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E is a collection of four ODD integers and the greatest [#permalink]

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25 Aug 2010, 06:41

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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?

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

Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}: 1. Find the mean, \(m\), of the values. 2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean. 3. Calculate the squares of these deviations. 4. Find the mean of the squared deviations. This quantity is the variance. 5. Take the square root of the variance. The quantity is th SD.

Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).

You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.

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

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

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.
_________________

as range is 4..we know there is a lowest and there is a highest number Now for the rest 2 numbers:

1) either they are equal to lowest number ... >> 1 SD 2) either they are equal to highest number... >> 1 SD 3) either one is equal to lowest number and one is equal to highest number... >> 1 SD 4) they are same but not equal to lowest or highest number... >> 1 SD

So 4 possibilities (please note that all the numbers can not be distinct..otherwise range will be greater than 4)

J is a collection of four odd integers whose range is 4. The standard deviation of J must be one of how many numbers?

a 3 b 4 c 5 d 6 e 7

Please explain your thought process on this one. Thanks

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.
_________________

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

Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}: 1. Find the mean, \(m\), of the values. 2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean. 3. Calculate the squares of these deviations. 4. Find the mean of the squared deviations. This quantity is the variance. 5. Take the square root of the variance. The quantity is th SD.

Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).

You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.

Answer: B.

Hope it's clear.

Hi, in the sets above why aren't sets [3,5,5,5] and [3,3,3,5] considered? Their is no limit on minimum range.

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

Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}: 1. Find the mean, \(m\), of the values. 2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean. 3. Calculate the squares of these deviations. 4. Find the mean of the squared deviations. This quantity is the variance. 5. Take the square root of the variance. The quantity is th SD.

Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).

You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.

Answer: B.

Hope it's clear.

Hi, in the sets above why aren't sets [3,5,5,5] and [3,3,3,5] considered? Their is no limit on minimum range.

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.
_________________

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

Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}: 1. Find the mean, \(m\), of the values. 2. For each value \(x_i\) calculate its deviation (\(m-x_i\)) from the mean. 3. Calculate the squares of these deviations. 4. Find the mean of the squared deviations. This quantity is the variance. 5. Take the square root of the variance. The quantity is th SD.

Expressed by formula: \(standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}\).

You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.

Answer: B.

Hope it's clear.

Hi, in the sets above why aren't sets [3,5,5,5] and [3,3,3,5] considered? Their is no limit on minimum range.

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.

I have seen that in all previous posts, consideration of sets that are all the same number such as [1,1,1,1] were not considered. Why is that?

Hi, in the sets above why aren't sets [3,5,5,5] and [3,3,3,5] considered? Their is no limit on minimum range.

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.

I have seen that in all previous posts, consideration of sets that are all the same number such as [1,1,1,1] were not considered. Why is that?

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.
_________________

The sets in my post are based on an assumption that the smallest integer is 1 to simplify finding a pattern. Your set is similar to {1, 1, 3, 5} in my solution.
_________________

Re: E is a collection of four ODD integers and the greatest [#permalink]

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19 Dec 2017, 21:19

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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