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Director  Joined: 09 Aug 2006
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A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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Question Stats: 30% (01:50) correct 70% (01:28) wrong based on 672 sessions

### HideShow timer Statistics 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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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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GK_Gmat 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

For GMAT purposes, Standard Deviation (SD) can often be thought of as "the average distance the data points are away from the mean."

So, with {0, 2, 4, 6, 8}, the mean is 4.
0 is 4 units away from the mean of 4.
2 is 2 units away from the mean of 4.
4 is 0 units away from the mean of 4.
6 is 2 units away from the mean of 4.
8 is 4 units away from the mean of 4.

So, the SD can be thought of as the average of 4, 2, 0, 2, and 4. The average of these values is 2.4, so we'll say that the SD is about 2.4

Note: This, of course, isn't 100% accurate, but it's all you should really need for the GMAT.

Okay, so which pair of new numbers, when added to the original 5, numbers will yield a new SD that is closest to 2.4?

Well, to begin, it's useful to notice that each pair consists of numbers that are equidistant from the original mean of 4.
For example, in answer choice A, -1 is 5 units less than 4, and 6 is 5 units more than 4.
As such, add the two values in each answer choice will yield a mean of 4.

Okay, let's see what happens if we add -1 and 9 (answer choice A).
Well, -1 is 5 units away from the mean of 4, and 9 is 5 units away from the mean of 4. So, 5 and 5 will be added to 4, 2, 0, 2, and 4 to get a new SD. As you can see, this will result in a much larger SD.

Now, let's examine D (2 and 6)
Well, 2 is 2 units away from the mean of 4, and 6 is 2 units away from the mean. So, we'll be adding 2 and 2 to the five original differences of 4, 2, 0, 2, and 4. Since the average of 4, 2, 0, 2, and 4 is 2.4, adding differences of 2 and 2 should have the least effect on the original SD.

As such, the correct answer must be

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UMB wrote:
D

From the given #s it is clear 4 is mean(average) of both sets.
If the variations of the sets are equal or close then their Stand Deviations are equal/close to each other too.

Variation of 1-set (5 #s) is 40/5=8
so Varition of set2 (7 #s) must be around 8 too, thus (40+ x)/7= around 8

x must be close to 16 so only D satisfies this value: (4-2)^2 +(4-6)^2=8
U can check others:
A) (4+1)^2+(4-9)^2=50
B) (4-4)^2+(4-4)^2=0
c) (4-3)^2+(4-5)^2=2
D) (4-2)^2 +(4-6)^2=8
E) (4-0)^2+(4-8)^2=32

This is a good explanation. You can also do it by looking at the numbers and using your common sense about standard deviation.

The average of the first list is 4 and the number deviate from 4 evenly. what i mean by that is, the next set of numbers (2,6) are both 2 away from 4, and the next set after that (0,8) are both 4 away from 4.

To have a similar deviation, we want the next set to look similar.

-1 and 9 will stretch the outter limits of the list, so that will increase the standard deviation significantly.
4 and 4 will add too much weight to the center of the list. It'll decrease the standard deviation.
3 and 5 may be right
2 and 6 may be right
0 and 8 will add weight to the outter limits again, and stretch the deviation.

So it's a toss up between (3 and 5) and (2 and 6). And my educated guess is on 2 and 6, since basically comes in right in the center of the previous standard deviation, it should change it the least.

For the record, I have never taught the standard deviation formula to any student of mine. I find it to be unnecessary for the GMAT when good, conceptual thinking can get you through.
##### General Discussion
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10
D

From the given #s it is clear 4 is mean(average) of both sets.
If the variations of the sets are equal or close then their Stand Deviations are equal/close to each other too.

Variation of 1-set (5 #s) is 40/5=8
so Varition of set2 (7 #s) must be around 8 too, thus (40+ x)/7= around 8

x must be close to 16 so only D satisfies this value: (4-2)^2 +(4-6)^2=8
U can check others:
A) (4+1)^2+(4-9)^2=50
B) (4-4)^2+(4-4)^2=0
c) (4-3)^2+(4-5)^2=2
D) (4-2)^2 +(4-6)^2=8
E) (4-0)^2+(4-8)^2=32
Manager  Joined: 13 Dec 2009
Posts: 108
Re: A set of data consists of the following 5 numbers: 0,2,4,6, and 8.  [#permalink]

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utin 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

as per rule: standard deviation is least affect by adding new element if the newly added elements are at equal distance from the mean of the set. so here mean is 4 => if 2 and 6 are added then SD will be least affected.
hope this will help
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: A set of data consists of the following 5 numbers: 0,2,4,6, and 8.  [#permalink]

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if that is true then why not 3 and 5?

