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GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4006
Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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vjsharma25 wrote:
Sets A, B and C are shown below. If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their standard deviation, from largest to smallest?

A {30, 50, 70, 90, 110}, B {-20, -10, 0, 10, 20}, C {30, 35, 40, 45, 50}

(A) A, C, B
(B) A, B, C
(C) C, A, B
(D) B, A, C
(E) B, C, A

----ASIDE-------------------------

For the purposes of the GMAT, it's sufficient to think of Standard Deviation as the Average Distance from the Mean.
Here's what I mean:

Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
The mean of set A = 10 and the mean of set B = 10
How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

Alternatively, let's examine the Average Distance from the Mean for each set.

Set A {7,9,10,14}
Mean = 10
7 is a distance of 3 from the mean of 10
9 is a distance of 1 from the mean of 10
10 is a distance of 0 from the mean of 10
14 is a distance of 4 from the mean of 10
So, the average distance from the mean = (3+1+0+4)/4 = 2

B {1,8,13,18}
Mean = 10
1 is a distance of 9 from the mean of 10
8 is a distance of 2 from the mean of 10
13 is a distance of 3 from the mean of 10
18 is a distance of 8 from the mean of 10
So, the average distance from the mean = (9+2+3+8)/4 = 5.5

IMPORTANT: I'm not saying that the Standard Deviation of set A equals 2, and I'm not saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A.
More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GMAT.

------NOW ONTO THE QUESTION!!!---------------

So, for this question, we have:

Mean of set A = 70
Mean of set B = 0
Mean of set C = 40

100 is furthest away from the mean of 0 in set B, so this will cause the GREATEST change in standard deviation.
100 is next furthest away from the mean of 40 in set C, so this will cause the 2nd greatest change in standard deviation.
100 is closest to the mean of 70 in set A, so this will cause the LEAST change in standard deviation.

Cheers,
Brent

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Joined: 05 Aug 2018
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Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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vjsharma25 wrote:
Sets A, B and C are shown below. If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their standard deviation, from largest to smallest?

A {30, 50, 70, 90, 110}, B {-20, -10, 0, 10, 20}, C {30, 35, 40, 45, 50}

(A) A, C, B
(B) A, B, C
(C) C, A, B
(D) B, A, C
(E) B, C, A

Dear VeritasPrepKarishma,
this is what i found in your post
"If you notice, we have seen two different cases (case 4 and case 5)– in one of them SD increases when you add two numbers to the set and in the other, SD decreases.
Case 4: S = {1, 3, 5} or T = {1, 1, 3, 5, 5} T has higher SD. It has two extra numbers far from the mean
Case 5: S = {1, 3, 5} or T = {1, 3, 3, 5} The standard deviation (SD) of T will be less than the SD of S
So how do you decide whether SD will increase or decrease? Say, what happens in case S = {3, 4, 5, 6, 7} and T = {3, 4, 4, 5, 6, 6, 7}? "
you also said that"If a new element is added which is far away from the mean, it will add much more to the deviations than if it were added close to the mean."

so in case 4: S = {1, 3, 5} ;mean =3 ; the difference of each number of S to mean is 2
T = {1, 1, 3, 5, 5} ; T =3; T added two numbers (1,5) the difference of 1 to the mean(T) of 3 is 2 ; the difference from 5 to the mean(T) of 3 is 2; my question is why It has two extra numbers far from the mean";
Intern  B
Joined: 05 Aug 2018
Posts: 7
Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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vjsharma25 wrote:
Sets A, B and C are shown below. If number 100 is included in each of these sets, which of the following
represents the correct ordering of the sets in terms of the absolute increase in their standard deviation, from
largest to smallest?
A {30, 50, 70, 90, 110}, B {-20, -10, 0, 10, 20}, C {30, 35, 40, 45, 50}
(A) A, C, B
(B) A, B, C
(C) C, A, B
(D) B, A, C
(E) B, C, A

As per me answer should be .
Wants to confirm or reject the OA.

Can you please post your source of this question?
It is a Veritas Prep Book X question for which the OA given is (E). The explanation clearly explains you why the answer is E.
You don't have to calculate anything. SD measures the distance between each element and mean. If a new element is added which is far away from the mean, it will distort the mean more than if it were added close to the mean.
The means of the 3 sets are 70, 0 and 40.
100 is farthest from 0 so it will change the SD of set B the most (in terms of absolute increase). It is closest to 70 so it will change the SD of set A the least. Hence answer is B, C, A

