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Retired Moderator B
Joined: 27 Aug 2012
Posts: 1081
Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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12 00:00

Difficulty:   95% (hard)

Question Stats: 42% (01:58) correct 58% (02:20) wrong based on 278 sessions

### HideShow timer Statistics 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. II only
B. III only
C. II and III
D. I, II and III
E. None of the above

Bunuel- How to identify the plug-ins for this type of questions?

Is there any other approach to solve this efficiently instead of 'plug-in and check'?

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Originally posted by bagdbmba on 07 Aug 2013, 21:49.
Last edited by Bunuel on 26 Dec 2018, 11:49, edited 2 times in total.
Edited the question.
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Joined: 02 Sep 2009
Posts: 56251
Re: Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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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.

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Re: Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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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.
Consider {2,5,6}.
II) Less than 50% of the numbers in Z are greater than the median.
Consider {1,4,7,8}
III) The median of Z is greater than the average of the largest and smallest numbers in Z.
Consider {1,2,4,5,7}

E)None of the above

I would say that plugging number is the fastest way to solve those, rather than a theoretical approach.

bagdbmba wrote:
Zarrolou - Great that you've inserted my query related to the above problem in the main post but I willingly posted it differently to keep the question tag neat & clean...!

P.S : May be I'm wrong,but I guess Bunuel doesn't prefer to see that sort of comments along with qs As long as the comment is not a spoiler (why A? for example) but it's a question, I think it's fine.
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##### General Discussion
Retired Moderator B
Joined: 27 Aug 2012
Posts: 1081
Re: Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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Zarrolou - Great that you've inserted my query related to the above problem in the main post but I willingly posted it differently to keep the question tag neat & clean...!

P.S : May be I'm wrong,but I guess Bunuel doesn't prefer to see that sort of comments along with qs _________________
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Re: Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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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 than the me  [#permalink]

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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

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Re: Z is a set of positive numbers. The median of Z is greater than the me  [#permalink]

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_________________ Re: Z is a set of positive numbers. The median of Z is greater than the me   [#permalink] 26 Dec 2018, 11:50
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