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Is the standard deviation of a certain set greater than 5,000?
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20 Oct 2015, 05:42
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Is the standard deviation of a certain set greater than 5,000? (1) The range of the set is greater than 6,000. (2) The range of the set is less than 8,000.
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Re: Is the standard deviation of a certain set greater than 5,000?
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30 Oct 2015, 20:55
Explanation
The concept used in this question is that Standard deviation of a certain list ≤ (1/2) range of that list
Statement 1) we do not have the information either about the terms in the list or about their range. So we cannot answer if the standard deviation of a certain set is greater than 5000. So this statement is insufficient
Statement 2) we know that range is < 8000
Based on the relation between range and standard deviation i.e. Standard deviation of a certain list ≤ (1/2) range of that list We can conclude that Standard deviation of this list < 4000
Though we do not know the exact standard deviation but still the information in statement is sufficient to conclude that Standard deviation of this list ≤ 5000
Hence we have a definite NO as an answer for the question asked
Hence the statement (2) is sufficient
Answer is option B
The following is the reasoning behind the concept
Standard deviation of a certain set ≤ (1/2) range of the list
Reasoning
One of the convenient 5step processes to calculate the numerical value of standard deviation is 1. Take the mean of the given terms 2. Take the difference of the terms from the mean 3. Take the Square of the differences 4. Take the average of the squares 5. Standard deviation = (average of the squares)1/2
Lets consider a list of two numbers (0, 10) and let’s calculate the standard deviation
1. Mean of the terms = (0 + 10)/2 = 5 2. On taking the difference of the terms from the mean we will get: 5, 5 3. On taking the Square of the differences we get: 25, 25 4. On taking the average of the squares we get: (25 + 25) / 2 = 25 5. So Standard deviation = (25)1/2 Standard deviation = 5
Conceptually we know that standard deviation signifies how deviated are the terms. So more deviated are the terms the more is the standard deviation
So if there are only two terms the list will have maximum standard deviation, as the terms will be the extreme or in others maximum deviated.
So we see that in the above list the
Standard deviation = (1/2) range
So if we add a number between the two numbers in the above list then the terms in general will become closer to the mean and hence the standard deviation will become smaller.
For better clarity lets add another term 5 in the list witch had two numbers (0, 10)
Now the new list will become (0, 5,10) and lets calculate the standard deviation
1. Mean of the terms = (0 + 5 +10)/3 = 5 2. On taking the difference of the terms from the mean we will get: 5,0, 5 3. On taking the Square of the differences we get: 25, 0, 25 4. On taking the average of the squares we get: (25 +0 + 25) / 3 = 50/3 5. So Standard deviation = (50/3) 1/2
Now the key point to notice is that the proportion with which the denominator has increased in the 4 Th step, the numerator has not increased.
As in this case if we add a number between the original numbers then we can say the numerator will never increase with the same proportion as the denominator will increase. Actually for the numerator to increase with the same proportion the 25 should have been added instead of ZERO, which will never be the case as we are adding a number between the previous range.
We can observe that Standard deviation of the second list ≤ Standard deviation of the first list
Given: Standard deviation of the first list = (1/2) range of the first list
Hence we can say Standard deviation of the second list ≤ (1/2) range of the first list
So we can conclude that
Standard deviation of a certain list ≤ (1/2) range of that list
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Alternate Reasoning
We can also say that standard deviation signifies the average distance of the terms from the mean. Please note that this average is not same as Arithmetic Average. Technically, it is called RootMeanSquared Value, but for GMAT we believe you should KEEP IT SIMPLE, so let us not go in its details.
If we take two terms (0,10) The average distance from the mean is 5.
If we add another term within 0 and 10 say 5 or 6 then the average distance of the new list will be LESS than the average distance of the terms in the previous list.
So we can say that
Standard deviation of the second list ≤ Standard deviation of the first list
Given: Standard deviation of the first list = (1/2) range of the first list
So we can conclude that
Standard deviation of a certain list ≤ (1/2) range of that list
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Is the standard deviation of a certain set greater than 5,000?
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20 Oct 2015, 06:37
AmitGoyalJamboree wrote: Is the standard deviation of a certain set greater than 5,000? (1) The range of the set is greater than 6,000. (2) The range of the set is less than 8,000. The OA should not be B. It should be E. It is not mentioned that the set has a particular # of elements. So consider the set {0,6500} and {0,7500}. SD of set 1 <5000 and SD of set 2 > 5000 and both these sets have 6000<range<8000 after you combine the 2 statements Statement 1 is shown to be not sufficient by {0,6500} and {0,7500}. Set 1 has SD <5000 and set 2 has SD > 5000. Not sufficient. Statement 2 is shown to be not sufficient by the same sets, {0,6500} and {0,7500}. Set 1 has SD <5000 and set 2 has SD > 5000. Not sufficient. Even after combining, you get a "yes" or "no" answer making E as the correct answer. Additionally, this is not a good GMAT question as GMAT will never (IMO) ask you to specifically calculate SD. You do need to know how SD of a set changes when you modify certain elements and this does not necessarily need you to know how to calculate SD.



