Set A, Set B, and Set C each contain only positive integers.

my answer is E

given: A = B + C
asking: medB > medA ?


(1) meanA > medB
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useless, the mean indicates nothing for finding medians
the mean can be easily misleading if one element is too large or too small

statement 1 is insufficient


(2) medA > medC
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although this could be a trap to make you think that the large medB cause medA > medC , the question says nothing about the number of elements in sets B and C. So, C could have like 10 elements while be has only 3 elements and thus the median of A could be larger than the median of B
on the other hand, B could have way more elements than C and thus resulting in a median of A smaller than the median of B

statement 2 is insufficient

(1) and (2) together
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is still not helpful as the mean in A leads to nothing

my answer is then E

OA ?

If Med(A) > Med(B), it tells me that the quantities in A are typically greater than B
For example Med(5,6,7,8,9) > Med(1,2,3,4,5,6,7,8,9)

Question to answer: is the median of Set B greater than the median of Set A?

(1) The mean of Set A is greater than the median of Set B.
This doesn't tell me much.
For example Mean(5,6,1000) > Med (7,8,9)

(2) The median of Set A is greater than the median of Set C.
Now this is something.
Lets take two lists.
Med(B)=Med(7,8,9,10,11) =9
Med (C) = Med(1,2,3,4,5,6,7) = 4
Now if we combine the two into set A, we should expect Med(C)<Med(A)<Med(B)
In our example, 6

Now lets change the balance between B and C
Med (B) = Med(7,8,9,10,11) =9
Med(C) = Med(12,13,14,15,16) = 14
Med (A) = Med(B,C) = 11.5
In other words, Med(B)<Med(A)<Med(C)

If we are told that Med(C) < Med(A), then Med(A) must be less than Med(B)
Sufficient.

I choose, B

OFFICIAL SOLUTION

From the question stem, we know that Set A is composed entirely of all the members of Set B plus all the members of Set C.

The question asks us to compare the median of Set A (the combined set) and the median of Set B (one of the smaller sets).

Statement (1) tells us that the mean of Set A is greater than the median of Set B. This gives us no useful information to compare the medians of the two sets. To see this, consider the following:

Set B: { 1, 1, 2 }
Set C: { 4, 7 }
Set A: { 1, 1, 2, 4, 7 }

In the example above, the mean of Set A (3) is greater than the median of Set B (1) and the median of Set A (2) is GREATER than the median of Set B (1).

However, consider the following example:

Set B: { 4, 5, 6 }
Set C: { 1, 2, 3, 21 }
Set A: { 1, 2, 3, 4, 5, 6, 21 }

Here the mean of Set A (6) is greater than the median of Set B (5) and the median of Set A (4) is LESS than the median of Set B (5).

This demonstrates that Statement (1) alone does is not sufficient to answer the question.

Let's consider Statement (2) alone: The median of Set A is greater than the median of Set C.

By definition, the median of the combined set (A) must be any value at or between the medians of the two smaller sets (B and C).

Test this out and you'll see that it is always true. Thus, before considering Statement (2), we have three possibilities

Possibility 1: The median of Set A is greater than the median of Set B but less than the median of Set C.

Possibility 2: The median of Set A is greater than the median of Set C but less than the median of Set B.

Possibility 3: The median of Set A is equal to the median of Set B or the median of Set C.

Statement (2) tells us that the median of Set A is greater than the median of Set C. This eliminates Possibility 1, but we are still left with Possibility 2 and Possibility 3. The median of Set B may be greater than OR equal to the median of Set A.

Thus, using Statement (2) we cannot determine whether the median of Set B is greater than the median of Set A.

Combining Statements (1) and (2) still does not yield an answer to the question, since Statement (1) gives no relevant information that compares the two medians and Statement (2) leaves open more than one possibility.

