Set Q consists of n integers, of which the standard

Question

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

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

Bunuel · Accepted Answer

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.

bagdbmba · Answer

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...?

Bunuel · Answer

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 .

bagdbmba · Answer

So why we're discarding the option III here?

Bunuel · Answer

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?

bagdbmba · Answer

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?

PiyushK · Answer

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.

Asifpirlo · Answer

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.........

VeritasPrepBrandon · Answer

This standard deviation question is a conceptual one (as are all standard deviation questions on the gmat). The fact that integer N being added to the set reduces the set's standard deviation means, in rough terms, that it is closer to the mean than the set's standard deviation. The addition of this new number will in reality change the mean, but we just need to know the fact that it basically means it is closer to the mean than the average number in the set. This, however, tells us nothing about whether it is greater than or less than the mean, as both are possible, eliminating numbers I and II. N and D are also only related in so far as N affects D, but neither of their values needs to be greater than the other, eliminating number III. It is fully possible that the numbers are extremely spread out and D is very high, and that N is very close to a mean with a low value and is therefore less than D. It is equally possible however that the numbers in the set are very large and not crazy spread out, meaning that the mean is much larger than the standard deviation, and N ends up being larger than D.

This all leaves us with our correct answer of E, "none of the above."

I hope this helps!

This all leaves us with our correct answer of E, "none of the above."

I hope this helps!

bumpbot · Answer

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Set Q consists of n integers, of which the standard

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