Is the standard deviation of a certain set greater than 5,000?

Question

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.

AmitGoyalJamboree · Accepted Answer

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 5-step 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

_____________________________________________________

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 Root-Mean-Squared 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

____________________________________________

ENGRTOMBA2018 · Answer

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 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.

mahakmalik · Answer

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 :-D

We know we can and we will.......

We know we can and we will.......

AmitGoyalJamboree · Answer

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.

ENGRTOMBA2018 · Answer

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.

MathRevolution · Answer

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).

saikiransomala · Answer

Hi,

Can you please elaborate the explanation using both the conventional and variable method?

Regards

Shekhar07 · Answer

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.

9SidMn · Answer

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 2008-9 ?

IanStewart · Answer

In case anyone is reading this thread, there's a lot of incorrect information in it - for example this is simply false:

There is no formula that lets you derive standard deviation from range. There is an inequality relationship that was used in the question in the OP, but you would never need to know it on a real GMAT question, so anyone reading this can ignore the question posted here.

Jinnraj · Answer

I am not sure if the rule of sd <= 1/2 range is correct because I applied the same logic in https://gmatclub.com/forum/set-x-and-set-y-have-10-numbers-each-is-standard-deviation-of-set-x-250884.html and got the wrong answer. If I had applied the same logic then in this (link) the answer to the question would be A but it is E.

I have my doubts on this rule of sd <= 1/2 range

I have my doubts on this rule of sd <= 1/2 range

IanStewart · Answer

The rule is correct. You definitely don't need to know the rule for any real GMAT question though, so you can safely forget about it.

There's no way to even apply this rule to the question you've linked to; I'm not sure how you've tried to use this rule when answering that question, but if you got the answer A, you've made an error somewhere.

Jinnraj · Answer

Thanks for clarifying however for the linked Question I applied the same logic. Statement A says range of X is greater than that of Y so doesn't it mean sd of X > sd of Y? Kindly advise if I am missing any fundamental here.

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IanStewart · Answer

I'm not following what you mean when you say you used the "same logic". The rule mentioned in this thread says that if you know a set's range, you know that the set's standard deviation is at most half the range. But it could be much less than half the range. If you have a set with a range of 10, then its standard deviation could be anything less than or equal to 5 (and greater than zero). And if you have another set with a range of 8, then its standard deviation could be anything less than or equal to 4. But that's all the rule tells you. You can't tell which standard deviation is larger just from the range -- the first set might have a standard deviation of 2 and the second set could have a standard deviation of 4.

k9527511797 · Answer

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egmat · Answer

Greater range does NOT mean greater standard deviation.

What Range tells us:
Range only measures the gap between the largest and smallest values. It says nothing about how the other values are distributed.

The actual relationship:
SD ≤ Range/2 (always!)

This means range gives us an upper limit on SD, not a lower limit. The SD can be anywhere from 0 up to Range/2.

Simple Example:
Consider two sets, both with Range = 10,000:

Set A: {0, 10000} → SD ≈ 5,000 (values spread apart)
Set B: {0, 5000, 5000, 5000, 10000} → SD ≈ 3,162 (values clustered in middle)

Same range, different SDs!

If we keep adding more 5000s to Set B, the SD gets smaller and smaller while the range stays at 10,000.

Back to this question:

Statement 1: Range > 6,000
This tells us SD could be as high as 3,000+ but could also be tiny (like 1). We can't tell if SD > 5,000. Not Sufficient

Statement 2: Range < 8,000
Since SD ≤ Range/2, we get SD < 4,000.
This means SD is definitely NOT greater than 5,000. Sufficient

Answer: B

Jinnraj

Thanks for clarifying however for the linked Question I applied the same logic. Statement A says range of X is greater than that of Y so doesn't it mean sd of X > sd of Y? Kindly advise if I am missing any fundamental here.

Posted from my mobile device

Is the standard deviation of a certain set greater than 5,000?

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Is the standard deviation of a certain set greater than 5,000?

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