Therefore, the correct answer is Choice (E): Statements (1) and (2) TOGETHER are NOT sufficient.
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I think the condition you mentioned that I have highlighted above doesn't satisfy the condition Median A > Median C

Consider these scenarios,

1) Set B = {3,3,3,3,3} Median of B = 3
Set C = {1,2,3} Median of C = 2
Set A = {1,2,3,3,3,3,3,3} Median of Set A = 3

Satisfies condition specified in Statement 2 i.e. Median of Set A > Median of Set C

But Median of Set B = Median of Set A

2) Set B = {3,4,5,6,7} Median of B = 5
Set C = {1,2,3} Median of C = 2
Set A = {1,2,3,3,4,5,6,7} Median of Set A = 3.5

Satisfies condition specified in Statement 2 i.e. Median of Set A > Median of Set C

But Median of Set B > Median of Set A

Two answers --- Hence Insufficient

Consider Kudos if the post helps!!!

There are 2 imp points here Mean and Median are not related and another is the elements in any of the sets are not fixed so its just impossible to arrive at the answer.

Hi experts, this question is indeed very interesting. Could you please share some of your knowledge on properties for statistics in combined sets.
I think I remember that for example the median of a combined set has to be between the median of of the subsets combined.
But are there any other properties that might be helpful to remember?

Please let us know, or anyone.
Will be happy to throw some kudos out there

Probably easiest way to solve would be to construct different sets. Try it.
_________________

Notice that the correct answer is E, not B.
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Agree with E here. The mean and median are not related in any way that you can devise from the information given, and S1 and S2 provide only comparisons between their supposed relation.

I got E as well. The mean tells us nothing about the median unless there is another specification (eg. consecutive numbers,etc.)
The median could be any number since the question stem does not specify how many numbers are in the set.
for example [1,2,3,4,100,1000], the mean can specify a very large number while the median would be 3.5.

This is a tricky question and statistics is my weakest area so far.
Bunuel Can you please solve it in your style.

Ah! did not notice that. Thanks.

I would prefer algebra here..
Take two sets..
X1, X2, X3 - B
Y1, Y2, Y3 -C

With the possibility for Set C integers being the greater ones or Set B..There are other but..considering these 2 and the equations formed..It is enough to select an E with a cent percent surity

Hi Bunuel,

Yes answer should be E.

First chose B, by misreading the question as Median of B > Median of C ? instead of Median of B > Median of A?

So if the question had been median of B > median of C?

Statement1: Mean of Set A > Median of Set B,
considering examples:
1. set B: {4,5,6}, set C:{1,2,2000} => mean of A > median of B, also median of B > median of C
2. set B: {1,2,6}, set C:{4,5,2000} => mean of A > median of B, also now median of B < median of C

So insuff

Statement 2: Median of A > Median of C.

This means that Set B has more elements whose values are greater than median of C than elements whose values is less than median of C.
So this implies that, Set B must have the middle element(i.e. median of B) definitely above median of C. So this statement is sufficent

so B, is my reasoning correct?

(1) The mean of Set A is greater than the median of Set B.

Mean can be made extremely large by having just one very large value which will not affect the median so let's ignore this statement.

(2) The median of Set A is greater than the median of Set C.

Case I:
It is easy to see that median of set B could be greater than the median of set A.
C = {1, 2, 3} Median = 2
B = {4, 4, 50} Median = 4
A = {1, 2, 3, 4, 4, 50} Median = 3.5

Case II:
But can median of set B be equal to (or less than) median of set A?
C = {1, 2, 3}
B = {3, 3, 50} Median = 3
A = {1, 2, 3, 3, 3, 50} Median = 3

Not sufficient.

Using both statements together, mean of set A is greater than the median of set B in both cases above.

Hence, not sufficient.

Answer (E)

Note that I had taken set B with small numbers initially when I was evaluating just statement 2. But when I needed to evaluate both statements together, I simply bumped the last number up to 50 to give a large mean of set A. The median of B remained unchanged so I had to do no other adjustment.


Now check out this question on mean, median and range: https://anaprep.com/sets-statistics-mea ... -question/