Math: Standard Deviation : GMAT Quantitative Section - Page 2
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# Math: Standard Deviation

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19 Jan 2011, 03:20
To confirm understanding: if we rephrase the example 4 to say:
Example #4
Q: There is a set A of 19 integers with mean 4 and standard deviation of 3. Now we form a new set B by adding 2 more elements to the set A. What two elements will increase the standard deviation the most?
A) 9 and 3
B) -3 and 3
C) 6 and 1
D) 4 and 5
E) 5 and 5

Then the solution will be B (gives 8 points increase to the variation)?
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05 Jun 2011, 07:45
Vorskl wrote:
To confirm understanding: if we rephrase the example 4 to say:
Example #4
Q: There is a set A of 19 integers with mean 4 and standard deviation of 3. Now we form a new set B by adding 2 more elements to the set A. What two elements will increase the standard deviation the most?
A) 9 and 3
B) -3 and 3
C) 6 and 1
D) 4 and 5
E) 5 and 5

Then the solution will be B
 (gives 8 points increase to the variation)
?

Your answer is correct but the "variance" need not be an increase of 8.
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09 Jun 2011, 20:52
Not seen any questions for Normal distribution, is it tested on GMAT?
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10 Jun 2011, 03:45
puneetj wrote:
Not seen any questions for Normal distribution, is it tested on GMAT?

You are right. I should change "you can rarely see" to "you will never see"
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16 Jun 2011, 17:09
Very very useful and easy to understand.Thanks so much
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13 Jul 2011, 06:43
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Could you pls explain why (2) alone is not insufficient? Is it because we don't have information of the number elements? that's why (2) is telling nothing useful? thanks....

Example #2
Q: There is a set of consecutive even integers. What is the standard deviation of the set?
(1) There are 39 elements in the set.
(2) the mean of the set is 382.
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13 Jul 2011, 08:35
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yes, you are right.

it can be {380,382,384} or {378, 380,382,384, 386} for example. Standard deviation of second set is greater than that of first set.
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13 Jul 2011, 17:26
thank you
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15 Oct 2012, 06:57
walker wrote:
yes, you are right.

it can be {380,382,384} or {378, 380,382,384, 386} for example. Standard deviation of second set is greater than that of first set.

I know this is an old post but need to clear this concept..... Please explain how statement 1 alone is sufficient as it gives only the number of elements....how can we only use that to answer the question as to what is standard deviation of the set....
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15 Oct 2012, 07:05
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avaneeshvyas wrote:
walker wrote:
yes, you are right.

it can be {380,382,384} or {378, 380,382,384, 386} for example. Standard deviation of second set is greater than that of first set.

I know this is an old post but need to clear this concept..... Please explain how statement 1 alone is sufficient as it gives only the number of elements....how can we only use that to answer the question as to what is standard deviation of the set....

Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set --> SD of a set of ANY 39 consecutive even integers will be the same, as any set of 39 consecutive even integers can be obtained by adding constant to another set of 39 consecutive integers. For example: set of 39 consecutive integers {4, 6, 8, ..., 80} can be obtained by adding 4 to each term of another set of 39 consecutive integers: {0, 2, 4, ..., 76}. So we can calculate SD of {0, 2, 4, ..., 76} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The mean of the set is 382 --> knowing mean gives us nothing, we must know the number of terms in the set, as SD of {380, 382, 384} is different from SD of {378, 380, 382, 384, 386}. Not sufficient.

Hope it's clear.
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15 Oct 2012, 07:30
Bunuel wrote:
avaneeshvyas wrote:
walker wrote:
yes, you are right.

it can be {380,382,384} or {378, 380,382,384, 386} for example. Standard deviation of second set is greater than that of first set.

I know this is an old post but need to clear this concept..... Please explain how statement 1 alone is sufficient as it gives only the number of elements....how can we only use that to answer the question as to what is standard deviation of the set....

Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set --> SD of a set of ANY 39 consecutive even integers will be the same, as any set of 39 consecutive even integers can be obtained by adding constant to another set of 39 consecutive integers. For example: set of 39 consecutive integers {4, 6, 8, ..., 80} can be obtained by adding 4 to each term of another set of 39 consecutive integers: {0, 2, 4, ..., 76}. So we can calculate SD of {0, 2, 4, ..., 76} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The mean of the set is 382 --> knowing mean gives us nothing, we must know the number of terms in the set, as SD of {380, 382, 384} is different from SD of {378, 380, 382, 384, 386}. Not sufficient.

