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The mode of a set of integers is x. what is the difference

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The mode of a set of integers is x. what is the difference [#permalink] New post 20 Sep 2012, 04:05
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The mode of a set of integers is x. what is the difference between the median of this set of integers and x?

(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers s x.
[Reveal] Spoiler: OA
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Re: The mode of a set of integers is x. what is the difference [#permalink] New post 20 Sep 2012, 10:01
ChrisGMATPrepster wrote:
This question seems to be flawed in my mind.

Statement 1 intends to tell you that this is a consecutive set of multiples of 3 (the consecutive nature would make the mean equal to the median)

Statement 2 intends to tell you that the mode is equal to the average.

This leads you to C

HOWEVER, this information cannot be true because all numbers would be the mode if they were each separated by 3. So the true answer is E, but the information is not presented correctly.


Statement 1 intends to tell you that this is a consecutive set of multiples of 3 - NO.
The statement is "the difference between any two integers in the set is less than 3"
Because all the numbers are integers, then the positive difference between any two can be 0,1 or 2.
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Last edited by EvaJager on 20 Sep 2012, 10:25, edited 1 time in total.
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Re: The mode of a set of integers is x. what is the difference [#permalink] New post 20 Sep 2012, 12:37
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ChrisGMATPrepster wrote:
HOWEVER, this information cannot be true because all numbers would be the mode if they were each separated by 3. So the true answer is E, but the information is not presented correctly.


I guess someone already pointed out that Statement 1 does not tell you your numbers are consecutive multiples of 3. It says that no two numbers are more than 2 apart. And in your comment following the 'HOWEVER', if every element in a set occurs exactly once, then the set is said to have no mode at all. For a set to have a mode, some element needs to appear more often than at least one other element. So sets like {1, 3, 5} and {3, 3, 4, 4} have no mode, because they do not contain any element which appears more often than some other element in the set.

In this question, neither statement is sufficient alone. For Statement 1, our set could be {1, 2, 2, 3}, and then the median and mode are equal, or it could be {1, 2, 3, 3}, and our median and mode are different. For Statement 2, again our set could be {1, 2, 2, 3}, and our median and mode can be equal, but our set could be {1, 2, 3, 4, 4, 10}, and our median and mode are different.

Now, using both statements together, if the mean is equal to the mode, then the mean must be equal to some value in the set. Technically, if all the values were the same, there'd be no mode, and we cannot have only two values exactly 1 apart, because then the mean would not be an integer, and thus would not be in the set. So we must have at least two values which are exactly 2 apart. Let's call them s-1 and s+1. So, our set has some elements equal to s-1, possibly some elements equal to s, and some elements equal to s+1, for some integer s. Notice now that the mean needs to be in the set, so must be s-1, s or s+1. But the mean can't be s-1, since s-1 is the smallest element in the set, and we have elements larger than s-1 in the set. Similarly the mean can't be s+1. So the mean, and therefore the mode, need to be equal to s. And for the mean to be equal to s, the number of elements equal to s-1 must be equal to the number of elements equal to s+1. So our set must be symmetric, and the median must also be s. So the median and mode are the same, and the two Statements together are sufficient.

edit: I'd add that I don't think I've ever seen a real GMAT question that even mentions the mode, let alone one as tedious to solve as this question, so it probably isn't all that important to study.
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Re: The mode of a set of integers is x. what is the difference [#permalink] New post 20 Sep 2012, 12:59
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Smita04 wrote:
The mode of a set of integers is x. what is the difference between the median of this set of integers and x?

1) the difference between any two integers in the set is less than 3.
2) the average of the set of integers s x.



(1) Consider the following two sets:
{x, x, x} and (x, x, x+1, x+2}
Not sufficient.

(2) Now consider the following two sets:
{x, x, x} and {x-3, x-2, x-1, x, x, x+6}
Again, not sufficient.