IMO it should be 4 and 4, as mean = new number, the change in deviation will be zero
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Re: A set of data consists of the following 5 numbers: 0,2,4,6, and 8.  [#permalink]

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gurpreetsingh wrote:
if that is true then why not 3 and 5?

IMO it should be 4 and 4, as mean = new number, the change in deviation will be zero

yes, i missed one thing, to have minimum impact on SD, new values should be at equal distance from the mean and
if new numbers are more farther from the mean, less impact will be on SD.
if u look at the formula for SD, it will be more clear. if the values are same in that case in formula numerator will add 0 values for the the number(as mean will not change). however, denominator will increase by 2, thus the SD will be reduced considerably.
hope this will help

Originally posted by einstein10 on 17 Apr 2010, 13:02.
Last edited by einstein10 on 17 Apr 2010, 18:31, edited 1 time in total.
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: A set of data consists of the following 5 numbers: 0,2,4,6, and 8.  [#permalink]

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sandeep25398 wrote:
gurpreetsingh wrote:
if that is true then why not 3 and 5?

IMO it should be 4 and 4, as mean = new number, the change in deviation will be zero

yes, i missed one thing, to have minimum impact on SD, new values should be at equal distance from the mean and
if new numbers are more farther from the mean, less impact will be on SD.
if u look at the formula for SD, it will be more clear. if the values are same in that case in formula denominator will add 0 values for the the number(as mean will not change). however, numerator will increase by 2, thus the SD will be reduced considerably.
hope this will help

oh yea its 2 am here...lol...out of senses i guess and u also seems to be....

lol pls change numerator with denominator

If the number are too far or too close to the mean the change will be bigger.
D is apparently not too far and too close to the means as compared to others.
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Originally posted by gurpreetsingh on 17 Apr 2010, 13:27.
Last edited by gurpreetsingh on 17 Apr 2010, 13:42, edited 1 time in total.
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Re: A set of data consists of the following 5 numbers: 0,2,4,6,  [#permalink]

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2
3
The above method explains it well , and if you are short of time and need to make an educated guess , this works perfect.

If you are in for some calculations , this is how I got to it

mean = 4
sd = \sqrt{8} = 2.8

Expected values for the SD to not change are - One value below SD from mean is (4 - 2.8) = 1.2 , and one value above SD is (4 + 2.8) = 6.8
This would mean , adding 1.2 ans 6.8 would have no impact on the SD . SD remains the same when these two numbers are added. Now for SD to change the least , we need to add two values that are closest to these two values.

Hence any two values that are closest to 1.2 and 6.8 would change the SD , the least.

1. -1 , 9
distance between (1,9) and (1.2 and 6.8) is 2.2 and 2.2

2. 4 , 4
distance etween (4,4) and (1.2 , 6.8) is 2.8 and 2.8

3. 3 , 5
Distance is - 1.8 and 1.8

4. 2 , 6
Distance is - 0.8 and 0.8

5. 0 , 8
Distnace is - 1.2 and 1.2

Hence from above , we see that adding 2 and 6 , results in a value that would change the SD to the least. Hence D

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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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1
1
Given: {0,2,4,6,8}
Mean : 4
Let us not exact calculate S.D., we will stop one step before calculation
$$((0 - 4)^2 + (2-4)^2 + (4-4)^2 + (6-4)^2 + (8-4)^2)$$ / 5 = 40/5 = 8

Let A = $$(0 - 4)^2 + (2-4)^2 + (4-4)^2 + (6-4)^2 + (8-4)^2$$

Examining the options, options are given thankfully so that mean remains the same, after adding the options. (reducing our burden to calc mean again)

Option 1: -1 and 9
$$( A + (-1-4)^2 + (9-4)^2)/7 => (40 + 25 + 25) / 7$$= well over 8

Option 2: 4 and 4

$$(A + (4-4)^2 + (4-4)^2)/7 = > 40/7$$ = well below 8 = 5.x

Option 3: 3 and 5
$$(A + (3-4)^2 + (5 - 4)^2)/7 => 42/7$$ = 6

Option 4: 2 and 6
$$(A + (2-4)^2 + (6-4)^2)/7 => 48/7$$ = 6.9x (closer to 8)

Option 5: 0 and 8
$$(A + (0-4)^2 + (8-4)^2)/7 => 72/7$$ => 10.x (above 8)

option D
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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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GK_Gmat 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

I chose answer C here, can anyone explain why C is wrong. I understand to have lesser impact on SD points added should be far away from mean. In that case along with D, E also explains the same thing. Why we rejected E?
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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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Cbirole wrote:
GK_Gmat 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

I chose answer C here, can anyone explain why C is wrong. I understand to have lesser impact on SD points added should be far away from mean. In that case along with D, E also explains the same thing. Why we rejected E?
Hello Cbirole. The SD of the original set of 5 is square root of 8 (I’m sending this from my phone, so I can’t the use the square root symbol). Thus you are looking or an answer that yields either the sq. root of 8 or that which is closest. Evaluating the answers individually, you will find by adding 0,8 to the original set yields the sq. root of 8. Hope that helps. As for trying to short-cut it through theoretical approximation, someone smarter than me in the SD field will have to step up.

