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Senior Manager  Joined: 28 Jul 2011
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Which of the following pairs of numbers, when added to the set above,  [#permalink]

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(9,12,15,18,21)

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Hi Bunnel,

This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning

I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1

I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

Regards
Srinath
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Joined: 28 Dec 2011
Posts: 4476
Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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3
4
mydreammba wrote:
This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?
1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning
I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1
I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

I am responding to a pm from mydreammba.

First of all, your reasoning as regards these three pairs of numbers is perfectly correct. Option #2 increases the SD, option #3 flamboyantly increases the SD, and option #1 decreases the SD.

Now, as far as generalizing --- what you are suggesting there is not correct.

Think about mean first. Suppose Set Q has a mean of 7. If I add a new number that's more than the mean, more than 7, adding it to the set will increase the mean of the set. If I add a number that's less than the mean, less than 7, adding it to the set will decrease the mean of the set.

Well, the Standard Deviation is a kind of mean, a kind of average. If you add a pair of numbers, equal and opposite distances from the mean, whose deviations from the mean is (in absolute values terms) more than the standard deviation of the set, then adding these numbers will increase the average among the deviations from the mean, that is to say, it will increase the SD of the set. If you add a pair of numbers, equal and opposite deviations from the mean, whose distance from the mean is less than the standard deviation of the set, then adding these numbers will decrease the average among the deviations from the mean, that is to say, it will decrease the SD of the set. If you add numbers that are not symmetrically distributed with respect to the mean, then that changes the mean itself, which means every single value has a new deviation from the mean, so all bets are off (the test will not ask you about this case, unless it's really obvious, as in Option #3).

In problems like this, no one actually expects you to calculate the SD. The GMAT will not expect you to do that. You are just be asked to estimate.

Now, when I look at the set {9,12,15,18,21}, it's symmetrically distributed. I know the mean is 15. The deviations from the mean are {-6, -3, 0, 3, 6} ----- none of them in this particular set has an absolute value greater than six, so the SD absolutely can't be greater than 6 --- in fact it has to be less than 6. (Any average over several values has to be lower than the largest value.) That's why six is a crucial number in this particular case.

In the set {30, 40, 50, 60, 70}, the mean is 50, and the largest deviations from the mean are +/-20, so the SD must be less than 20. Adding {47, 53} would definitely decrease the SC. Adding {30, 70) would definitely increase the SD. Adding {30, 500) would also increase the SD.

Does all this make sense?

Mike _________________
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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mydreammba wrote:
Hi Bunnel,

This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning

I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1

I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

Regards
Srinath

Mean of the above set would be = 15

so if the number added are closer to the mean then SD will decrease and vice-versa.
=> 14,16 will lower SD and other two would increase as copared to 14,16, but i think 15,100 would increase SD to the most, so i got confused as in which set to should be take as base to compare.

Thanks
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Which of the following pairs of numbers, when added to the set above,  [#permalink]

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1
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

i Mikemcgarry,

This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning

I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1

I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

Regards
Srinath
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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My two cents on this....

more the values in the sets are away from the mean more is the standard deviation. I think as in the option 3 the number hundred will shoot up the standard deviation significantly and hence it would increase the standard deviation...Now we can see that the average of the set is 15 and option 1 and option 2 both has an average of 15 so both of them should not affect the SD...hence the answer should be the 3) 15, 100

Mike am I correct ??
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Think like that:

is the average total deviation of the two numbers to the mean of the set higher than the mean deviation of the numbers in the set?

or just take the mean of the lower half and the upper half of the set. if a number is outside this interval it will increase sd otherwise decrease
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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mikemcgarry wrote:
mydreammba wrote:
This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?
1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning
I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1
I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

I am responding to a pm from mydreammba.

First of all, your reasoning as regards these three pairs of numbers is perfectly correct. Option #2 increases the SD, option #3 flamboyantly increases the SD, and option #1 decreases the SD.

Now, as far as generalizing --- what you are suggesting there is not correct.

Think about mean first. Suppose Set Q has a mean of 7. If I add a new number that's more than the mean, more than 7, adding it to the set will increase the mean of the set. If I add a number that's less than the mean, less than 7, adding it to the set will decrease the mean of the set.