(1) and (2) together:
If all the numbers in the set are equal to x, then the difference between the median and the mode x is 0.
If not all the numbers are equal to x and because the average is x, there must be some numbers below as well as above the average. Since the range cannot be greater than 2, additional values in the set must be x-1 and x+1. In order to obtain the average x, we must have the number of terms equal to x-1 the same as the number of terms equal to x+1. So, our set consists of a symmetrical set of numbers around x.
Therefore, the median is x, and the requested difference is 0.

Sufficient.

Answer C.
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Re: The mode of a set of integers is x. what is the difference [#permalink] New post 21 Sep 2012, 03:32
St1:
Let set of integers be {x,x,x+1,x+2,x+2}
mode=x
median=x+1
diff=1
Another set of intergers {x-1,x,x,x,x+1}
mode=x
median=x
diff=0-----two diff values(Hence Insufficient)
St2:
Let set of integers be {x-5,x,x,x,x+5}
mode=x
median=x
diff=0
Another set of integers be {x-15,x,x,x+1,x+2,x+3,x+9}
mode=x
median=x+1
diff=1---two diff values(Hence Insufficient)

St1&St2:Let the set of integers be {x-1,x,x,x,x+1}
mode=x
median=x
diff=0
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Mode - level:700-800 [#permalink] New post 13 Nov 2012, 21:19
The mode of a set of integers is x. What is the difference between the median of this set of integers and x?
(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers is x.

Please explain. Thanks !

Last edited by mneeti on 13 Nov 2012, 23:30, edited 1 time in total.
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Re: Mode - level:700-800 [#permalink] New post 13 Nov 2012, 22:27
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mneeti wrote:
The mode of a set of integers is x. What is the difference between the median of this set of integers and x?
(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers is x.

The OA is C, but I do not get how. Please explain. Thanks !


1) Set could be (1,2,3,3), Difference = 3 - 2.25 = 0.75
or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

2)Set could be (-10,0,0,1,2,3,4), Difference = 1
or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

1 & 2 together Mean and mode are the same, so the mean has to be a number in the set. Also, the range cannot be more than 2. Hence only a set with the mode and the median being the same can satisfy such a condition. ie a set with the same number repeted in the middle with the highest and lowest being at the same distance from the middle number.

Sufficient. Answer is C. Although I doub whther I would have come up with this if i had not seen the OA. Would help if you could spoiler hide it.

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Last edited by MacFauz on 14 Nov 2012, 03:40, edited 2 times in total.
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Re: Mode - level:700-800 [#permalink] New post 13 Nov 2012, 23:08
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MacFauz wrote:
mneeti wrote:
The mode of a set of integers is x. What is the difference between the median of this set of integers and x?
(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers is x.

The OA is C, but I do not get how. Please explain. Thanks !


1) Set could be (1,2,3,3), Difference = 3 - 2.25 = 0.75or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

2)Set could be (-10,0,0,1,2,3,4), Difference = 1
or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

1 & 2 together Mean and mode are the same, so the mean has to be a number in the set. Also, the range cannot be more than 2. Hence only a set with the mode and the median being the same can satisfy such a condition. ie a set with the same number repeted in the middle with the highest and lowest being at the same distance from the middle number.

Sufficient. Answer is C. Although I doub whther I would have come up with this if i had not seen the OA. Would help if you could spoiler hide it.

Kudos Please... If my post helped.


Hii MacFauz...
I guess there is a typo in the red part. The median of a set consisting of even number of elements is the average of (n/2) th term + {(n/2)+1} th term.
So 2.25 must be 2.5.
Rest of the solution is just fantastic.

I went with an alternative approach.
1)The difference between any two integers in the set is less than 3. This can only happen in two cases.
i) When the set consists of same element.
ii) When the set consists of 3 consecutive integers. Note that when the set consists of only distinct integers, then all the integers are modes.
Two cases insufficient.
2) The average of the set of integers is x.
Not sufficient to answer the asked question.
On combining ,
either the set consists of same elements with the same number being the mode
OR
the set consists of consecutive elements. ex-{1,2,3}. In the given example, the modes are 1, 2 and 3. But as per the statement, the mode is x. What can be x?
Hence the previous case is considerd where the elements are same. Therefore difference would be 0.
C.
Hope that helps.

