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

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Z is a set of positive numbers. The median of Z is greater [#permalink]

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New post 17 Jul 2014, 11:26
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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
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Re: Z is a set of positive numbers. The median of Z is greater [#permalink]

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New post 17 Jul 2014, 11:43
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goodyear2013 wrote:
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


I. At least 50% of the numbers in Z are smaller than the median. Not always true. Consider {1, 2, 2} --> (median=2) > (mean=5/3) --> only one number (1, so 33%) is less than the median. Discard.

II. Less than 50% of the numbers in Z are greater than the median. Not always true. Consider {1, 2, 3, 3} --> (median=2.5) > (mean=9/4) --> 50% of the numbers (3 and 3) are greater than the median. Discard.

III. The median of Z is greater than the average of the largest and smallest numbers in Z. Not always true. Consider {1, 1, 3, 3, 5} --> (median=3) > (mean=13/5) --> (median=3) = (1+5)/2=3. Discard.

Answer: E.
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Re: Z is a set of positive numbers. The median of Z is greater [#permalink]

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New post 24 Oct 2014, 08:53
Is this a valid train of thought?

Condition 3 can be invalidated by: (1, 3, 4, 5)--median is equal to average of 1 and 5 and average is greater than 3.

Then, don't conditions 1 and 2 have to occur simultaneously? I reasoned this based on the word... maybe that's incorrect.

Even if not, we can rule them both out. So, E.
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Re: Z is a set of positive numbers. The median of Z is greater [#permalink]

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New post 13 Feb 2017, 12:59
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goodyear2013 wrote:
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



What does this imply - "The median of Z is greater than the mean of Z"?
Median will be the middle number. If mean is to the left of the median, it means the sum of deviations of the smaller terms is more than the sum of deviations of the greater terms.

I. At least 50% of the numbers in Z are smaller than the median.
Not necessary. Some numbers could be equal to the median so we may not have 50% numbers smaller.

II. Less than 50% of the numbers in Z are greater than the median.
Again not necessary. 50% of the numbers in Z could be greater than the median If Z has even number of numbers, the median would be the avg of the middle two numbers and if the middle two numbers are distinct, the median will be less than 50% of the numbers.

III. The median of Z is greater than the average of the largest and smallest numbers in Z.
Again not necessary. The greatest number could be much greater pulling the average up even though the sum of deviations of smaller numbers may be smaller.
e.g.

1, 2, 2, 8, 9, 9, 18

Mean is 7.
Median is 8.
Avg of smallest and greatest numbers is (1+18)/2 = 9.5

Answer (E)
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Re: Z is a set of positive numbers. The median of Z is greater [#permalink]

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Re: Z is a set of positive numbers. The median of Z is greater   [#permalink] 08 Apr 2018, 06:22
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