Is standard deviation of Set A > the standard deviation of Set B? 1)

Hi
The answer should be E.

if set A = {1,2,2,3} B ={1,2,3}, SD = SD of A = 0.707, SD of B = 0.816

if set A = {1,1,2,3,3} B ={1,2,3}, SD of A = 0.894, SD of B = 0.816

Hence even after combining Statement A & B, it is not sufficient to answer .

Answer is E

Please clarify and change the OA

I think it should be Answer E.

Standard deviation is the dispersion around the mean. Knowing the relative ranges and/or number of terms does not help us find the mean, nor the dispersion around it.

Let's take 2 examples:
Set A can have 2 data elements, 1 at each extreme, and the rest of the data elements can be right at the mean.
Set B can have all of its data elements at its 2 extremes.
This means (standard deviation of A) < (standard deviation of B)

Or

Set B can have 2 data elements, 1 at each extreme, and the rest of the data elements can be right at the mean.
Set A can have all of its data points at its 2 extremes.
This means (standard deviation of A) > (standard deviation of B)

Hi gmatbusters

Usually in a set when we say number of terms, it refers to set's cardinal number. so as per statement 2, set's A cardinal number is more than that of set B.

However the illustration that you took, the cardinal number of Set A=Set B (i.e. 1,2,3 in both cases), hence you are getting S.D of A more than S.D of B.

Hi GMATinsight

Kindly clarify Statement 2 of the question because as per your OA it should refer to set's cardinal number (i.e unique elements). In my solution I have assumed that the terms are different and hence got the OA as C

Dear Sir

1) In statistics question involving mean, standard deviation etc, existence of repeated elements is very common.
We have to take care of repeated elements. Hence I feel we need to take the case of repeated elements also.

2) Secondly if we take all elements different, then also We can have SD of Set A>SD of Set B
eg:
If Set A = {1,1.1,1.2,2,2.8,2.99,3} ; SD = 0.851
If Set B = { 1,2,3} ; SD = 0.816

Hence SD of Set A> SD of Set B

Answer Should be E.

We can see this like this :
The standard deviation is a widely used concept in statistics and it tells how much variation (spread or dispersion) is in the data set.
Even we have the same range, but since we have number of elements greater in Set A, the new elements in Set A can be
1) Closer to mean : it gives lower SD
2) Closer to the extreme value of set: It gives higher SD.

Hence the SD of Set A can be lower or higher than SD of Set B

Hi GMATinsight

I think the answer should be E.

Hi gmatbusters

At first I am no "Sir" buddy, don't make me so old
This is where the confusion is about (and forget about my logic on "number of terms", I guess I was sleeping when I did not realize about non-integer numbers )

Usually in Statistics we use the formula for Standard Deviation of a "Sample" which is given below -


Whereas GMAT uses the formula for Standard Deviation of a "Population" which ignores "Bessel's Correction" i.e the use of \(n-1\), instead of \(n\)


So if you are using S.D for population then mathematically you Can get higher S.D for set A than for Set B. But here the author is using S.D for Sample.

unfortunately Mr. Bill Gates (i.e M.S Excel) also uses S.D for sample formula . Generally S.D calculation is not asked in GMAT and while solving this question I used Descriptive statistics tool in excel to calculate S.D, hence I was always getting S.D of A< S.D of B.

Here's your example and you can see how the solution changes -


Logically speaking as the number of terms in Set A & B are not mentioned, that is they could be a infinite set, so S.D of Sample formula should be used. Also as the boundary of both the sets is fixed and Set A has more elements than Set B so ideally S.D of A<S.D of B

Now it is up to owner of the question GMATinsight to clarify the logic behind the question and solution.

In GMAT although the calculation of S.D is not asked but if required we will have to use formula given in OG i.e. S.D for Population. Hence Your calculation is correct, but the logic behind the question and solution is debatable.

