Z is a set of positive numbers. The median of Z is greater

[BTGmoderatorDC]

Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
II. Less than 50% of the numbers in Z are greater than the median.
III. The median of Z is greater than the average of the largest and smallest numbers in Z.

A. I only
B. II only
C. III only
D. I and III only
E. None of the above

OA E

Source: https://gmat.economist.com/btg?gsrc=btg

[Jay@ManhattanReview]

Z is a set of positive numbers. The median of Z is greater than the mean of Z. Which of the following must be true?

I. At least 50% of the numbers in Z are smaller than the median.
II. Less than 50% of the numbers in Z are greater than the median.
III. The median of Z is greater than the average of the largest and smallest numbers in Z.

A. I only
B. II only
C. III only
D. I and III only
E. None of the above

OA E

Source: Economist GMAT

Since this is a roman numeral question and each of the three statements are independent of each other, we must think of Set Z such that the statements are challenged.

Let's see each statement one by one.

I. At least 50% of the numbers in Z are smaller than the median.

Say Z : {0, 2, 2, 4, 6, 6, 6}
Median = 4; Mean = 26/7 = 3.7...; we see that Median (4) > Mean (3.7), so this is a valid set.
The numbers smaller than Median (4) are 0, 2, and 2; there are 3 out of 7 numbers (3/7 * 100% < 50%) smaller than Median.
Statement I must not be true.

II. Less than 50% of the numbers in Z are greater than the median.

Say Z : {0, 1, 1, 3, 3, 3}
Median = (1 + 3)/2 = 2; Mean = 11/6 = 1.8...; we see that Median (2) > Mean (1.8), so this is a valid set.
The numbers greater than Median (2) are 3, 3, and 3; there are 3 out of 6 numbers (3/6 * 100% = 50%) greater than Median.
Statement II must not be true.

III. The median of Z is greater than the average of the largest and smallest numbers in Z.

Say Z : {0, 1, 1, 3, 3, 3}
Median = (1 + 3)/2 = 2; Mean = 11/6 = 1.8...; we see that Median (2) > Mean (1.8)
The the average of the largest and smallest numbers = (0 + 3)/2 = 2 = Median (2)
Statement III must not be true.

The correct answer: E

Hope this helps!

-Jay
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