset $A$ .Answer
E
The median of Dataset $B$ is strictly greater than the median of Dataset $A$ .

Answer

The statements asserting that the 25th percentile remains equal, the interquartile range remains equal, and the range is strictly smaller for Dataset B compared to Dataset A are all correct.

The statements confirming that the 25th percentile remains equal, the interquartile range remains equal, and the range decreases are correct. The lower 75% of the ordered dataset is untouched, leaving

P_{25}

and

P_{75}

unchanged, which preserves the IQR. Furthermore, replacing the top 25 distinct values with

P_{75}

lowers the maximum value while keeping the minimum value the same, strictly decreasing the range.

Step-by-Step Solution

Analyze how Dataset B is constructed from Dataset A

Since all 100 values in Dataset A are distinct, exactly 25 values are strictly greater than the 75th percentile (

P_{75}

). In Dataset B, these top 25 values are replaced by

P_{75}

, while the bottom 75 values remain unchanged.

Understanding which specific data points change allows us to determine positional and dispersion metrics.

Evaluate positional statistics (25th percentile, median, 75th percentile)

The lower 75% of the ordered dataset is identical in Dataset A and Dataset B. Thus,

P_{25}(B) = P_{25}(A)

\text{Median}(B) = \text{Median}(A)

, and

P_{75}(B) = P_{75}(A)

Percentiles at or below the 75th percentile depend only on the values at or below those percentile ranks.

Evaluate Interquartile Range (IQR) and Range

\text{IQR}(B) = P_{75}(B) - P_{25}(B) = P_{75}(A) - P_{25}(A) = \text{IQR}(A)

. For range,

\text{Min}(B) = \text{Min}(A)

, but

\text{Max}(B) = P_{75}(A) < \text{Max}(A)

. Therefore,

\text{Range}(B) < \text{Range}(A)

IQR depends on

P_{75}

and

P_{25}

, which are unchanged. Range depends on Max and Min; decreasing the maximum value decreases the range.

Evaluate Standard Deviation

All altered values were in the upper tail and were moved closer to the center of the distribution. Reducing extreme values decreases variance and standard deviation.

Standard deviation measures average squared distance from the mean; pulling upper extreme values inward reduces standard deviation.

Key Concept

Effect of upper-tail data transformation on measures of position (percentiles, median) and dispersion (range, IQR, standard deviation).

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