Sent from my iPhone using GMAT Club Forum mobile app
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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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UMB wrote:
D

From the given #s it is clear 4 is mean(average) of both sets.
If the variations of the sets are equal or close then their Stand Deviations are equal/close to each other too.

Variation of 1-set (5 #s) is 40/5=8
so Varition of set2 (7 #s) must be around 8 too, thus (40+ x)/7= around 8

x must be close to 16 so only D satisfies this value: (4-2)^2 +(4-6)^2=8
U can check others:
A) (4+1)^2+(4-9)^2=50
B) (4-4)^2+(4-4)^2=0
c) (4-3)^2+(4-5)^2=2
D) (4-2)^2 +(4-6)^2=8
E) (4-0)^2+(4-8)^2=32

What exactly are variations? Could you please elaborate on that?
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A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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Pardon me but I am still not able to get this question...

Bunuel chetan2u

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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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lets start by finding mean of given data, its 4
now if we add each option one by one in the given series we will have same mean (4)
lets calculate SD of 5 given numbers
SD = (16+4+0+4+16)/5 = 40/5 = 8

it means we are looking for adding 2 numbers in the given series of 5 numbers to have SD very close to 8.
lets plug in all the options one by one to calculate SD of 7 numbers.

A) ((5)^2 + 40 + (-5)^2)/7 = 12.---
B) (0+40+0)/7 = 5.---
C) (1+40+1)/7 = 6
D) (4+40+4)/7 = ~7
E) (16+40+16)/7 = 10.---

As Option (D) is very close to 8
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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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GMATPrepNow wrote:
GK_Gmat 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

For GMAT purposes, Standard Deviation (SD) can often be thought of as "the average distance the data points are away from the mean."

So, with {0, 2, 4, 6, 8}, the mean is 4.
0 is 4 units away from the mean of 4.
2 is 2 units away from the mean of 4.
4 is 0 units away from the mean of 4.
6 is 2 units away from the mean of 4.
8 is 4 units away from the mean of 4.

So, the SD can be thought of as the average of 4, 2, 0, 2, and 4. The average of these values is 2.4, so we'll say that the SD is about 2.4

Note: This, of course, isn't 100% accurate, but it's all you should really need for the GMAT.

Okay, so which pair of new numbers, when added to the original 5, numbers will yield a new SD that is closest to 2.4?

Well, to begin, it's useful to notice that each pair consists of numbers that are equidistant from the original mean of 4.
For example, in answer choice A, -1 is 5 units less than 4, and 6 is 5 units more than 4.
As such, add the two values in each answer choice will yield a mean of 4.

Okay, let's see what happens if we add -1 and 9 (answer choice A).
Well, -1 is 5 units away from the mean of 4, and 9 is 5 units away from the mean of 4. So, 5 and 5 will be added to 4, 2, 0, 2, and 4 to get a new SD. As you can see, this will result in a much larger SD.

Now, let's examine D (2 and 6)
Well, 2 is 2 units away from the mean of 4, and 6 is 2 units away from the mean. So, we'll be adding 2 and 2 to the five original differences of 4, 2, 0, 2, and 4. Since the average of 4, 2, 0, 2, and 4 is 2.4, adding differences of 2 and 2 should have the least effect on the original SD.

As such, the correct answer must be

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Why cant it be 3 and 5 as the distannce frm mean is less?
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Re: A set of data consists of the following 5 numbers: 0, 2, 4  [#permalink]

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A very important question to understand Standard deviation. The 5 numbers on average are 2.4 points away from the mean...ie 12/5 =2.4.

Now we need to find a pair of 2 numbers such that their average deviation from mean is closest to 2.4. Or the sum of their deviation is closest to 4.8

Following are the average deviation in case of each option
A) -1, 9 --> Sum of deviation from mean(4) is equal to (5+5=10)
B) 4, 4 --> Sum of deviation from mean(4) is equal to (0+0=0)
C) 3,5 --> Sum of deviation from mean(4) is equal to (1+1=2)
D) 2,6 --> Sum of deviation from mean(4) is equal to (2+2=4) Closest to 4.8 . Hence, this is the correct answer
E) 0,8 --> Sum of deviation from mean(4) is equal to (4+4=8)
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