Is the median greater than the mean? (GMAT PREP 1)

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alex.gellatly: Master | Next Rank: 500 Posts; Posts: 435; Joined: Wed Nov 16, 2011 7:27 amFollowed by: 16 members; Thanked: 48 times

by alex.gellatly » Fri Aug 03, 2012 11:04 pm

x,3,1,12,8

If x is an integer, is the median of the 5 numbers shown greater than the average (arithmetic mean) of the 5 numbers?

1. x>6
2. x is greater then the median of the 5 numbers.

Thanks

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Source: Beat The GMAT — Data Sufficiency | Fri Aug 03, 2012 11:04 pm

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Aug 04, 2012 3:24 am

alex.gellatly wrote:x,3,1,12,8

If x is an integer, is the median of the 5 numbers shown greater than the average (arithmetic mean) of the 5 numbers?

1. x>6
2. x is greater then the median of the 5 numbers.

Thanks

Mean of 5 numbers = (x + 1 + 3 + 8 + 12)/5 = (x + 24)/5
Question is: Is Median > (x + 24)/5?

(1) x > 6
If x = 7, then the set of numbers is 1, 3, 7, 8, 12. Here mean = (7 + 24)/5 = 6.2 and median = 7. So, here median (= 7) > mean (= 6.2)
If x = 26, then the set of numbers is 1, 3, 8, 12, 26. Here mean = (26 + 24)/5 = 10 and median = 8. So, here median (= 8) < mean (= 10)
No definite answer; NOT sufficient.

(2) x is greater then the median of the 5 numbers implies median = 8
If x = 11, then mean = (11 + 24)/5 = 7. Here median (= 8) > mean (= 7).
If x = 26, then mean = (26 + 24)/5 = 10. Here median (= 8) > mean (= 10).
No definite answer; NOT sufficient.

Combining (1) and (2), we can take the same examples as in statement 2; NOT sufficient.

The correct answer is E.

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Nina1987: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Tue Dec 15, 2015 3:56 amFollowed by: 1 member; GMAT Score:750

by Nina1987 » Wed Dec 23, 2015 8:11 am

Compared to the median the mean has greater flexibility in terms of how big it can get. So for the median to be greater than the mean we need to limit the upward movement of the mean by restricting the value of x. Thus we need a constraint such as x<(certain number) and since neither of the choices does so answer is E.

For instance, even if we were given x>0
then say x=1 {1 1 3 8 12} 3>25/5? NO
now say x=8 {1 3 7 8 12} 8>(24+8)/5? Yes

on the other hand, say if we were given x<6, the answer would be a definitive No

Does that make sense?
What do others/experts think?

Last edited by Nina1987 on Wed Dec 30, 2015 5:41 am, edited 1 time in total.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Sun Dec 27, 2015 6:02 pm

Nina1987 wrote:Mean has a greater flexibility in terms of how big it can get. So for the median to be greater than the mean we need to limit the upward movement of the mean by restricting the value of x. Thus we need a constraint such as x<(certain number) and since neither of the choices does so answer is E.

For instance, even if we were given x>0
then say x=1 {1 1 3 8 12} 3>25/5? NO
now say x=8 {1 3 7 8 12} 8>(24+8)/5? Yes

on the other hand, say if we were given x<6, the answer would be a definitive No

Does that make sense?
What do others/experts think?

I think you've got the right idea, but there might be easier way of putting this.

We know the mean is (24 + x)/5, or 4.8 + (x/5).

We know the median is either 3, x, or 8.

So our question becomes

"Is {3, x, or 8} > 4.8 + x/5?"

If x is the median, the question becomes

"Is x > 4.8 + x/5?"

or

"Is x > 6?"

But if 8 is the median, the question becomes

"Is 8 > 4.8 + x/5?"

or

"Is 16 > x?"

S1 tells us that either x or 8 is the median, but this only gets us partly there; NOT SUFFICIENT.

S2 tells us that 8 is the median, so we're closer: now our question is "Is 16 > x?"; NOT SUFFICIENT.

Together, we still want to know if 16 > x, and we still can't say; NOT SUFFICIENT.

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