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Difficult Math Question #50 - Standard Deviation

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Difficult Math Question #50 - Standard Deviation

by 800guy » Wed Nov 08, 2006 3:54 pm
A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?

A). -1 and 9
B). 4 and 4
C). 3 and 5
D). 2 and 6
E). 0 and 8
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answer

by gamemaster » Mon Nov 13, 2006 3:22 am
the average of the series is 4
the standard deviation is: sqrt [(40)/5] = sqrt [8]

we would like the new sum to be ~56 to get sqrt[8] so the closest will be:

2 and 6 then the standard deviation would be: sqrt[48/7]

D
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by rajs.kumar » Mon Nov 13, 2006 10:43 am
game I get a different answer. This is what I did.

Answer is C.

SD = srqt( sum(xi - x)^2/n)

SD = standard deviation
(xi - x)^2 = square of the difference between sample and the mean
n = number of samples in the population

the denominator n increases from 5 to 7. This means the new sum(xi - x)^2 should increase by a factor 7/5 = 1.4 to get the same value for the SD.

-1 and 9 the sum increases by 50 (no)
4 and 4 the sum remains the same (no)
3 and 5 the sum increases by 2 (yes)
2 and 6 the sum increases by 8 (no)
0 and 8 the sum increases by 32 (no)
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.

by gamemaster » Mon Nov 13, 2006 11:12 am
well in case its 3 and 5, the standard deviation would be

sqrt[42/7] = sqrt[6]

in case its 2 and 6 it will be:

sqrt[48/7] which is bigger than sqrt[6] but smaller than sqrt[8] so closer to sqrt[8] than sqrt[6] is

so i still think its 2 and 6
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OA

by 800guy » Mon Nov 13, 2006 4:57 pm
OA:

SD = Sqrt(Sum(X-x)^2/N)
Since N is changing from 5 to 7. Value of Sum(X-x)^2 should change from 40(current) to 48. So that SD remains same.

so due to new numbers it adds 8. Choice D only fits here.
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by deya1107 » Sun Oct 20, 2013 11:54 pm
The answer would be (E) 0 and 8. The standard deviation of the original series is ~3.16. I have calculated each option and tally with the original one. The standard deviation of the series including 0 and 8 is ~3.464 which is the closest among all.

There is another option (B) 4 and 4. Here the standard deviation are not changing. So, it is moot whether (B) or (E).
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by sahilchaudhary » Mon Oct 21, 2013 12:51 am
The original S.D. is sqrt(8) = 2.8.
So, I think the two closest options to this are 2 & 6.
So, D.
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by rakeshd347 » Mon Oct 21, 2013 1:17 am
800guy wrote:A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?

A). -1 and 9
B). 4 and 4
C). 3 and 5
D). 2 and 6
E). 0 and 8
Answer to this question should be A. Why

Standard deviation tells you how wide spread numbers are from the mean. If you add 0 or 8 it won't change anything because they are already in the set. If you add 4,4 it will reduce the SD because we will have 3 4's in the set. Same is C and D. 3 &5 and a2&6 are already in the set so adding them won't make any difference but if you widen the range or add new numbers outside the range then it will change the SD.

One more thing you don't need to calculate SD here. Gmat will never give you question in which you have to calculate SD.

Thanks,
Rakesh
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by theCodeToGMAT » Mon Oct 21, 2013 1:31 am
Rakesh, SD varies with Mean i.e. the farther the numbers from mean.. greater will be SD.

So, {A} & {E} are surely out

I am stuck between {B}, {C} & {D}.

I know we can solve and find, but GMAC doesn't want us to do so.

However, if the question had been "which of the combination will result in lowest SD" then it answer would have been {B} 4,4
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by rakeshd347 » Mon Oct 21, 2013 1:59 am
theCodeToGMAT wrote:Rakesh, SD varies with Mean i.e. the farther the numbers from mean.. greater will be SD.

So, {A} & {E} are surely out

I am stuck between {B}, {C} & {D}.

I know we can solve and find, but GMAC doesn't want us to do so.

However, if the question had been "which of the combination will result in lowest SD" then it answer would have been {B} 4,4
Hi Rahul,

Your reasoning is wrong. Let me elaborate it here.
If you add -1,9 to the set the mean will still be 4 ( the mean for the current set is 4). however 9 will be farther from 4 than all other number and hence SD will rise. So answer is 100% A.
If you add 4,4...then SD will reduce as 4 is already in the set and mean won't change it will be 4 itself so 3 numbers are on the mean.
If you add 3 and 5....then SD will not reduce rather than increase because 3 and 5 are within the number set given.
same is with 2&6 and 0&8.

