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# Sets A, B and C are shown below. If number 100 is included

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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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A {30, 50, 70, 90, 110}, B {-20, -10, 0, 10, 20}, C {30, 35, 40, 45, 50}

Initially SD(A) > SD(B) > SD(C). When 100 is added. The range of A is unchanged, so least change. But to calculate any useful relationship between the modified A Vs B we have to know the fact that B contains negative numbers. So we will get new sds as follows SD(B) > SD(C) > SD(A). Pls verify this reasoning.

To prove this inference let me calculate change in mean for sets B and C -
m(B) changes by (100 - 0)/6 = 50/3 = 16.67 hence the new mean of set B is 16.67 + Old mean = 16.67
m(C) changes by (100 - 40)/6 = 10. Hence the new mean of set C is 10 + Old mean = 10 + 40 = 50

Now the new distances from their respective means of set B (mean 16.67) and set C (mean 50)
B = {36.67, 26.67,16.67, 6.67, 3.33, 83.33}
C = {20,15,10,5,0,50}

Hence SD(B) > SD(C) > SD (A). Answer E. So this is not a 120 sec question. How to save time ?
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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VeritasPrepKarishma 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.

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

Thanks Karishma. You saved a lot of time and effort !!
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
VeritasPrepKarishma 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

Hey Karishma.... I am stuck.... I dealt with it in another way..... for any series.... If we draw Gaussian curve.... https://upload.wikimedia.org/wikipedia/c ... am.svg.png
the min and the max are 8sigma interval difference... where sigma is standard deviation.... in other terms... min and max are 4sigma intervals away from mean.... So if we calculate sigma for the sets as max-min/8....
Set A will be 110-30/8 which is 80/8
Set B will be 100-(-20)/8 which is 120/8
Set C will be 100-30/8 which is 70/8

So in order of highest to lowest wouldn't it be B,A,C?
Where am I going wrong? thank you
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
Amateur wrote:
VeritasPrepKarishma 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

Hey Karishma.... I am stuck.... I dealt with it in another way..... for any series.... If we draw Gaussian curve.... https://upload.wikimedia.org/wikipedia/c ... am.svg.png
the min and the max are 8sigma interval difference... where sigma is standard deviation.... in other terms... min and max are 4sigma intervals away from mean.... So if we calculate sigma for the sets as max-min/8....
Set A will be 110-30/8 which is 80/8
Set B will be 100-(-20)/8 which is 120/8
Set C will be 100-30/8 which is 70/8

So in order of highest to lowest wouldn't it be B,A,C?
Where am I going wrong? thank you

First of all, this is not a normal distribution. In a normal distribution, the values are concentrated around the mean (as is obvious from the normal distribution curve). You cannot calculate the SD of these sets based on the ND curve.
Secondly, you have to order them in terms of the absoluteincrease in their standard deviation, not in terms of their new SD.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
VeritasPrepKarishma 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.

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

Hi Karishma,

I can understand that by adding 100 to the three sets the extent to which the S.D changes is based on the absolute difference b/w the mean and 100. But based on this, how can you conclude that the new SD will be in B, C & A order??

Thanks.
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Sets A, B and C are shown below. If number 100 is included [#permalink]
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aalriy wrote:
VeritasPrepKarishma 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.

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

Hi Karishma,

I can understand that by adding 100 to the three sets the extent to which the S.D changes is based on the absolute difference b/w the mean and 100. But based on this, how can you conclude that the new SD will be in B, C & A order??

Thanks.

Notice that the denominator in the calculation of SD will be the same in the case of all the 3 sets (since they all have 5 elements each). When you add 100 to each one of them, they will have 6 elements each and hence the denominator will still stay the same.

In case of set B, the numerator increases by 100^2 (before you take the root)
In case of set C, the numerator increases by 60^2 (before you take the root)
In case of set A, the numerator increases by 30^2 (before you take the root)
So in absolute terms, B will see the most effect and A will see the least. You can look at the actual calculation to understand exactly why this happens. The formula for SD is discussed in the first post below.

Originally posted by KarishmaB on 23 Nov 2012, 13:05.
Last edited by KarishmaB on 10 Oct 2022, 23:00, edited 1 time in total.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
Quote:
Notice that the denominator in the calculation of SD will be the same in the case of all the 3 sets (since they all have 5 elements each). When you add 100 to each one of them, they will have 6 elements each and hence the denominator will still stay the same.

In case of set B, the numerator increases by 100^2 (before you take the root)
In case of set C, the numerator increases by 60^2 (before you take the root)
In case of set A, the numerator increases by 30^2 (before you take the root)
So in absolute terms, B will see the most effect and A will see the least. You can look at the actual calculation to understand exactly why this happens. The formula for SD is discussed in the first post below.

For more on SD, check out these posts:

Hi Karishma,
From what I can infer, it seems that the order of 'increase in the SD' and the order of 'new SD' will always be the same ..
Please correct me if its incorrect.
Regards,
Sach
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
Sachin9 wrote:

Hi Karishma,
From what I can infer, it seems that the order of 'increase in the SD' and the order of 'new SD' will always be the same ..
Please correct me if its incorrect.
Regards,
Sach

Actually no, that may not be the case.

