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# If set S consists of even number of integers, is the median

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Re: If set S consists of even number of integers, is the median [#permalink]
If set S consists of even number of integers, is the median of set S negative?

1)Exactly half of all elements of set S are positive.
2)The largest negative element of set S is -1.

median for a set made of even number of intigers is the average of the middle 2

from 1

insuff

from 2

insuff

both
to get the median one has to arrange the set in ascending order then take the average of the middle 2

ie: the largest -ve + the least +ve / 2 , as long as we dont know the least +ve we 9 could be at least 1 or 2

median = (1-1)/2 = 0 or (2-1)/2 = 1/2 so we can say that the median is not -ve...suff

C
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Re: If set S consists of even number of integers, is the median [#permalink]
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definition of median:
In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is not unique, so one often takes the mean of the two middle values.

so in this case, we can have a set of -3 -2 -1 1 2 3, hence going by the rule stated above, the median of the set is (-1+1)/2 = 0, and 0 is neither positive nor negative, how is this one C?
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Re: If set S consists of even number of integers, is the median [#permalink]
sprtng wrote:
definition of median:
In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is not unique, so one often takes the mean of the two middle values.

so in this case, we can have a set of -3 -2 -1 1 2 3, hence going by the rule stated above, the median of the set is (-1+1)/2 = 0, and 0 is neither positive nor negative, how is this one C?

combining 1&2 we can get the median is either 0 or positive. for any of these case, median is non negative.
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Re: If set S consists of even number of integers, is the median [#permalink]
Guys,I am not understanding why you people are saying that the set consists of only 2 elements..the answer stands at C only:
eg.: {-2, -1 ,1 ,2} median=0. Is the median -ive?NO (o is neither positive nor negative)

{-3,-2,-1,2,3,4}Median=1/2.NO
{-3,-1,5,8} Median=2.NO. SUFFICIENT>C.

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Re: If set S consists of even number of integers, is the median [#permalink]
Agreed, its C

I thought it was B initially until you read the statement more carefully.

It says it consist of even number of intergers. So the set can have either 2,4,6,8 numbers.

1) insuff
2) insuff

1) + 2) = -1 is the highest negative value. So the next highest value thats positive is 1. In that case the median is 0. So no the median is not negative.

If you work with any number higher than 1, the median is positive. Therefore the median of set S is not negative.

C
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Re: If set S consists of even number of integers, is the median [#permalink]
 NOTE: 0 is neither positive nor negative..

That is the crucial piece of knowledge to remember. W/o it you'll be keep getting the median of -1/2, as I did
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Re: If set S consists of even number of integers, is the median [#permalink]
Hi Brunel,

In the underlined;
1) + (2) half positive, biggest negative -1 --> median is either (0 + positive)/2=positive (so not negative) or (-1 + at least smallest positive integer, which is 1)/2=0 (min value of median in this case) also not negative. Q was is the median negative answer NO. SUFFICIENT

The minumum positive number in the set is not mentioned in the question so I am thinking it could be any number for example 5. If so then the median becomes (-1+5)/2, a positive value.

Could you kindly clarify
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Re: If set S consists of even number of integers, is the median [#permalink]
manishgeorge wrote:
Hi Brunel,

In the underlined;
1) + (2) half positive, biggest negative -1 --> median is either (0 + positive)/2=positive (so not negative) or (-1 + at least smallest positive integer, which is 1)/2=0 (min value of median in this case) also not negative. Q was is the median negative answer NO. SUFFICIENT

The minumum positive number in the set is not mentioned in the question so I am thinking it could be any number for example 5. If so then the median becomes (-1+5)/2, a positive value.

Could you kindly clarify

Yes, that's the point the median could be positive but in no case it could be negative. So, we can answer NO to the question whether the median is negative.

Hope it's clear.
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Re: If set S consists of even number of integers, is the median [#permalink]
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tejal777 wrote:
Guys,I am not understanding why you people are saying that the set consists of only 2 elements..the answer stands at C only:
eg.: {-2, -1 ,1 ,2} median=0. Is the median -ive?NO (o is neither positive nor negative)

{-3,-2,-1,2,3,4}Median=1/2.NO
{-3,-1,5,8} Median=2.NO. SUFFICIENT>C.

I'm not quite sure I've understood your question. But I'll try to answer it anyways.

We are looking for the median, or more specifically - We are curious if it is negative(note at this point that 0 is neither negative nor positive).

We know from the original question that there is an even ammount of elements in the set. Since we are looking for the median we and the ammount of elements is even, then we are just interested in the middle two elements.

From statement one we can draw the following conclusion:
Half of the elements are positive, which means that the other half is either negative or zero(since zero is not positive).

With this information we know that the larger one of the middle numbers is 1 or higher. Now we need to know what the lower one of the two numbers is.

Which means (1) is insufficient. As the middle of the set could look like this: {..., -10, 1, ...} making the median negative.

With statement two(2), we find out that the largest of the negative numbers is -1. (Which alone is insufficient as we don't know if there's any positive numbers at all at this point)

But combined with (1), which told us that half was positive, we can start pinpointing the lower of the two middle numbers.

Posibility one - One half is positive, the rest of the numbers are negative (which would make the -1 the lower of the middle numbers, making the median 0(Which is not negative))

Posibility two - One half is positive, the rest consists of atleast one zero and negative numbers. This would make the zero larger than -1 and therefore the number we would use when calculating the median. For example{..., 0, 1, ...} making the median 0.5.

But given a range for the two numbers that actually matters to us we can solve this question.

(1) The larger of the two elements making up the median is larger than 0(since we know it's just integers that means 1 or higher).
(2) The smaller of the two elements making up the median is either -1 or 0.