P.S. Please use spoiler to hide the OA.
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Last edited by Marcab on 13 Nov 2012, 23:12, edited 1 time in total.
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Re: Mode - level:700-800 [#permalink] New post 13 Nov 2012, 23:12
Marcab wrote:
MacFauz wrote:
mneeti wrote:
The mode of a set of integers is x. What is the difference between the median of this set of integers and x?
(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers is x.

The OA is C, but I do not get how. Please explain. Thanks !


1) Set could be (1,2,3,3), Difference = 3 - 2.25 = 0.75or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

2)Set could be (-10,0,0,1,2,3,4), Difference = 1
or set could be (1,1,1), Difference = 1 - 1 = 0
Insufficient.

1 & 2 together Mean and mode are the same, so the mean has to be a number in the set. Also, the range cannot be more than 2. Hence only a set with the mode and the median being the same can satisfy such a condition. ie a set with the same number repeted in the middle with the highest and lowest being at the same distance from the middle number.

Sufficient. Answer is C. Although I doub whther I would have come up with this if i had not seen the OA. Would help if you could spoiler hide it.

Kudos Please... If my post helped.


Hii MacFauz...
I guess there is a typo in the red part. The median of a set consisting of even number of elements is the average of \frac{n}{2 th term + {(n/2)+1}th term}.
So 2.25 must be 2.5.
Rest of the solution is just fantastic.

I went with an alternative approach.
1)The difference between any two integers in the set is less than 3. This can only happen in two cases.
i) When the set consists of same element.
ii) When the set consists of 3 consecutive integers. Note that when the set consists of only distinct integers, then all the integers are modes.
Two cases insufficient.
2) The average of the set of integers is x.
Not sufficient to answer the asked question.
On combining ,
either the set consists of same elements with the same number being the mode
OR
the set consists of consecutive elements. ex-{1,2,3}. In the given example, the modes are 1, 2 and 3. But as per the statement, the mode is x. What can be x?
Hence the previous case is considerd where the elements are same. Therefore difference would be 0.
C.
Hope that helps.

P.S. Please use spoiler to hide the OA.

Thanks Marcab... I've changed that now.. And it was not a typo.. I tend to make these silly mathematical mistakes :oops:
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Re: The mode of a set of integers is x. What is the difference [#permalink] New post 13 Nov 2012, 23:37
Expert's post
1)The difference between any two integers in the set is less than 3. This can only happen in two cases.
i) When the set consists of same element.
ii) When the set consists of 3 consecutive integers. Note that when the set consists of only distinct integers, then all the integers are modes.
Two cases insufficient.
2) The average of the set of integers is x.
Not sufficient to answer the asked question.
On combining ,
either the set consists of same elements with the same number being the mode
OR
the set consists of consecutive elements. ex-{1,2,3}. In the given example, the modes are 1, 2 and 3. But as per the statement, the mode is x. What can be x?
Hence the previous case is considerd where the elements are same. Therefore difference would be 0.
C.
Hope that helps.
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Re: Mode - level:700-800 [#permalink] New post 13 Nov 2012, 23:53
Thanks MacFauz and Marcab.. both the approaches are pretty helpful !! :-D
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Re: Mode - level:700-800 [#permalink] New post 14 Nov 2012, 01:35
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mneeti wrote:
The mode of a set of integers is x. What is the difference between the median of this set of integers and x?
(1) The difference between any two integers in the set is less than 3.
(2) The average of the set of integers is x.

Please explain. Thanks !


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Re: The mode of a set of integers is x. what is the difference [#permalink] New post 25 Jun 2013, 23:38
Hi

I read in one of my quant preparation book for CAT (a management test conducted in India) that
mode = 3 Median - 2 Mean

If we go by that rule:
mode + 2 mean = 3 Median

Mode = x (given in question)
Mean = x (given in statement b)
thrfore : x +2x = 3 Median
hence Median = x
so stateent B alone is sufficient to answer the question.

Correct me if I am wrong.
Re: The mode of a set of integers is x. what is the difference   [#permalink] 25 Jun 2013, 23:38
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