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Set Q consists of n integers, of which the standard

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Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 22:01
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Set Q consists of n integers, of which the standard deviation is D and the average (arithmetic mean) is M. If integer N is added to set Q, the standard deviation of the new set is less than D. Which of the following must be true?

I. N > M
II. N < M
III. N < D

A. I only
B. II only
C. III only
D. II & III only
E. None of the above
[Reveal] Spoiler: OA

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Last edited by Bunuel on 10 Jul 2013, 22:03, edited 1 time in total.
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 22:15
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bagdbmba wrote:
Set Q consists of n integers, of which the standard deviation is D and the average (arithmetic mean) is M. If integer N is added to set Q, the standard deviation of the new set is less than D. Which of the following must be true?

I. N > M
II. N < M
III. N < D

A. I only
B. II only
C. III only
D. II & III only
E. None of the above


Say Q={2, 3, 4} (consider a simple set).
The standard deviation (D) will be very small (~1).
The mean (M) is 3;

We want to add new element so that the standard deviation to decrease. Add the element which is equal to the mean. So, if N=3:

D will decrease --> condition satisfied.
N=M --> discard I and II.
N>D --> discard III.

Answer: E.

Hope it's clear.
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 22:51
Expert's post
Bunuel wrote:
bagdbmba wrote:
Set Q consists of n integers, of which the standard deviation is D and the average (arithmetic mean) is M. If integer N is added to set Q, the standard deviation of the new set is less than D. Which of the following must be true?

I. N > M
II. N < M
III. N < D

A. I only
B. II only
C. III only
D. II & III only
E. None of the above


Say Q={2, 3, 4} (consider a simple set).
The standard deviation (D) will be very small (~1).
The mean (M) is 3;

We want to add new element so that the standard deviation to decrease. Add the element which is equal to the mean. So, if N=3:

D will decrease --> condition satisfied.
N=M --> discard I and II.
N>D --> discard III.

Answer: E.

Hope it's clear.


Thanks Bunuel...
Just couple of quick clarifications required:

1. For N>D--> S.D decreases but does that mean for N<D it won't be true always? Please clarify it.

2.What should be the approach for this type of problems? I mean is there any other way we can do it or the one you've shown is the most generic approach to solve...?
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 23:03
Expert's post
bagdbmba wrote:
Bunuel wrote:
bagdbmba wrote:
Set Q consists of n integers, of which the standard deviation is D and the average (arithmetic mean) is M. If integer N is added to set Q, the standard deviation of the new set is less than D. Which of the following must be true?

I. N > M
II. N < M
III. N < D

A. I only
B. II only
C. III only
D. II & III only
E. None of the above


Say Q={2, 3, 4} (consider a simple set).
The standard deviation (D) will be very small (~1).
The mean (M) is 3;

We want to add new element so that the standard deviation to decrease. Add the element which is equal to the mean. So, if N=3:

D will decrease --> condition satisfied.
N=M --> discard I and II.
N>D --> discard III.

Answer: E.

Hope it's clear.


Thanks Bunuel...
Just couple of quick clarifications required:

1. For N>D--> S.D decreases but does that mean for N<D it won't be true always? Please clarify it.

2.What should be the approach for this type of problems? I mean is there any other way we can do it or the one you've shown is the most generic approach to solve...?


1. The red part is wrong. If N=1000, the standard deviation will increase.

2. I think approach shown in my post is the easiest for this particular problem.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 23:07
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 23:10
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 10 Jul 2013, 23:26
Expert's post
Bunuel wrote:
bagdbmba wrote:
So why we're discarding the option III here?


The question is which of the options must be true?

Consider the example in my post. Does any of the options hold true for it?


Got it!

How I could know that which example set would be best to plug-in for soln? I mean is there any baseline to determine which set would be ideal to consider?
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 21 Aug 2013, 13:23
This question is dealing with conceptual knowledge about standard deviation (SD):
If SD is decreasing after adding a new number it means a number is added closer to the average arithmetic mean (AM) and that could be on the right side or left side on the number line of the AM, thus we can not say that this new number is less than or greater than M only we know that this number is closer to AM, further standard deviation is the measure of range +- density distribution of numbers around AM like most of the numbers are under umbrella of radius SD, it basically tells us that most of the numbers are around AM proximity in rage like 1.2 etc, so still we can not predict that new number is greater than SD or not, actually they are not comparable, and provided data is insufficient to calculate any number so 50 50 chances. Therefore answer should be E.
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 23 Aug 2013, 11:53
@PiyushK, I solved it without pen and paper myself having a similar thought process as yours.

Bunuel,

Can you please confirm this thought process for this question? Or are there any exceptional cases that we are missing?

Thanks.
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Re: Set Q consists of n integers, of which the standard [#permalink] New post 23 Aug 2013, 16:33
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bagdbmba wrote:
Bunuel wrote:
bagdbmba wrote:
So why we're discarding the option III here?


The question is which of the options must be true?

Consider the example in my post. Does any of the options hold true for it?


Got it!

How I could know that which example set would be best to plug-in for soln? I mean is there any baseline to determine which set would be ideal to consider?



Details:

Q= {2,3,4}
Standard deviation :
1. mean = 3
2. difference: 1,0,-1
3. squaring: 1,0,1
4. mean again = (1+0+1)/3 = 2/3
5. root over (2/3) = standard deviation.

Now, Adding the man to the set Q . So {2,3,3,4}
same way standard deviation: root over (1/2)

Certainly standard deviation decreases after adding mean 3.

The mean was M=3 and added number N=3 too and first standard deviation D= root over (2/3)

So N=M ( I and II both eliminated}
we can see, N>D . So III eliminated. Nothing is must be true here...............

The example here {1,2,3} works well too, we can use it here conveniently....
start something like {1,2,3} or {2,3,4}...

Basics of this math is, add mean to a set and standard deviation will decrease certainly.........
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Re: Set Q consists of n integers, of which the standard   [#permalink] 23 Aug 2013, 16:33
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