Set S

[mmslf75]

Source:
GMATCLub
If set S consists of even number of integers, is the median of set negative?
1. Exactly half of all elements of set S are positive.
2. The largest negative element of set S is -1.


OA is C

Query :

In this type can i take S to be { -6, -5,1,2} or {-2,-1,1,2} or {1,-5,2,-6}

Is it that in SET type questions asking MEDIAN, the numbers chosen should be arranaged in ASCENDING order ?? before proceeding ???
How does statement 2 help here.?

[mmslf75]

IMO E
However OA C


My approach.. testing for C and E options
0 cannot be the 3rd element as question is Postive=Negative and 0 doesnot fit the criteria...


{ -8 , -1 ,2, 1 } = -0.5 = is negative ... ( note : the question doesnot say "IS MEDIAN a NEGATIVE INTEGER.. implies, it is a
negative number )
therefore for above set meeting st1 and st2 Answer is YES

consider,
{-3,-1,1,2} = 0 = Neither +ve nor -ve

therfoer for above set meeting st1 and st 2... Answer is NO

Wat am I missing here,...

[Stuart@KaplanGMAT]

{ -8 , -1 ,2, 1 } = -0.5 = is negative ... ( note : the question doesnot say "IS MEDIAN a NEGATIVE INTEGER.. implies, it is a
negative number )
therefore for above set meeting st1 and st2 Answer is YES

Wat am I missing here,...

The problem is that the median of that set is NOT -0.5.

{-8, -1, 2, 1} needs to be reordered to:

{-8, -1, 1, 2} and has a median of 0.

If you meant 2 to be the smallest positive, then we could look at the set:

{-8, -1, 2, 11} which has a median of +0.5, not -0.5.

When we combine the statements, we know that the number to the right of the median is positive (since exactly half of the terms are positive and since there are an even number of terms) and that the term to the left of the median is either -1 or 0 (since -1 is the biggest negative number in the set).

So, there are only two possibilities:

(1) the median is the average of -1 and a positive integer, giving us a value >=0; and

(2) the median is the average of 0 and a positive integer, giving us a value > 0.

In BOTH cases, the median is definitely non-negative, providing a definite NO answer to the original question, and thereby sufficient.

[mmslf75]

{ -8 , -1 ,2, 1 } = -0.5 = is negative ... ( note : the question doesnot say "IS MEDIAN a NEGATIVE INTEGER.. implies, it is a
negative number )
therefore for above set meeting st1 and st2 Answer is YES

Wat am I missing here,...

The problem is that the median of that set is NOT -0.5.

{-8, -1, 2, 1} needs to be reordered to:

{-8, -1, 1, 2} and has a median of 0.

If you meant 2 to be the smallest positive, then we could look at the set:

{-8, -1, 2, 11} which has a median of +0.5, not -0.5.

When we combine the statements, we know that the number to the right of the median is positive (since exactly half of the terms are positive and since there are an even number of terms) and that the term to the left of the median is either -1 or 0 (since -1 is the biggest negative number in the set).

So, there are only two possibilities:

(1) the median is the average of -1 and a positive integer, giving us a value >=0; and

(2) the median is the average of 0 and a positive integer, giving us a value > 0.

In BOTH cases, the median is definitely non-negative, providing a definite NO answer to the original question, and thereby sufficient.

But stuart, 0 is neither positive nor neagtive na ?

[Stuart@KaplanGMAT]

But stuart, 0 is neither positive nor neagtive na ?

That's correct. 0 is non-negative and non-positive.

The question is : is the median negative?

0 is non-negative, so, the median is definitely NOT negative.

[mmslf75]

But stuart, 0 is neither positive nor neagtive na ?

That's correct. 0 is non-negative and non-positive.

The question is : is the median negative?

0 is non-negative, so, the median is definitely NOT negative.

You have mentioned : """""(2) the median is the average of 0 and a positive integer, giving us a value > 0. """"""

How can we consider 0 and a + ve... wont this mean that the set S is { -1(say),0,1(say),2(say) } ??

Also, why we take the case of 0 here, ?

[Stuart@KaplanGMAT]

You have mentioned : """""(2) the median is the average of 0 and a positive integer, giving us a value > 0. """"""

How can we consider 0 and a + ve... wont this mean that the set S is { -1(say),0,1(say),2(say) } ??

Also, why we take the case of 0 here, ?

Correct, that's one possible set that fits both statements; that particular set has a median of +0.5.

We take the case of 0 because in data sufficiency, we need to consider all possibilities and whether different cases lead to different answers to the question.

Since this is a positive/negative question, the three cases that are of most interest are negative, 0 and positive.

Based on statement (1), we know that half of the elements are positive, which does NOT necessarily mean that half of them are negative, since 0 is neither positive nor negative. So, I looked at the two possibilities: the number to the left of the median being negative and the number to the left of the median being 0.

In this instance they both lead to the same answer to the original question ("no"), which is why combined, the statements are sufficient.

To explicitly answer a question from your original post (which I only addresses implicitly): Yes, when working with properties of sets (median, mode, range, standard deviation, etc...) we always want to reorder the set in either ascending or descending order before attacking the question.

[adamsmith2009]

What if you have two integers? The question stem said only even number of integers.

Let's say (-1,10) or (-5,5) then you would have a negative median, no?

[Stuart@KaplanGMAT]

What if you have two integers? The question stem said only even number of integers.

Let's say (-1,10) or (-5,5) then you would have a negative median, no?

The median of a set with an even number of terms is the average (mean) of the two middle terms.

The mean/median of {-1, 10} is 9/2 = 4.5; the mean/median of {-5, 5} is 0/2 = 0. Neither of those medians are negative.