Well, the Standard Deviation is a kind of mean, a kind of average. If you add a pair of numbers, equal and opposite distances from the mean, whose deviations from the mean is (in absolute values terms) more than the standard deviation of the set, then adding these numbers will increase the average among the deviations from the mean, that is to say, it will increase the SD of the set. If you add a pair of numbers, equal and opposite deviations from the mean, whose distance from the mean is less than the standard deviation of the set, then adding these numbers will decrease the average among the deviations from the mean, that is to say, it will decrease the SD of the set. If you add numbers that are not symmetrically distributed with respect to the mean, then that changes the mean itself, which means every single value has a new deviation from the mean, so all bets are off (the test will not ask you about this case, unless it's really obvious, as in Option #3).

In problems like this, no one actually expects you to calculate the SD. The GMAT will not expect you to do that. You are just be asked to estimate.

Now, when I look at the set {9,12,15,18,21}, it's symmetrically distributed. I know the mean is 15. The deviations from the mean are {-6, -3, 0, 3, 6} ----- none of them in this particular set has an absolute value greater than six, so the SD absolutely can't be greater than 6 --- in fact it has to be less than 6. (Any average over several values has to be lower than the largest value.) That's why six is a crucial number in this particular case.

In the set {30, 40, 50, 60, 70}, the mean is 50, and the largest deviations from the mean are +/-20, so the SD must be less than 20. Adding {47, 53} would definitely decrease the SC. Adding {30, 70) would definitely increase the SD. Adding {30, 500) would also increase the SD.

Does all this make sense?

Mike Thanks Mike for a wonderful reply, now i get it
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Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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1
1) The SD decreases

2) adding two equal terma at the edges of the distribution the SD increases

3) adding 100 at the end the SD increase

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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I think all three increase the Stand deviation.

I went the traditional method with a bit of guessing. Please let me know if correct.

Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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2
4
Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.

CONCEPT:Standard Deviation represents Average Deviation of the terms from the mean of the set

Rule 1: If the new terms included in the set are at a Higher distance from mean than the Standard Deviation will INCREASE
Rule 2: If the new terms included in the set are at a Lower distance from mean than the Standard Deviation will DECREASE

Set - {9, 12, 15, 18, 21} i.e. Mean = 15 (Middle Term as all terms are equally separated) and Deviation between consecutive terms = 3

I - New Set Becomes {9, 12, 14, 15, 16, 18, 21} i.e. New Terms (14,16) are closer to the Mean of the set Hence Standard Deviation will DECREASE

II - New Set Becomes {9, 9, 12, 15, 18, 21} i.e. New Terms (9,21) are at MORE Distance (greater than 3) from the Mean of the set Hence Standard Deviation will INCREASE

III - New Set Becomes {9, 12, 15, 18, 21, 51, 100} i.e. New Terms (51,100) are at FAR MORE Distance from the Mean of the set Hence Standard Deviation will INCREASE

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Hi GMATInsight,

Please could you help me with this question...as I am a bit weak in stats.

Rule 1: If the new terms included in the set are at a Higher distance from mean than the Standard Deviation will INCREASE

I want to know do we need to calculate the mean again after adding the new variables. I ask this question because you mentioned 'New Terms (9,21) are at MORE Distance (greater than 3)' , which are actually 6 units away from the mean.

GMATinsight wrote:
Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.

CONCEPT:Standard Deviation represents Average Deviation of the terms from the mean of the set

Rule 1: If the new terms included in the set are at a Higher distance from mean than the Standard Deviation will INCREASE
Rule 2: If the new terms included in the set are at a Lower distance from mean than the Standard Deviation will DECREASE

Set - {9, 12, 15, 18, 21} i.e. Mean = 15 (Middle Term as all terms are equally separated) and Deviation between consecutive terms = 3

I - New Set Becomes {9, 12, 14, 15, 16, 18, 21} i.e. New Terms (14,16) are closer to the Mean of the set Hence Standard Deviation will DECREASE

II - New Set Becomes {9, 9, 12, 15, 18, 21} i.e. New Terms (9,21) are at MORE Distance (greater than 3) from the Mean of the set Hence Standard Deviation will INCREASE

III - New Set Becomes {9, 12, 15, 18, 21, 51, 100} i.e. New Terms (51,100) are at FAR MORE Distance from the Mean of the set Hence Standard Deviation will INCREASE

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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shriramvelamuri wrote:
Hi GMATInsight,

Please could you help me with this question...as I am a bit weak in stats.

Rule 1: If the new terms included in the set are at a Higher distance from mean than the Standard Deviation will INCREASE

I want to know do we need to calculate the mean again after adding the new variables. I ask this question because you mentioned 'New Terms (9,21) are at MORE Distance (greater than 3)' , which are actually 6 units away from the mean.

