Set S contains more than one element. Is the range S bigger

Question

Set S contains more than one element. Is the range S bigger than its mean?

A. Set S does not contain positive elements 
B. The median of the set S is negative

walker · Answer

B

A. Set S does not contain positive elements

S={-2,-1} ==> Range=1 > Mean=-1.5 but
S={0,0} ==> Range=0 > Mean=0
insufficient.

B. The median of the set S is negative
Let X is the larger number in S and Y is the smallest number in S.

If X<0 then Mean<0 but Range is always >=0
if X>=0 then Range>=X+1 but Mean is always < X
sufficient.

sreehari · Answer

D

A - Range is always positive. Since all elements are negative, mean will be negative and will always be lessthan range
B - Two scenarios - all elements are negative in which case range is always greaterthan mean. if half the elements are negative, mean will be either negative or lessthan the highest number which will be lessthan range again.

GMAT TIGER · Answer

does the question say "all elements of S are -ve"?

sreehari · Answer

My understanding is, by definition set contains unique elements. And the questons said S has morethan one element. So for A, S must have contain negative number.

sreehari · Answer

Actually I checked Wikipedia when I ran into such situation before. Here is a copy and paste...

Unlike a multiset, every element of a set must be unique; no two members may be identical.

Now it's getting interesting... what is OA?

prov · Answer

I'd say D too
I don't think {0,0} can be considered as set with 2 elements
so in 1) the set must contain negative element(s)

walker · Answer

It is stange... in OG11th p.155 in Math Review the authors use a few times "set" to refer to lists of numbers with repetitions... 
I prefer OG

walker · Answer

I found this: 7-t24270
I understand that set has to be contain different numbers according to the Set Theory. But I'm not sure what exactly GMAT think about that. And how we can explain "free" use of "set" in OG.....

Tiger, What is source of the question?

GMAT TIGER · Answer

wikipidia says {6, 11} is same as {11, 6} or as {6, 6, 6, 6, 6, 6, 11, 11}.

but do they ave same mean, range or median?

walker · Answer

more info: OG11 PS#65 a list = a list of numbers with repetitions OG11 PS#212 a data set = a list of numbers with repetitions I confused

Any thoughts?

hobbit · Answer

"data set" can indeed contain repetitions....but "set of numbers" never contain repetitions. as noted before, as sets all of the following are equal: {6,11} {6,6,11} {11,6} {11,6,11,6,6,6,11}

the answer is D, and actually walker's brilliant and correct proof of why (2) is sufficient, works for (1) as well. I'll just repeat it in more general terms:

let X be the maximal number in S. if X<0 then the mean is -ve and range is always +ve.
if X>0 and if there is a -ve number in S, then range is bigger than X, but mean is always less than X
both(1) and (2) in the question independently tells us that there is at least 1 negative element in S, and hence sufficient.

GMAT TIGER · Answer

hobbit has a very good point and walker has some OG examples. I also believe we should follow what OG says but here one doubt I have.

Does the current gmat vender (Pearson VUE) follow OG rules? Since OG is not a product of Pearson VUE, how strictly it follows OG rules while crafting the real Gmat questions?

OA: I have B and D, both, in my note.

sreehari · Answer

I haven't opened my OG yet, so can't comment on what's in there. But obviously if OG says duplication is allowed, that's what we should do in GMAT

Being from engineering background, I always considered set by definition as limited set of distinct objects, with an exception of list of numbers specifically listed in the problem (e.g {1,2,2,3,4}) in which case they are technically called multiset. Another example is something like a set defined by {x|1<=x<=10, x is even} - the correct way of interpreting this is to list down all even integers lessthan 10 {2,4,6,8,10}. It should not be assumed that it's an infinite list of elements. With that being said, I haven't seen any GMAT match question yet that refered a set as multiset, so probably it's not in GMAT's dictionary. So trust OG and move forward.

Set S contains more than one element. Is the range S bigger

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