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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: E is a collection of four ODD integers and the greatest  [#permalink]

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Hi zs2,

IF you want to use the numbers +1 and -3 as two of your four odd numbers, then that's absolutely fine; by extension though, since those two values differ by 4, each of the OTHER two ODD numbers in the set would have to be +1, -1 or -3.

Thus, your possible Standard Deviations would be...
-3, 1, 1, 1.... which has the SAME S.D. as -3, -3, -3, 1, so we only count this option ONCE.
-3, -3, 1, 1
-3, -3, -1, 1... which has the SAME S.D as -3, -1, 1, 1, so we only count this option ONCE.
-3, -1, -1, 1

You still end up with 4 DIFFERENT possible S.D.s

This same pattern will occur if you use the numbers 13 and 17 too.

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

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Hi Bunuel,

Is there a simple algebraic approach to solve this within 2 minutes? I am really getting stuck on this one. Thank you!!

-KHow

Bunuel wrote:
I know this question, I've posted it in my topic: http://gmatclub.com/forum/ps-questions- ... 85897.html

But there is a typo, it should be:

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:

1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).

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.

Hope it's clear.
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Re: E is a collection of four ODD integers and the greatest  [#permalink]

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why are we not using negative integers such as (-3,1,-1,1)
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: E is a collection of four ODD integers and the greatest  [#permalink]

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1
Hi shivmsp,

IF you want to use the numbers +1 and -3 as two of your four odd numbers, then that's absolutely fine; by extension though, since those two values differ by 4, each of the OTHER two ODD numbers in the set would have to be +1, -1 or -3.

Thus, your possible Standard Deviations would be...
-3, 1, 1, 1.... which has the SAME S.D. as -3, -3, -3, 1, so we only count this option ONCE.
-3, -3, 1, 1
-3, -3, -1, 1... which has the SAME S.D as -3, -1, 1, 1, so we only count this option ONCE.
-3, -1, -1, 1

You still end up with 4 DIFFERENT possible S.D.s

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

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Bunuel wrote:
I know this question, I've posted it in my topic: http://gmatclub.com/forum/ps-questions- ... 85897.html

But there is a typo, it should be:

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:

1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).

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.

Hope it's clear.

Hi Bunuel,

Just want to clarify, we are told that greatest difference between any two integers is 4. this means that they are odd seines of integers with common difference 4

But in the sets
1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).

if we take difference between any two integers its not 4

Had it been that range is 4 then above set does work .

What am i missing?
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Re: E is a collection of four ODD integers and the greatest  [#permalink]

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1
Bunuel wrote:
cumulonimbus wrote:
Bunuel wrote:
I know this question, I've posted it in my topic: http://gmatclub.com/forum/ps-questions- ... 85897.html

But there is a typo, it should be:

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:

1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).

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.

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.

Doesn't The greatest difference between any two integers means the maximum difference of any two integers in this set can have is 4. So why cant we consider sets like {1,1,1,1} and {1,1,3,3} as the difference between any two integers in this set is not exceeding 4.

Can you please help to prove how this interpretation is wrong?
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: E is a collection of four ODD integers and the greatest  [#permalink]

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Hi Sreeragc,

The examples that you noted do not fit the 'restrictions' given by the prompt. We're told that the greatest difference between any two of the four numbers IS 4... meaning that the RANGE of the set is 4.

IF the question stated that the difference between any two of the numbers was 'no more than 4', then both of your examples would be permissible (since in that situation we'd want a range of 4 OR LESS).

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

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I assumed the numbers to be 2n-1,2n+1,2n+3,2n+5 and ended up with distances from the mean as 1,1,3,3 and their variance was 5 .Where is the mistake ?

Thanks a lot Bunuel and VeritasKarishma for clarifying
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: E is a collection of four ODD integers and the greatest  [#permalink]

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Hi GMATestaker,

We're told that the greatest difference between any two of the four numbers IS 4... meaning that the RANGE of the set is 4. In your approach, the range of your four values is 6.

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

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Why can't we have a set like (3,3,3,7) in consideration for this question? The greatest difference between any 2 integers is 7-3=4
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: E is a collection of four ODD integers and the greatest  [#permalink]

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Hi DarkHorse2019,

You can certainly have a set of numbers as you've described (with the lowest value as 3 and the highest value as 7). The specific group that you listed would have a specific standard deviation. How many OTHER groups of 4 odd numbers can you list that would also have a lowest value of 3 and a highest value of 7 though? The question is asking for the number of different possible Standard Deviations that would fit a particular group - and you've named just one of the options so far.

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

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DarkHorse2019 wrote:

Why can't we have a set like (3,3,3,7) in consideration for this question? The greatest difference between any 2 integers is 7-3=4

This set is considered.

Check:
https://gmatclub.com/forum/e-is-a-colle ... ml#p815649

{1, 1, 1, 5} is exactly the same as {3,3,3,7} as far as SD is considered. Distance from the mean for each element is the same for both sets.

{1, 1, 1, 5}
Mean = 2
Distance of 2 from mean = 1
Distance of 5 from mean = 3

{3,3,3,7}
Mean = 4
Distance of 3 from mean = 1
Distance of 7 from mean = 3

But this is just one of the 6 possible sets. Check the link above for the detailed solution.
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