Dear VeritasPrepKarishma,
this is what i found in your post
"If you notice, we have seen two different cases (case 4 and case 5)– in one of them SD increases when you add two numbers to the set and in the other, SD decreases.
Case 4: S = {1, 3, 5} or T = {1, 1, 3, 5, 5} T has higher SD. It has two extra numbers far from the mean
Case 5: S = {1, 3, 5} or T = {1, 3, 3, 5} The standard deviation (SD) of T will be less than the SD of S
So how do you decide whether SD will increase or decrease? Say, what happens in case S = {3, 4, 5, 6, 7} and T = {3, 4, 4, 5, 6, 6, 7}? "
you also said that"If a new element is added which is far away from the mean, it will add much more to the deviations than if it were added close to the mean."

so in case 4: S = {1, 3, 5} ;mean =3 ; the difference of each number of S to mean is 2
T = {1, 1, 3, 5, 5} ; T =3; T added two numbers (1,5) the difference of 1 to the mean(T) of 3 is 2 ; the difference from 5 to the mean(T) of 3 is 2; my question is why It has two extra numbers far from the mean";
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9705
Location: Pune, India
Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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xiaoxueren wrote:
vjsharma25 wrote:
Sets A, B and C are shown below. If number 100 is included in each of these sets, which of the following
represents the correct ordering of the sets in terms of the absolute increase in their standard deviation, from
largest to smallest?
A {30, 50, 70, 90, 110}, B {-20, -10, 0, 10, 20}, C {30, 35, 40, 45, 50}
(A) A, C, B
(B) A, B, C
(C) C, A, B
(D) B, A, C
(E) B, C, A

As per me answer should be .
Wants to confirm or reject the OA.

Can you please post your source of this question?
It is a Veritas Prep Book X question for which the OA given is (E). The explanation clearly explains you why the answer is E.
You don't have to calculate anything. SD measures the distance between each element and mean. If a new element is added which is far away from the mean, it will distort the mean more than if it were added close to the mean.
The means of the 3 sets are 70, 0 and 40.
100 is farthest from 0 so it will change the SD of set B the most (in terms of absolute increase). It is closest to 70 so it will change the SD of set A the least. Hence answer is B, C, A

Dear VeritasPrepKarishma,
this is what i found in your post
"If you notice, we have seen two different cases (case 4 and case 5)– in one of them SD increases when you add two numbers to the set and in the other, SD decreases.
Case 4: S = {1, 3, 5} or T = {1, 1, 3, 5, 5} T has higher SD. It has two extra numbers far from the mean
Case 5: S = {1, 3, 5} or T = {1, 3, 3, 5} The standard deviation (SD) of T will be less than the SD of S
So how do you decide whether SD will increase or decrease? Say, what happens in case S = {3, 4, 5, 6, 7} and T = {3, 4, 4, 5, 6, 6, 7}? "
you also said that"If a new element is added which is far away from the mean, it will add much more to the deviations than if it were added close to the mean."

so in case 4: S = {1, 3, 5} ;mean =3 ; the difference of each number of S to mean is 2
T = {1, 1, 3, 5, 5} ; T =3; T added two numbers (1,5) the difference of 1 to the mean(T) of 3 is 2 ; the difference from 5 to the mean(T) of 3 is 2; my question is why It has two extra numbers far from the mean";

When we add numbers far from the mean, they add much more to the numerator and relatively less to the denominator. Though in this question, T does end up having the same SD as S (and that is the caveat I mentioned on my post).

But this is the general concept. So a number added at extreme {1, 3, 5, 100} will increase the SD much more than say {1, 3, 5, 6}
_________________
Karishma
Veritas Prep GMAT Instructor

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Intern  B
Joined: 21 Feb 2019
Posts: 22
Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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Hi. I solved this Q by a different method, not sure if this is correct or a fluke. Can you please confirm?

Initial SD - A > B > C because of their range (80 >40 >20)
New SD - B > A > C becasue of the new range (120 > 80 > 70).

Since the Q is asking the absolute increase in their standard deviation, I subtracted the new and old SD which led to B (120 - 40)> C(70-20) >A(80-80).

Thanks
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9705
Location: Pune, India
Re: Sets A, B and C are shown below. If number 100 is included  [#permalink]

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kanikab wrote:

Hi. I solved this Q by a different method, not sure if this is correct or a fluke. Can you please confirm?

Initial SD - A > B > C because of their range (80 >40 >20)
New SD - B > A > C becasue of the new range (120 > 80 > 70).

Since the Q is asking the absolute increase in their standard deviation, I subtracted the new and old SD which led to B (120 - 40)> C(70-20) >A(80-80).

Thanks

In this question, range does help us since it is related to the SD in the same way. 100 is the greatest number in each set and the sets are evenly distributed around the mean so 100 increases the range in the same way in which it increases the SD. But for most SD questions, I wouldn't relate range to SD since SD depends on lots of other attributes too.
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Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > Re: Sets A, B and C are shown below. If number 100 is included   [#permalink] 21 Aug 2019, 21:55

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