The increase in SD depends on the distance between the number added (100 here) and the mean. Set B has the smallest mean (0) so it is farthest from 100 hence it will see maximum increase.
The 'new SD' depends on the difference between all the elements (including the new one) and the mean. If the rest of the numbers are very close to the mean, it is certainly possible that the new SD does not have the same ordering.

e.g.

Set A = {-1, 0, 1}
Set B = (0, 20, 40}
If you add another number say, 30, the increase in SD of set A will be substantial because 30 is far from mean
but increase in SD of set B will not be very much.
Nevertheless, new SD of set A will be less than the new SD of set B.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
VeritasPrepKarishma wrote:
Sachin9 wrote:

Hi Karishma,
From what I can infer, it seems that the order of 'increase in the SD' and the order of 'new SD' will always be the same ..
Please correct me if its incorrect.
Regards,
Sach

Actually no, that may not be the case.

The increase in SD depends on the distance between the number added (100 here) and the mean. Set B has the smallest mean (0) so it is farthest from 100 hence it will see maximum increase.
The 'new SD' depends on the difference between all the elements (including the new one) and the mean. If the rest of the numbers are very close to the mean, it is certainly possible that the new SD does not have the same ordering.

e.g.

Set A = {-1, 0, 1}
Set B = (0, 20, 40}
If you add another number say, 30, the increase in SD of set A will be substantial because 30 is far from mean
but increase in SD of set B will not be very much.
Nevertheless, new SD of set A will be less than the new SD of set B.

I guess the new SD of A will be more than the new SD of B..
Numbers in A would be more dispersed than those in B..
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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Sachin9 wrote:
I guess the new SD of A will be more than the new SD of B..
Numbers in A would be more dispersed than those in B..

No. You can use a fin calc to find that the SD of A is 13 and that of B is 14.8. The difference isn't much but still the new SD of A is less than the new SD of B. As I said, what matters is that how far apart are all the elements from the mean in case of new SD. One element can have a huge impact but it still may not be sufficient. So you cannot infer that the new SD will be in the same order.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
VeritasPrepKarishma wrote:
Sachin9 wrote:
I guess the new SD of A will be more than the new SD of B..
Numbers in A would be more dispersed than those in B..

No. You can use a fin calc to find that the SD of A is 13 and that of B is 14.8. The difference isn't much but still the new SD of A is less than the new SD of B. As I said, what matters is that how far apart are all the elements from the mean in case of new SD. One element can have a huge impact but it still may not be sufficient. So you cannot infer that the new SD will be in the same order.

Thanks a lot, Karishma..

From what I understand,highest effect would depend on how far is the new no. from the mean in all the sets..

and Actual order of SD among sets would depend on how dispersed all elements are from the mean..
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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Sachin9 wrote:
VeritasPrepKarishma wrote:
Sachin9 wrote:
I guess the new SD of A will be more than the new SD of B..
Numbers in A would be more dispersed than those in B..

No. You can use a fin calc to find that the SD of A is 13 and that of B is 14.8. The difference isn't much but still the new SD of A is less than the new SD of B. As I said, what matters is that how far apart are all the elements from the mean in case of new SD. One element can have a huge impact but it still may not be sufficient. So you cannot infer that the new SD will be in the same order.

Thanks a lot, Karishma..

From what I understand,highest effect would depend on how far is the new no. from the mean in all the sets..

and Actual order of SD among sets would depend on how dispersed all elements are from the mean..

Yes, that's correct.
'Change' in SD depends on how far the new no is from the mean. If the new no is close to the mean, the change in SD is very little because it adds very little dispersion to the scenario. If the new no is far from the mean, the change in SD is significant because it adds a lot more dispersion. The mean changes, it becomes farther than the previous mean and hence overall dispersion in a lot higher.

Actual SD depends in big part on how the previous numbers were dispersed around the mean. So it is hard to say what the new order will be based on just the new no. and the previous mean.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
5 sec. approach:

SD is the deviation of the mean to the smallest and biggest number in the set. So if a number is added inside the boundaries of that set, there will be no changes in the SD of that set. So no absolute increase will occur in set A.

A has to be the set with the smallest increase, what's bigger than 0 right? And hence E is the only right answer.

Correct me if I'm wrong
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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.

For evenly spaced sets, median = mean

so mean of 3 sets are 70, 0 & 40.

see for yourself 100 is farthest from which mean
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
of all sets the SD of set A will either see no change or decrease because 100 is less than 110, whereas B and C will definitely see the increase in SD.

the only answer that has A at the end is E
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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Nicely explained by Karishma. A good question but One needs to know How mean and distance from mean changes SD.
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Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
Definition of Standard deviation =>
It is a Statistical tool that is used to measure the average dispersion/deviation/variation of individual data element around the arithmetic mean.
A low value of standard deviate indicates that data elements are close to the mean.
A high value of standard deviation indicates that data elements are far away from the mean.

Mean =>
A-> 70
B->0
C->40

Hence B>C>A is the order

Hence E
Re: Sets A, B and C are shown below. If number 100 is included [#permalink]
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