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# What is the median of the numbers 4, 5, 6, 7, 9, and x?

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What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  01 Nov 2012, 14:43
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52% (02:11) correct 48% (01:08) wrong based on 130 sessions
What is the median of the numbers 4, 5, 6, 7, 9, and x?

(1) x > 7
(2) The mean of the six numbers is equal to their median.

[Reveal] Spoiler:
The solution states that the answer is A but in my opinion the answer is D.

The mean of a sequence is equal to the median of that sequence. Which means that x can only be 8 according to Stat(2)- (in order to make the mean of that set the same as the median). The numbers that the solution cites, to me, seems invalid. For example, the solution states x<5. Let's take 4 then. If x=4, the mean becomes 5.83, but the median is 5.5, which doesn't satisfy Stat(2). Could someone clarify this?

Here is the official solution:
This is a "what is the value of..." DS question. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) that you're asked about.

Remember:

The median is the middle number in a set of numbers, arranged in ascending or descending order.
To find the median consider the number of elements:
If the number of elements is odd, the median is the middle number.
If the number of elements is even the median is the average of the middle two elements.
Together with x, there are 6 numbers; therefore, the median will be calculated as the average of the two middle numbers. Therefore, the real issue of the question is the values of the two middle numbers.

According to Stat. (2),

The average of 4,5,6,7,9, and x is equal to the median. The median, which is also the average, will vary according to the value of x:

If x<5, the median is equal to the average of the two middle numbers --> 5 and 6 = 5.5. But,

If x>7, the median is equal to the average of the two middle numbers --> 6 and 7 = 6.5. No single value can be determined for the median of the set, so Stat.(2)->IS.
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Nov 2012, 14:47, edited 1 time in total.
Renamed the topic and edited the question.
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Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  01 Nov 2012, 15:10
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Expert's post
What is the median of the numbers 4, 5, 6, 7, 9, and x?

The median of a set with even number of terms is the average of two middle terms when arranged in ascending/descending order.

(1) x > 7 --> two middle terms are 6 and 7, thus median=(6+7)/2=6.5. Sufficient.

(2) The mean of the six numbers is equal to their median.

If $$x\leq{5}$$, then two middle terms are 5 and 6, thus median=(5+6)/2=5.5. In this case mean=(4+5+6+7+9+x)/6=5.5 --> x=2. Possible scenario.
If $$x\geq{7}$$, then two middle terms are 6 and 7, thus median=(6+7)/2=6.5. In this case mean=(4+5+6+7+9+x)/6=6.5 --> x=8. Possible scenario.
Not sufficient.

P.S. For (2): if $$5<x<7$$, then two middle terms are x and 6, thus median=(x+6)/2. In this case mean=median=(4+5+6+7+9+x)/6=(x+6)/2 --> x=6.5. Also, possible scenario.
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Kudos [?]: 62445 [0], given: 9455

Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  01 Nov 2012, 15:13
Expert's post
anon1 wrote:
What is the median of the numbers 4, 5, 6, 7, 9, and x?

(1) x > 7
(2) The mean of the six numbers is equal to their median.

[Reveal] Spoiler:
The solution states that the answer is A but in my opinion the answer is D.

The mean of a sequence is equal to the median of that sequence. Which means that x can only be 8 according to Stat(2)- (in order to make the mean of that set the same as the median). The numbers that the solution cites, to me, seems invalid. For example, the solution states x<5. Let's take 4 then. If x=4, the mean becomes 5.83, but the median is 5.5, which doesn't satisfy Stat(2). Could someone clarify this?

Here is the official solution:
This is a "what is the value of..." DS question. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) that you're asked about.

Remember:

The median is the middle number in a set of numbers, arranged in ascending or descending order.
To find the median consider the number of elements:
If the number of elements is odd, the median is the middle number.
If the number of elements is even the median is the average of the middle two elements.
Together with x, there are 6 numbers; therefore, the median will be calculated as the average of the two middle numbers. Therefore, the real issue of the question is the values of the two middle numbers.

According to Stat. (2),

The average of 4,5,6,7,9, and x is equal to the median. The median, which is also the average, will vary according to the value of x:

If x<5, the median is equal to the average of the two middle numbers --> 5 and 6 = 5.5. But,

If x>7, the median is equal to the average of the two middle numbers --> 6 and 7 = 6.5. No single value can be determined for the median of the set, so Stat.(2)->IS.

