E is a collection of four ODD integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

I dont think this is a hard question, expecially if it only asks for odd integers. You do not need to plug in any values and certainly no calculations needed. All you need is a fundamental understanding of what standard deviation means. It is a measure of variation in the set or the distribution of numbers. So without loss of generality if you know the range you can easily enumerate the numbers. Let the 5 dashes below represent the range within which our four integers lie and I will use x's to denote the place of each constituent of the set :

- - - - -

Now, I know the range is 4, so there must be an "x" at the beginning and at the end :

x - - - x

I also know all numbers are odd so the other two numbers can only lie on either the first middle or last place giving me the arrangements :

xx - - - xx
x - xx - x
xx - x - x
xxx - - - x

Note that since standard deviation is a second order measure which measures the distribution of numbers it will be exactly the same for the sets "xx - x - x" and "x - x - xx". So we don't need to enumerate symmetric cases

Answer is 4
_________________

Sets with distinct SD:
1. {1, 1, 1, 5};
2. {1, 1, 3, 5};
3. {1, 1, 5, 5};
4. {1, 3, 3, 5};

So 4 different values of SD.

5. SD of {1, 3, 5, 5} equals to SD of 2. {1, 1, 3, 5};
6. SD of {1, 5, 5, 5} equals to SD of 1. {1, 1, 1, 5}.

Hope it's clear.
_________________

I'd like to clear something for the people who are preparing for GMAT: this might not be a hard question for professional statistician but if such question ever appear on GMAT it'll be considered 750+, so very hard.

Usually GMAT SD questions are fairly straightforward and don't require actual calculation of SD, they are about the general understanding of the concept.

So don't be scared: it's really unlikely you'll see such a question on GMAT and if you will, then you must know that you are doing very well and are probably very close to 51 on quant.
_________________

All you need is a fundamental understanding of standard deviation to solve this question, plugging in values is painful and not required. Standard deviation measures how the elements of a set are distributed around the mean, or the "deviation" of the elements in other words. If you have just 4 elements in which the first and last are fixed relative to each other it just boils down to how you can distribute the other two to form different amounts of deviation.

The actual enumeration of this is shown above, but all you you need to note is that the deviation is symmetric cases is just the same :

{1,1,3,5}
{1,3,5,5}
OR
xx - x - x
x - x - xx

The deviation is exactly the same, its just the mean which is shifted.

Keeping this in mind there are only 4 possibilities with 4 odd numbers of range 4.
_________________

This is a good question though I did not like the wording very much. Instead of 'SD of J must be one one how many numbers', 'How many distinct values can SD of J take' is better. Anyway,

First I thought J is a set of four odd integers with range 4 so I said J = {1, x, y, 5}
Now x and y can take 3 different values: 1, 3 or 5
Either both x and y are same. This can be done in 3 ways.
Or x and y are different. This can be done in 3C2 ways = 3 ways
Total x and y can take values in 3 + 3 = 6 ways
Let me enumerate them for clarification:
{1, 1, 1, 5}, {1, 3, 3, 5}, {1, 5, 5, 5}, {1, 1, 3, 5}, {1, 1, 5, 5}, {1, 3, 5, 5}
These are the 6 ways in which you can choose the numbers.
Important thing: SD of {1, 1, 1, 5} and {1, 5, 5, 5} is same. Why?
SD measures distance from mean. It has nothing to do with the actual value of mean and actual value of numbers.
In {1, 1, 1, 5}, mean is 2. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.
In {1, 5, 5, 5}, mean is 4. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.

Similarly, {1, 1, 3, 5} and {1, 3, 5, 5} will have the same SD.

Then, {1, 3, 3, 5} will have a distinct SD and {1, 1, 5, 5} will have a distinct SD.
In all, there are 4 different values that SD can take in such a case.

Note: It doesn't matter what the actual numbers are. SD of 1, 3, 5, 7 is the same as SD of 12, 14, 16, 18. For detailed explanation of SD and how to calculate it, check the theory or Stats.
_________________

This cases are not possible since "the greatest difference between any two integers in E is 4" means that the range of the set is 4.
_________________

The greatest difference between any two integers in E is 4 means that the range of the set is 4 and the range of {1, 1, 1, 1} is 0, not 4.
_________________

Hi

Isn't a set with values 3,3,5, 7 viable??

Hi All,

While this question requires some specific knowledge about Standard Deviation, you don't actually have to do much math to solve it.

To start, it's worth noting that the GMAT will NEVER ask you to calculate the Standard Deviation of a group using the S.D. formula, so that is NOT what this question is actually about.

We're told that there are four ODD integers and the greatest difference between ANY TWO of them is 4. This significantly limits the range of values.

For example, the group: 1, 1, 1, 5 fits everything that we were told. If you take ANY TWO of those values, then the greatest difference is 4. That group would have a certain S.D., but if we change any of those individual numbers, then the S.D. will also change. Thus, we really just have to think in terms of how many changes we could make while still keeping the greatest difference as 4. We also have to be careful about not creating groups that have the SAME S.D.

For example, the group: 1, 1, 1, 5 has the same S.D. as the group: 1, 5, 5, 5.... The prompt asks us for the number of different S.D.s that are possible here, so we would count this option just once (and not twice).

The other possibilities would be....
1, 1, 5, 5
1, 1, 3, 5 which has the same S.D. as 1, 3, 5, 5
1, 3, 3, 5

That gives us a total of four possibilities.

Final Answer:

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