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E is a collection of four ODD integers and the greatest 15 Aug 2018, 20:51
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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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E is a collection of four ODD integers and the greatest 05 Feb 2019, 07:13
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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
I know this question, I've posted it in my topic: https://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.

Answer: B.

Hope it's clear.
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E is a collection of four ODD integers and the greatest 17 Feb 2019, 03:21
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why are we not using negative integers such as (-3,1,-1,1)
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E is a collection of four ODD integers and the greatest 17 Feb 2019, 14:40
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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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E is a collection of four ODD integers and the greatest 07 May 2019, 14:25
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Bunuel
I know this question, I've posted it in my topic: https://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.

Answer: B.

Hope it's clear.
Show more
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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E is a collection of four ODD integers and the greatest 21 Jun 2019, 05:09
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Bunuel
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I know this question, I've posted it in my topic: https://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.

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.
Show more


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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E is a collection of four ODD integers and the greatest 21 Jun 2019, 13:57
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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 18 Jul 2019, 11:38
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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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E is a collection of four ODD integers and the greatest 18 Jul 2019, 11:52
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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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E is a collection of four ODD integers and the greatest 22 Jul 2019, 20:20
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chetan2u, Gladiator59, VeritasKarishma

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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E is a collection of four ODD integers and the greatest 22 Jul 2019, 20:28
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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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E is a collection of four ODD integers and the greatest 22 Jul 2019, 22:55
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chetan2u, Gladiator59, VeritasKarishma

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
Show more

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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E is a collection of four ODD integers and the greatest 11 Sep 2020, 06:27
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Bunuel
I know this question, I've posted it in my topic: https://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.

Answer: B.

Hope it's clear.
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I have a doubt Bunuel, when the greatest difference between any 2 integers is 4, that means the difference could be 0,1,2,or 3 as well. Naturally, 1 and 3 will not hold for odd numbers, but we can have sets of odd numbers with elements having differences between them as 0 or 2.
In that case why are we not considering the sets {3,3,5,5}, {5,5,5,5} and so on? I got stuck trying to write out all these combinations.
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E is a collection of four ODD integers and the greatest 11 Sep 2020, 19:51
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Hi ShreyasJavahar,

Since the prompt tells us that the GREATEST difference between any two of the integers is 4, that means that there MUST be a pair of integers in the set that differ by 4.

The sets {3,3,5,5} and {5,5,5,5} don't 'fit' with that piece of information, so they're not sets that you should be considering.

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E is a collection of four ODD integers and the greatest 12 Sep 2020, 01:01
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Hi ShreyasJavahar,

Since the prompt tells us that the GREATEST difference between any two of the integers is 4, that means that there MUST be a pair of integers in the set that differ by 4.

The sets {3,3,5,5} and {5,5,5,5} don't 'fit' with that piece of information, so they're not sets that you should be considering.

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Rich
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Thanks for the response Rich, I understood the question to mean, "the greatest POSSIBLE difference between any two integers" and that introduced the ambiguity for me. Guess I should've just stopped at the conclusion that the range of the set is 4, instead of reading too much into it.
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E is a collection of four ODD integers and the greatest 01 Oct 2020, 13:19
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Bunuel
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I know this question, I've posted it in my topic: https://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.

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.
Show more


The difference between the highest number and lowest number is called range and the words used in the question are “ the greatest difference between ANY two numbers in E is 4 “

So can we consider ( 1, 3, 3, 3 )?

I think ANY is the problem because if we read the sentence without ANY, it can be read as “ the greatest difference between two numbers in E is 4 “ and greatest difference can only be achieved in a set when we subtract lowest number from the highest one, which is range.

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E is a collection of four ODD integers and the greatest 01 Oct 2020, 16:43
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chikki420
Bunuel



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.


The difference between the highest number and lowest number is called range and the words used in the question are “ the greatest difference between ANY two numbers in E is 4 “

So can we consider ( 1, 3, 3, 3 )?

I think ANY is the problem because if we read the sentence without ANY, it can be read as “ the greatest difference between two numbers in E is 4 “ and greatest difference can only be achieved in a set when we subtract lowest number from the highest one, which is range.

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Hi chikki420,

Since the prompt tells us that the GREATEST difference between any two of the integers is 4, that means that there MUST be a pair of integers in the set that differ by 4.

The set {1,3,3,3} does not 'fit' with that piece of information, so that's not a set that you should be considering.

GMAT assassins aren't born, they're made,
Rich
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pudu
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E is a collection of four ODD integers and the greatest 04 Apr 2023, 02:56
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shrouded1
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Very good but time consuming question.

Bunuel do you think it is a Gmat Question.?

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
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couldn't understand it...

also how we can limit numbers to 5? (11,11,11,15) is also possible ...someone please help me to understand it

if possible to calcualate without calcualtions then please explain that :)
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Akanksha270897
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E is a collection of four ODD integers and the greatest 24 Oct 2023, 13:40
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I have one question - (also) an alternate way to solve this question (according to me) - kindly correct me if I am wrong.

I am aware of the relationship between Standard Deviation (SD) and Range wherein S.D. <= Range/2.

Given the range for this question is 4, S.D. <= 2, which has three possible values 0,1,2 instead of the 4 illustrated above. Can you please guide me as to what I am doing wrong?
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E is a collection of four ODD integers and the greatest 25 Oct 2023, 11:38
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Akanksha270897
I have one question - (also) an alternate way to solve this question (according to me) - kindly correct me if I am wrong.

I am aware of the relationship between Standard Deviation (SD) and Range wherein S.D. <= Range/2.

Given the range for this question is 4, S.D. <= 2, which has three possible values 0,1,2 instead of the 4 illustrated above. Can you please guide me as to what I am doing wrong?
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Hi Akanksha270897,

When calculating the Standard Deviation of a group of number, the resulting S.D. is often NOT an integer. In your example, you've named three non-negative integers that are less than or equal to 2 -> but that does NOT mean that they are all possible S.D.s given what the prompt tells us. The actual number of possible S.D.s depends on the possible "spreads" of the four numbers involved - and that is FOUR.

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

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