Similar questions to practice:
if-set-s-consists-of-the-numbers-1-5-2-8-and-n-is-132570.html
what-is-the-value-of-n-in-the-list-above-137225.html
if-the-range-of-the-set-containing-the-numbers-x-y-and-z-127089.html
the-sum-of-the-integers-in-list-s-is-the-same-as-the-sum-of-127755.html
given-the-ascending-set-of-positive-integers-a-b-c-d-e-115675.html

Hope it helps.
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Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  01 Nov 2012, 15:33
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Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  24 Oct 2013, 06:21
Bunuel wrote:
What is the median of the numbers 4, 5, 6, 7, 9, and x?

The median of a set with even number of terms is the average of two middle terms when arranged in ascending/descending order.

(1) x > 7 --> two middle terms are 6 and 7, thus median=(6+7)/2=6.5. Sufficient.

(2) The mean of the six numbers is equal to their median.

If $$x\leq{5}$$, then two middle terms are 5 and 6, thus median=(5+6)/2=5.5. In this case mean=(4+5+6+7+9+x)/6=5.5 --> x=2. Possible scenario.
If $$x\geq{7}$$, then two middle terms are 6 and 7, thus median=(6+7)/2=6.5. In this case mean=(4+5+6+7+9+x)/6=6.5 --> x=8. Possible scenario.
Not sufficient.

P.S. For (2): if $$5<x<7$$, then two middle terms are x and 6, thus median=(x+6)/2. In this case mean=median=(4+5+6+7+9+x)/6=(x+6)/2 --> x=6.5. Also, possible scenario.

why have you taken mean =median i.e 5.5 and 6.5 .
Mean =Median is only in evenly spaced sets.
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Posts: 31297
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Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  24 Oct 2013, 06:35
Expert's post
kop wrote:
Bunuel wrote:
What is the median of the numbers 4, 5, 6, 7, 9, and x?

The median of a set with even number of terms is the average of two middle terms when arranged in ascending/descending order.

(1) x > 7 --> two middle terms are 6 and 7, thus median=(6+7)/2=6.5. Sufficient.

(2) The mean of the six numbers is equal to their median.

If $$x\leq{5}$$, then two middle terms are 5 and 6, thus median=(5+6)/2=5.5. In this case mean=(4+5+6+7+9+x)/6=5.5 --> x=2. Possible scenario.
If $$x\geq{7}$$, then two middle terms are 6 and 7, thus median=(6+7)/2=6.5. In this case mean=(4+5+6+7+9+x)/6=6.5 --> x=8. Possible scenario.
Not sufficient.

P.S. For (2): if $$5<x<7$$, then two middle terms are x and 6, thus median=(x+6)/2. In this case mean=median=(4+5+6+7+9+x)/6=(x+6)/2 --> x=6.5. Also, possible scenario.

why have you taken mean =median i.e 5.5 and 6.5 .
Mean =Median is only in evenly spaced sets.

That's not true. Consider: {0, 1, 1, 2} --> mean=1=median.

So, if a set is evenly spaced, then mean=median, but if mean=median, then it's not necessary the set to be evenly spaced.

Does this make sense?
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Re: What is the median of the numbers 4, 5, 6, 7, 9, and x? [#permalink]  24 Oct 2013, 09:36
Bunuel wrote:
kop wrote:
Bunuel wrote:
What is the median of the numbers 4, 5, 6, 7, 9, and x?

The median of a set with even number of terms is the average of two middle terms when arranged in ascending/descending order.

(1) x > 7 --> two middle terms are 6 and 7, thus median=(6+7)/2=6.5. Sufficient.

(2) The mean of the six numbers is equal to their median.

If $$x\leq{5}$$, then two middle terms are 5 and 6, thus median=(5+6)/2=5.5. In this case mean=(4+5+6+7+9+x)/6=5.5 --> x=2. Possible scenario.
If $$x\geq{7}$$, then two middle terms are 6 and 7, thus median=(6+7)/2=6.5. In this case mean=(4+5+6+7+9+x)/6=6.5 --> x=8. Possible scenario.
Not sufficient.

P.S. For (2): if $$5<x<7$$, then two middle terms are x and 6, thus median=(x+6)/2. In this case mean=median=(4+5+6+7+9+x)/6=(x+6)/2 --> x=6.5. Also, possible scenario.

why have you taken mean =median i.e 5.5 and 6.5 .
Mean =Median is only in evenly spaced sets.

That's not true. Consider: {0, 1, 1, 2} --> mean=1=median.

So, if a set is evenly spaced, then mean=median, but if mean=median, then it's not necessary the set to be evenly spaced.

Does this make sense?

Thanks Bunnel. I understood it now.
Re: What is the median of the numbers 4, 5, 6, 7, 9, and x?   [#permalink] 24 Oct 2013, 09:36
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