Dataset S consists of 100 distinct positive numbers. The 25th percentile of dataset S is 4… — GRE General Test

Dataset $S$ consists of $100$ distinct positive numbers. The 25th percentile of dataset $S$ is $40$ , the median is $60$ , and the 75th percentile is $80$ . A new dataset $T$ is formed by multiplying every number in dataset $S$ that is strictly greater than the 80th percentile by $2$ , while keeping all other numbers unchanged. Which of the following statistics MUST be identical for dataset $S$ and dataset $T$ ?

A
The mean
B
The standard deviation
The interquartile rangeAnswer
D
The range
E
The 90th percentile

Answer

The interquartile range

The interquartile range is the difference between the 75th percentile (

Q_3

) and the 25th percentile (

Q_1

). Because only values strictly greater than the 80th percentile are scaled, the values defining

Q_1

and

Q_3

are unaffected. Therefore,

Q_1 = 40

and

Q_3 = 80

remain the same in both datasets, making the interquartile range (

80 - 40 = 40

) identical.

Step-by-Step Solution

Identify which values in the dataset are modified by the transformation

Only numbers strictly greater than the 80th percentile are multiplied by 2. All numbers at or below the 80th percentile remain unchanged.

The transformation criteria specifies that values below or equal to the 80th percentile threshold undergo no change.

Evaluate the effect on the 25th percentile (

Q_1

) and 75th percentile (

Q_3

)

Since

Q_1

(25th percentile) and

Q_3

(75th percentile) are position metrics located at or below the 80th percentile mark, neither

Q_1

nor

Q_3

changes in value.

Modifying elements only above the 80th percentile leaves the ordering and exact values of elements up to the 80th percentile completely intact.

Determine the impact on the Interquartile Range (IQR)

The interquartile range is defined as

IQR = Q_3 - Q_1

. Because both

Q_1

and

Q_3

remain constant,

IQR

is unchanged.

A metric defined solely by unchanged percentile boundaries must itself remain constant.

Key Concept

Effect of Upper-Tail Data Transformations on Measures of Position and Dispersion

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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