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Re: Is the standard deviation of a certain set greater than 5,000?
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20 Oct 2015, 06:52
Standerd deviation is not dependent on range.....we can sometimes deduce standard deviation through range like if the data set is {45,45,45,45}.The range is zero and S.D will also be Zero. But if set is {45,30,10,5}Then range will be 40 while s.d will be around 6. for dataset{45,0,5}range will be 40 while S.D will be greater than 10. So,OA should be E If liked the post then give +1 Kudos We know we can and we will.......
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Re: Is the standard deviation of a certain set greater than 5,000?
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20 Oct 2015, 07:36
hi Engr2012
thanks for responding to the question . i would request you to reconsider the calculations you are doing . in the second set { 0,7500} the Standard deviation is 3750 and not > 5000 as suggested by you .
also there is a relation between range and standard deviation . i will post my explanation in about next 20  25 hours . i want to see the response i get .
also i agree GMAT will not ask you to calculate the Standard deviation . this question also does not expect you to calculate the exact standard deviation . GMAT tests you on the reasoning so this this question.



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Is the standard deviation of a certain set greater than 5,000?
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20 Oct 2015, 07:43
AmitGoyalJamboree wrote: hi Engr2012
thanks for responding to the question . i would request you to reconsider the calculations you are doing . in the second set { 0,7500} the Standard deviation is 3750 and not > 5000 as suggested by you .
also there is a relation between range and standard deviation . i will post my explanation in about next 20  25 hours . i want to see the response i get .
also i agree GMAT will not ask you to calculate the Standard deviation . this question also does not expect you to calculate the exact standard deviation . GMAT tests you on the reasoning so this this question. I understand what you meant with your question but for SD you have 2 different formulae, one for samples SD and 1 for population SD. Both the formulations give you different results. Population SD gives you 3750, while sample SD gives you 5303. I belive for your question, we are referring to population SD and not the sample SD which will make B the answer. Thanks for the reply.



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Re: Is the standard deviation of a certain set greater than 5,000?
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22 Oct 2015, 13:35
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. Is the standard deviation of a certain set greater than 5,000? (1) The range of the set is greater than 6,000. (2) The range of the set is less than 8,000. If we let standard deviation: d, and range: r, the inequality d<=r/2 can be established. From the original condition, we can see that we want to know whether d>5,000. Condition 1 gives that r=20,000, but d<=20,000/2=10,000. This is not a sufficient condition, as the range of the question does not include that of this condition. For question 2, if we say r=8,000, d<=8,000/2=4,000. The answer to what the question asks is always 'no'; the condition is sufficient, and the answer becomes (B).
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Re: Is the standard deviation of a certain set greater than 5,000?
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26 Oct 2015, 07:36
Hi,
Can you please elaborate the explanation using both the conventional and variable method?
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Re: Is the standard deviation of a certain set greater than 5,000?
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30 Oct 2015, 10:57
AmitGoyalJamboree wrote: Is the standard deviation of a certain set greater than 5,000? (1) The range of the set is greater than 6,000. (2) The range of the set is less than 8,000. The Range and Standard Deviation are related by a formula which is : Standard Deviation = Range/4 Taking the first statement..it gives that Range is > 6000, which takes me to: Let us take range as 10000(Since it is greater than 6000) so 10000/4 is 2500 which is less than 5000( Hence No). Again let us take range as 24000( Since it is greater than 6000) so 24000/4 is 6000 which is greater than 5000 (Hence Yes) So First statement is not sufficient. Now the second statement is Range is less than 8000 (Take 7999 for example which is the maximum value less than 8000) Here the Standard Deviation would be 7999/4 which is less than 5000. Hence B is correct.



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Re: Is the standard deviation of a certain set greater than 5,000?
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04 Nov 2015, 07:06
AmitGoyalJamboree wrote: Explanation
The concept used in this question is that Standard deviation of a certain list ≤ (1/2) range of that list
Its not very GMATy to depend on a formula especially in DS. SD questions are usually play with the data consistency and "density" or amplitudes of a set" Is this a new question or one from 20089 ?



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Re: Is the standard deviation of a certain set greater than 5,000?
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17 Jun 2018, 21:32
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