Sorry about the long post and if I'm totally off the ball. It's my first post and I'm not much of a mathematician :p
Cheers

Edit: Noticed it had several pages just now, lol. Sorry about that tejal. I'll just leave this here if anyone finds my explanation useful.
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Re: If set S consists of even number of integers, is the median [#permalink]
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Official Explanation.

If set S consists of even number of integers, is the median of set S negative?

Notice that since set consists of even number of integers then the median will be the average of two middle elements.

(1) Exactly half of all elements of set S are positive --> either all other elements are negative for example S can be {-30, -20, -10, 1, 2, 3} or all other elements but 0 are negative, for example S can be {-3, -2, -1, 0, 1, 2, 3, 4}. Not sufficient.

(2) The largest negative element of set S is -1. Not sufficient

(1)+(2) We have that exactly half of all elements are positive and the largest negative element is -1. Two cases for the median:
If zero is in S the median equals to $$\frac{0 + positive}{2}=positive$$;
If zero is NOT is S the median equals to $$\frac{-1+positive \ integer}{2}\geq{0}=non \ negative$$.

So, the answer to the question whether the median is negative is NO. Sufficient.

Hope it helps.
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Re: If set S consists of even number of integers, is the median [#permalink]
I'm still a bit confused because the question stem says that Set S is comprised of even integers, so why would the smallest negative element in S be -1? Wouldn't the smallest element have to be at least -2 because it is even?
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Re: If set S consists of even number of integers, is the median [#permalink]
pitpat wrote:
I'm still a bit confused because the question stem says that Set S is comprised of even integers, so why would the smallest negative element in S be -1? Wouldn't the smallest element have to be at least -2 because it is even?

We are not told that set S consists of even integers, we are told that the number of integers in S is even (so there could be 2, 4, 6, ... elements in set S).

Hope it's clear.
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Re: If set S consists of even number of integers, is the median [#permalink]
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I am confused with the second statement:- The largest negative element in the set is -1. Shouldn't it be smallest negative integer?

When the largest negative integer is mentioned, does is it not mean that the set can't have a larger (than -1) integer, but have a smaller(-2,-100 etc) negative integer?
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Re: If set S consists of even number of integers, is the median [#permalink]
Hello tejal777,

I do not believe that the set has to consist of only 2 elements. The question states that the half of the elements in set S are positive and that the largest negative value is -1. However, it does not mention anything about the least possible negative value which can be -1,000,000 for all we care.
So the set can be {-x,-y,-1,0,a,b,c,d}
or {-x,-y,-z,-w,-1,a,b,c,d,e)

Both these sets satisfy all the conditions in the statements and the question. The sets consists of even number of integers. Exactly half of the elements in the sets are positive. The greatest possible negative integer in the sets are -1.

Hope this helps! Let me know if you have any further questions.

tejal777 wrote:
Guys,I am not understanding why you people are saying that the set consists of only 2 elements..the answer stands at C only:
eg.: {-2, -1 ,1 ,2} median=0. Is the median -ive?NO (o is neither positive nor negative)

{-3,-2,-1,2,3,4}Median=1/2.NO
{-3,-1,5,8} Median=2.NO. SUFFICIENT>C.

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Re: If set S consists of even number of integers, is the median [#permalink]
mirhaque wrote:
If set S consists of even number of integers, is the median of set S negative?

(1) Exactly half of all elements of set S are positive
(2) The largest negative element of set S is -1

(1) Exactly half of all elements of set S are positive
{-1,1} or {-1,10} or {-10,1} => imply median can be zero or positive or negative. S with 4 or more will make median positives though; so, I omit those cases.
=> Insufficient
(2) The largest negative element of set S is -1
S can be {-1,0} or {-1,-1} or {-1,1} or {-1,2}. Median can be negative, zero (or even positive).

Now, using both statements:
S can be {-1,1} or {-1,-1,1,1} or {-10,-1,1,10}, {-1,10} and so on... Median, as we see can be either zero or positive but not negative. Hence, C.
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Re: If set S consists of even number of integers, is the median [#permalink]
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I'm getting hung up on the phrase "Largest Negative number"

Interpretation of that phrase is why people are arguing either E or C.

Does that mean that the possible negative numbers are less than -1? That's how I translate it. In other words, if -1 is the largest negative number, then we could also have -2, -3, -4 etc.. in the set. In essence, it's a meaningless statement since the number has to be an integer anyway and there is no negative integer "larger" than -1. Treating it this way would lead to answer E.

If, however, you are interpretting it as the largest integer that is then made negative (IE the the rest of the numbers in the set must be >-1), then that would lead to answer C.

The literal meaning, as i take it, leads to answer E... but I misread the question the first time around and answered C, and only noticed the difference in the different poster's responses.
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Re: If set S consists of even number of integers, is the median [#permalink]
dave785 wrote:
I'm getting hung up on the phrase "Largest Negative number"

Interpretation of that phrase is why people are arguing either E or C.

Does that mean that the possible negative numbers are less than -1? That's how I translate it. In other words, if -1 is the largest negative number, then we could also have -2, -3, -4 etc.. in the set. In essence, it's a meaningless statement since the number has to be an integer anyway and there is no negative integer "larger" than -1. Treating it this way would lead to answer E.

If, however, you are interpretting it as the largest integer that is then made negative (IE the the rest of the numbers in the set must be >-1), then that would lead to answer C.

The literal meaning, as i take it, leads to answer E... but I misread the question the first time around and answered C, and only noticed the difference in the different poster's responses.

OA for this question is C, not E.

-1 is the largest negative integer: all other negative integers are less than -1.
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