A dataset consists of 15 distinct positive integers. Is the median of the dataset greater … — GMAT

A dataset consists of $15$ distinct positive integers. Is the median of the dataset greater than its arithmetic mean?

(1) The sum of the $7$ smallest integers in the dataset is $42$ , and the sum of the $7$ largest integers in the dataset is $140$ .
(2) The dataset forms an arithmetic progression.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

The option stating that Statement (2) alone is sufficient while Statement (1) alone is not sufficient is correct because Statement (2) establishes that the dataset is an arithmetic progression, where median always equals mean. This provides a definitive 'No' to the question of whether the median is strictly greater than the mean, satisfying sufficiency. Statement (1) allows values of the median both above and below

13

, yielding both 'Yes' and 'No' outcomes.

Step-by-Step Solution

Rephrase the question stem mathematically.

Let the 15 distinct positive integers be ordered as

x_1 < x_2 < \dots < x_7 < x_8 < x_9 < \dots < x_{15}

. The median is

x_8

, and the arithmetic mean is

\mu = \frac{\sum_{i=1}^{15} x_i}{15}

. We need to determine whether

x_8 > \mu

Simplifying the question target clarifies what information is necessary for sufficiency.

Evaluate Statement (1) independently.

Statement (1) gives

\sum_{i=1}^7 x_i = 42

and

\sum_{i=9}^{15} x_i = 140

. The total sum is

42 + x_8 + 140 = 182 + x_8

, so the mean is

\mu = \frac{182 + x_8}{15}

. The inequality

x_8 > \frac{182 + x_8}{15}

simplifies to

14 x_8 > 182

, or

x_8 > 13

. If

x_8 = 10

(with

x_1 \dots x_7 = \{3,4,5,6,7,8,9\}

and

x_9 \dots x_{15} = \{11,12,13,14,15,16,59\}

), then

x_8 \le 13

yields 'No'. If

x_8 = 15

(with

x_1 \dots x_7 = \{1,2,3,4,5,13,14\}

and

x_9 \dots x_{15} = \{16,17,18,19,20,21,29\}

), then

x_8 > 13

yields 'Yes'.

Since Statement (1) allows for both 'Yes' and 'No' answers, it is NOT sufficient.

Evaluate Statement (2) independently.

Statement (2) states that the elements form an arithmetic progression. In any set of numbers that forms an arithmetic progression, the distribution is perfectly symmetric, so the arithmetic mean is exactly equal to the median (

x_8 = \mu

). Therefore,

x_8 > \mu

is false, giving a definitive answer of 'No'.

In Data Sufficiency Yes/No questions, a definitive 'No' answer establishes that the statement IS sufficient.

Key Concept

Data Sufficiency evaluation of data distributions, comparing median and arithmetic mean in symmetric versus asymmetric datasets.

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