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# Which of the following pairs of numbers, when added to the

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Director
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Which of the following pairs of numbers, when added to the [#permalink]  05 Dec 2012, 19:35
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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 [#permalink]  06 Dec 2012, 02:39
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 [#permalink]  06 Dec 2012, 05:15
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 [#permalink]  06 Dec 2012, 14:37
Expert's post
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 [#permalink]  08 Dec 2012, 22:29
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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Re: Which of the following pairs of numbers, when added to the   [#permalink] 08 Dec 2012, 22:29
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