Hi

Actually, I have taken non integer values arbitrarily. But we can prove the same by integers value also.

Secondly the formula as per GMAT purview is for denominator " n" only.

It was really a nice discussion, it really refreshed my old school concepts.

Thanks to the question.

It seems no one likes my insights, it is quite evident from the absence of any kudos.



Posted from my mobile device

Bunuel chetan2u

Can you please provide the detailed explanation for this question?

Thanks.

The answer would be E..
SD is the spread of terms around the mean...
It is either 0, when all terms are same or >0..

1) Range of set A = Range of set B
Let the numbers be 1,3,5 in set A and 5,7,9
Range is same and even SD is same as same spread
A is 1,2,3,4,5 and B is 5,7,9
SD of A>SD of B
A is 5,7,9 and B is 1,2,3,4,5
SD of A<SD of B
Insufficient

2) numbers of term in A>number of terms in B
Range not known
1,1,1,1,1 in A and 1,2,3 in B
SD of A<SD of B
1,1,1,1,1 in A and 1,1,1 in B
SD are same
1,2,3,4,5 in A and 1,1,1 in B
SD of A>SD of B
Insufficient

Combined
1,1,1,1,1 in A and 1,1,1 in B.......same SD
1,3,3,3,3,3,5 in A and 1,1,2,2,4,5 in B.....SD of A<SD of B
1,2,3,4,5 in A and 1,3,5 in B.......SD of A> SD of B
Insufficient

E

Note :- number of terms in sets and same range will not be sufficient to talk about spread of terms and in turn relations of SD of sets. SD is dependent on spread of numbers

Basically, we are trying to find out how closely spaced are the elements of set A compared to those of set B.

Statement 1 gives us that the range of set A and that of set B is the same. But it does not tell us about the number of elements in Set A or Set B; it also does not give any information on Mean of Set A and Set B. Therefore, insufficient.

Statement 2 gives that the number of elements in Set A is greater than that in Set B. However, it does not tell us about any range of elements in Sets A and B or about the mean of respective sets. Hence, insufficient.

Both Statement 1 & 2 combined: We know the range of set A and set B is the same. We know that the number of elements in set A is greater than that in set B. This shows that within the same maximum and minimum elements, i.e. within the same boundary limits, the number of elements in set A is greater than that in set B. This naturally shows that elements in A are more closely grouped than the elements in B. Thus, the Standard Deviation of set A is smaller than that of set B. This gives us a definite NO answer.
Thus, sufficient.

Answer C.

Please see above discussions, the Answer shall be E.

The standard deviation is a widely used concept in statistics and it tells how much variation (spread or dispersion) is in the data set.
Even we have the same range, but since we have number of elements greater in Set A, the new elements in Set A can be
1) Closer to mean : it gives lower SD
2) Closer to the extreme value of set: It gives higher SD.

Hence the SD of Set A can be lower or higher than SD of Set B.

Hi chetan2u

In my opinion this is an ambiguous question and should not be part of GMAT discussions. As per data analysis tool SD of set A will never be greater than SD of set B. Here are the SD's of your illustration as per Descriptive statistics which is generally used in businesses because its provided in MS Excel



Even if you plot the data of your second example in a scatter plot you will get something like this which illustrates that SD of B> SD of A because data are more spread in Set B than Set A



So if we use MS excel then in no case SD of A>SD of B which gives us the answer as Option C

BUT if we use the formula for SD as used in GMAT then we can get SD of A> SD of B which will lead to Option E as an answer.

There are two different formula for SD, SD for sample and SD for Universe as explained in my earlier post, hence there will be different results.

Hence I feel this is NOT A GMAT question as it leaves ambiguity and GMAT never gives a question which could be challenged, hence should be archived

In the hindsight suppose, if we have a situation where we have two identical rooms (boundaries are same i.e the range), in one room there are 10 people and in the other room there are 3 people, then obviously there will be more spread in the second room than in the first room.

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