We don't have to calculate the New mean but look at the new set from the perspective of Mean of Old set only. The fact to observe is

Old set had consecutive terms at a gap of 3

The new terms 9 and 21, are at a gap higher than 3 from the previous mean (gap between 9 and 15=6 and similarly gap between 15 and 21 = 6), then the average deviation will automatically increase thereby increasing the standard deviation.

I hope it clears your doubt!
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.

The set is evenly spaced with an average and median of 15. To increase the standard deviation of the set, we need to have numbers deviating as much as possible from the mean (15).

I. 14, 16 > they actually decrease the SD, being very close to the mean (i.e. having a smaller diffrence to the mean than 3)
II. 9, 21 > Clearly increase SD because 9 and 21 are farther away from 15
III. 15, 100 > Clearly increases SD with 100

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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1
SD = deviation from the mean!
Added values closer to the mean/=mean = SD decreases
Added values if far away from the mean = SD Increases
if the set has same/equal values and you add more same values to the set = SD = 0
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

Fortunately, you do not need to perform any calculations to answer this question. The mean of the set is 15. Take a look at each Roman Numeral:

I. The numbers 14 and 16 are both very close to the mean (15). Additionally, they are closer to the mean than four of the numbers in the set, and will reduce the spread around the mean. This pair of numbers will reduce the standard deviation of the set.

II. The numbers 9 and 21 are relatively far away from the mean (15). Adding them to the list will increase the spread of the set and increase the standard deviation.

III. While adding the number 15 to the set would actually decrease the standard deviation (because it is the same as the mean of the set), the number 100 is so far away from the mean that it will greatly increase the standard deviation of the set. This pair of numbers will increase the standard deviation.

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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Bunuel wrote:
Bunuel wrote:
{9, 12, 15, 18, 21}
Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

Fortunately, you do not need to perform any calculations to answer this question. The mean of the set is 15. Take a look at each Roman Numeral:

I. The numbers 14 and 16 are both very close to the mean (15). Additionally, they are closer to the mean than four of the numbers in the set, and will reduce the spread around the mean. This pair of numbers will reduce the standard deviation of the set.

II. The numbers 9 and 21 are relatively far away from the mean (15). Adding them to the list will increase the spread of the set and increase the standard deviation.

III. While adding the number 15 to the set would actually decrease the standard deviation (because it is the same as the mean of the set), the number 100 is so far away from the mean that it will greatly increase the standard deviation of the set. This pair of numbers will increase the standard deviation.

Check other Standard Deviation Questions in our Special Questions Directory.
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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mydreammba wrote:
Hi Bunnel,

This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning

I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1

I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

Regards
Srinath

Option 3: 15 and 100 will increase the SD the most. Option 2 would also increase the SD but not as much as Option 3 would.
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Re: Which of the following pairs of numbers, when added to the set above,  [#permalink]

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mydreammba wrote:
(9,12,15,18,21)

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III

Hi Bunnel,

This is a problem from Manhattan 13th Edition

consider Set {9,12,15,18,21}

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

1. 14, 16
2. 9, 21
3. 15, 100

Here is my reasoning

I know that with Options 2 & Option 3 the SD will increase as we are adding numbers whose difference between Mean the corresponding new adding number is maximum i.e the difference is >= the max difference between any two numbers prior to the adding of the two new numbers.

Now with Option 1

I know that the numbers are closer to mean so SD will decrease, but here can i generalize always that for SD to increase the two new added numbers must be such that the difference between the two new added numbers and the mean must always be >=6 or it is enough if the difference between one new added number (among the two numbers) and the mean is >=6 ?

I think i am making things complex, but don't know somehow got confused....

Regards
Srinath

we have an evenly spaced set.
SD is 3.

ok, so first one..we add 2 numbers that are less than 1 SD away from the mean (15). This one will decrease the SD.
second one - both numbers are far away from the mean more than 1 SD. thus, it will increase.
third one - first one is the mean, thus, it will not change the SD at all. second one, on the contrary, it is very very far away from the mean. the result - will increase the SD.

ok, so first one doesn't work, but second and third works - D
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Which of the following pairs of numbers, when added to the set above,  [#permalink]

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Hi guys,

I am a little bit confused here. Why does the II variant increase the SD? we already have 9 and 21 in our set, so what difference will the same numbers make? I am using the Manhattan Word Problems book and i couldn't find the answer there. Heeeelp! Which of the following pairs of numbers, when added to the set above,   [#permalink] 09 Dec 2017, 00:48

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