So final answer is definitely
[/spoiler]A[spoiler]. I don't know why you crossed out A. It is indeed the OA.[/spoiler]
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by rakeshd347 » Mon Oct 21, 2013 2:03 am
theCodeToGMAT wrote:Rakesh, SD varies with Mean i.e. the farther the numbers from mean.. greater will be SD.

So, {A} & {E} are surely out

I am stuck between {B}, {C} & {D}.

I know we can solve and find, but GMAC doesn't want us to do so.

However, if the question had been "which of the combination will result in lowest SD" then it answer would have been {B} 4,4
To even give better understanding if you would have add -2 and 10 then this new set will have even higher SD than adding -1,9. Why? because 10 is farthest from mean 4 and if we remove 10 then numbers will come closer and SD will reduce.

I hope this makes sense let me know if you still have confusion.
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by rakeshd347 » Mon Oct 21, 2013 2:06 am
theCodeToGMAT wrote:Rakesh, SD varies with Mean i.e. the farther the numbers from mean.. greater will be SD.

So, {A} & {E} are surely out

I am stuck between {B}, {C} & {D}.

I know we can solve and find, but GMAC doesn't want us to do so.

However, if the question had been "which of the combination will result in lowest SD" then it answer would have been {B} 4,4
ZEEEEE....i have been calculating set which will have higher than the current set. Not close to the current one.
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by theCodeToGMAT » Mon Oct 21, 2013 2:13 am
Rakesh, I think you must misunderstood the question. The question says:

"will result in a new standard deviation that is close to the standard deviation for the original 5 numbers"

This means that closest difference between Original & New Set

Aside, just to be concrete on my answer:

ORIGINAL SET: SD = 3.16
A). -1 and 9 => SD = 3.87 ==> Difference = 0.71
B). 4 and 4 => SD = 2.58 ==> Difference = 0.56 # Lowest SD; I hope my point is clear now :)
C). 3 and 5 => SD = 2.64 ==> Difference = 0.50
D). 2 and 6 => SD = 2.828 ==> Difference = 0.332
E). 0 and 8 => SD = 3.464 ==> Difference = 0.324

As I said, GMAC doesn't want us to do so much calculation.. So, i doubt on this question
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by Brent@GMATPrepNow » Mon Oct 21, 2013 6:46 am
theCodeToGMAT wrote:Rakesh, I think you must misunderstood the question. The question says:

"will result in a new standard deviation that is close to the standard deviation for the original 5 numbers"

This means that closest difference between Original & New Set

Aside, just to be concrete on my answer:

ORIGINAL SET: SD = 3.16
A). -1 and 9 => SD = 3.87 ==> Difference = 0.71
B). 4 and 4 => SD = 2.58 ==> Difference = 0.56 # Lowest SD; I hope my point is clear now :)
C). 3 and 5 => SD = 2.64 ==> Difference = 0.50
D). 2 and 6 => SD = 2.828 ==> Difference = 0.332
E). 0 and 8 => SD = 3.464 ==> Difference = 0.324

As I said, GMAC doesn't want us to do so much calculation.. So, i doubt on this question
Be careful, theCodeToGMAT. You have use the formula for the sample standard deviation, and you need to use the formula for the population standard deviation.

For more see: https://www.mathsisfun.com/data/standard ... mulas.html

You can plug in values here to get population SDs: https://easycalculation.com/statistics/s ... iation.php


Having said that, I do agree with you that the question required FAR TOO many calculations to be GMAT-worthy.

Cheers,
Brent
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by theCodeToGMAT » Mon Oct 21, 2013 6:54 am
Brent@GMATPrepNow wrote: Be careful, theCodeToGMAT. You have use the formula for the sample standard deviation, and you need to use the formula for the population standard deviation.

For more see: https://www.mathsisfun.com/data/standard ... mulas.html

You can plug in values here to get population SDs: https://easycalculation.com/statistics/s ... iation.php


Having said that, I do agree with you that the question required FAR TOO many calculations to be GMAT-worthy.

Cheers,
Brent
Hello Brent, I didn't do the calculations manually .. But yes.. I use the formula on the link you shared :)

The link I used for calculations: https://www.calculator.net/standard-devi ... lator.html

Ok, now i see why there were different types of SDs :) ..

Thanks!
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