Hope it's clear.

Extending your logic can we go on to say that the sum of 39 consecutive even integers will be the same as that of a set with 39 consecutive odd integers and a constant added to it.....E.G: a set {4, 6, 8, ..., 80} can be obtained by adding 3 to a set {1, 3, 5, ..., 77}....
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15 Oct 2012, 07:32
avaneeshvyas wrote:
Bunuel wrote:
avaneeshvyas wrote:
Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set --> SD of a set of ANY 39 consecutive even integers will be the same, as any set of 39 consecutive even integers can be obtained by adding constant to another set of 39 consecutive integers. For example: set of 39 consecutive integers {4, 6, 8, ..., 80} can be obtained by adding 4 to each term of another set of 39 consecutive integers: {0, 2, 4, ..., 76}. So we can calculate SD of {0, 2, 4, ..., 76} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The mean of the set is 382 --> knowing mean gives us nothing, we must know the number of terms in the set, as SD of {380, 382, 384} is different from SD of {378, 380, 382, 384, 386}. Not sufficient.

Hope it's clear.

Extending your logic can we go on to say that the sum of 39 consecutive even integers will be the same as that of a set with 39 consecutive odd integers and a constant added to it.....E.G: a set {4, 6, 8, ..., 80} can be obtained by adding 3 to a set {1, 3, 5, ..., 77}....

Yes, that's correct.
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15 Oct 2012, 08:05
So for GMAT sake all we should be concerned about is the number of terms, if a question of Determining the Standard deviation crops up...

Thank you for the brilliant explanation Bunuel.....
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15 Oct 2012, 08:08
avaneeshvyas wrote:
So for GMAT sake all we should be concerned about is the number of terms, if a question of Determining the Standard deviation crops up...

Thank you for the brilliant explanation Bunuel.....

Well, that's not correct. For this question yes, knowing that the set is composed of even integers and knowing the number of the terms in the set is sufficient to determine the standard deviation. But just knowing the number of the terms in a set is certainly not enough.
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15 Oct 2012, 09:08
Bunuel wrote:
avaneeshvyas wrote:
So for GMAT sake all we should be concerned about is the number of terms, if a question of Determining the Standard deviation crops up...

Thank you for the brilliant explanation Bunuel.....

Well, that's not correct. For this question yes, knowing that the set is composed of even integers and knowing the number of the terms in the set is sufficient to determine the standard deviation. But just knowing the number of the terms in a set is certainly not enough.

could you put in an example for the same and also how then to go about such problems
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15 Oct 2012, 09:13
avaneeshvyas wrote:
Bunuel wrote:
avaneeshvyas wrote:
So for GMAT sake all we should be concerned about is the number of terms, if a question of Determining the Standard deviation crops up...

Thank you for the brilliant explanation Bunuel.....

Well, that's not correct. For this question yes, knowing that the set is composed of even integers and knowing the number of the terms in the set is sufficient to determine the standard deviation. But just knowing the number of the terms in a set is certainly not enough.

could you put in an example for the same and also how then to go about such problems

Check our question banks (viewforumtags.php) for more questions on SD.

DS questions on SD: search.php?search_id=tag&tag_id=34
PS questions on SD: search.php?search_id=tag&tag_id=55

Also, from my signature:
PS SD-problems: [PS Standard Deviation Problems]
DS SD-problems: [DS Standard Deviation Problems]

Hope it helps.
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10 Jul 2013, 23:07
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09 Apr 2014, 00:05
Under the "Properties "section , when a new element is added to a set ,it says , newer standard deviation is greater than the older standard deviation if | y - mean| >older standard deviation. Which mean is it alluding to? The mean after having a new element in the set or the old mean without the new element?

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12 May 2014, 04:38
Bunuel, so even for a set of consecutive odd integers, the S.D will be the same for a particular number of integers, whatever the integers may be?
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12 May 2014, 05:07
gaurav1418z wrote:
Bunuel, so even for a set of consecutive odd integers, the S.D will be the same for a particular number of integers, whatever the integers may be?

For equal number of terms, yes. For example, {1, 3, 5, 7} and {11, 13, 15, 17} have the same standard deviation: $$\sqrt{5}$$.
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