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

tagged by: M7MBA

### Top Member

M7MBA Moderator
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#### If set S consists of even number of integers, is the median

Sat Dec 02, 2017 5:24 am
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

The OA is C.

I don't know how to solve this DS question. I am confused.

### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
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Mon Dec 04, 2017 11:46 am
Hi M7MBA,

We're told that set S consists of EVEN number of INTEGERS. We're asked if the MEDIAN of set S negative. This is a YES/NO question. This is actually a great 'concept' question (meaning that you don't actually have to do a lot of 'math' to answer it IF you understand the concepts involved. We can solve it by TESTing VALUES. It's worth noting that the MEDIAN of an even number of terms is the AVERAGE of the two 'middle' terms (when the terms are placed in order from least to greatest).

1) Exactly half of all elements of set S are positive

Since there are an even number of terms - and Fact 1 tells us that HALF of the terms are positive - the other half of the terms are either negatives or 0s.
IF...
Set S = {0, 1} then the median = 1/2 and the answer to the question is NO.
Set S = {-3, 1} then the median = -1 and the answer to the question is YES.
Fact 1 is INSUFFICIENT

2) The largest negative element of set S is -1

IF...
Set S = {-1, 1} then the median = 0 and the answer to the question is NO.
Set S = {-3, -1} then the median = -2 and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we know:
-Exactly half of all elements of set S are positive
-The largest negative element of set S is -1

Half of the numbers are POSITIVE and the LARGEST negative element is -1. If the smallest positive element was +1, then the median would either be the average of +1 and -1 (which is 0 and the answer to the question is NO) or the average of +1 and 0 (which is 1/2 and the answer to the question is still NO). As the smallest positive number of the set increases, the answer to the question remains the same (the median will NEVER be negative), so the answer to the question is ALWAYS no.
Combined, INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

### Top Member

Marty Murray Legendary Member
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Sat Dec 02, 2017 11:50 am
Question: If set S consists of even number of integers, is the median of set S negative?

First, we have to notice that set S consists of

- an even number of elements.

- integers.

Now, regarding the median of set S, since S has an even number of elements, the median of set S will be the mean, aka the average, of the two middle elements.

For example, if set S were {2, 4, 8, 10}, the median would be the mean of the two middle values, 4, and 8. So the median would be (4+8)/2 = 6.

Now , let's go to the statements.

Statement 1: Exactly half of all elements of set S are positive

In order to determine the median of set S, we would use the two middle values. This statement does not provide information sufficient for determining whether the median determined using the two middle values would be negative. Either of the following sets would fit this statement and the information provided in the question.

{- 100, 0, 1, 4}

Median = (0 + 1)/2 = 1/2

The median is positive.

{-13, -3, 1, 76}

Median = (-3 + 1)/2 = -1

The median is negative.

Insufficient.

Statement 2: The largest negative element of set S is -1

The following two sets would fit the parameters given in the question and the information provided in this statement.

{-10, -8, -4, -1}

Median = (-8 -4)/2 = -6

The median is negative.

{-1, 2, 4, 9}

Median = (2 + 4)/2 = 3

Median is positive.

Insufficient.

Statements Combined:

When we combine the statements, we know the following.

Half of the elements are positive. So, the other half of the elements must be either negative or 0.

The greatest negative element is -1.

Here's what you have to see to get this one right.

If half of the elements are positive, and they are integers, as the question says that they are, then the lowest possible element of half the elements is 1.

Since the rest of the elements are either negative or zero, and the greatest negative value is -1, then the greatest element of the rest of the elements is either 0 or -1.

Since the lowest element of the top half and the greatest element of the bottom half of the elements are the two middle values of our set, we can use them to calculate the median.

The lowest possible median will occur when the lowest value in the top half of the elements and the highest value in the bottom half of the elements are as low as possible.

Lowest possible top half bottom value: 1

Lowest possible bottom half top value: -1

Median: (-1 +1)/2 = 0

The lowest possible median is not negative.

Sufficient.

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