bad at statistics :(

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mehaksal: Master | Next Rank: 500 Posts; Posts: 122; Joined: Tue Jul 10, 2012 9:12 amFollowed by: 8 members; Thanked: 2 times

by mehaksal » Sat Aug 25, 2012 6:17 am

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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Source: Beat The GMAT — Problem Solving | Sat Aug 25, 2012 6:17 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Aug 25, 2012 6:33 am

mehaksal 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

To begin, the SD formula looks like this:

The set {0, 2, 4, 6, 8} has a mean of 4.
So, first we need to find the difference between each value and the mean.

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 will equal the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2)/5.
In other words, the SD = the square root of 40/5.
= the square root of 8

Okay, so which pair of new numbers, when added to the original 5 numbers will yield a new SD that is closest to the square root of 8?

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, the new SD = the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2 + 5^2 + 5^2)/7.
= the square root of 90/7.
= the square root of approximately 13
This is considerably larger than the original SD of sqrt(8)

Now let's skip a few answers and try answer choice D.
Here, 2 is 2 units away from the mean of 4, and 6 is 2 units away from the mean.
So, the new SD = the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2 + 2^2 + 2^2)/7.
= the square root of 48/7.
= the square root of approximately 7
This one is pretty close to the original SD of sqrt(8).

In fact, if we check the other answer choices (lots of work!), we'll see that answer choice D is the best answer.

Cheers,
Brent

Last edited by Brent@GMATPrepNow on Sat Aug 25, 2012 7:15 am, edited 1 time in total.

Brent Hanneson - Creator of GMATPrepNow.com

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neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 amFollowed by: 52 members; Location: Utrecht, The Netherlands; Thanked: 318 times

by neelgandham » Sat Aug 25, 2012 6:59 am

Average of the set 0,2,4,6 and 8 = (0+2+4+6+8)/5 = 4
Standard deviation of the set 0,2,4,6, and 8 = âˆš(((0-4)^2+(2-4)^2+(4-4)^2+(6-4)^2+(8-4)^2)/5) = âˆš(40/5) = âˆš8

If you observe carefully, you will find that the sum of the number(-1,9),(4,4).. is 8 and the average is 4. So the average of the set 0,2,4,6,8,A,B will still be 4.

Standard deviation of the set 0,2,4,6,8,-1,9 = âˆš((40+(-1-4)^2 + (9-4)^2)/7) = âˆš90/7 ~âˆš(12.x)
Standard deviation of the set 0,2,4,6,8,4,4 = âˆš((40+(4-4)^2 + (4-4)^2)/7) = âˆš40/7 ~âˆš(5.x)
Standard deviation of the set 0,2,4,6,8,3,5 = âˆš((40+(3-4)^2 + (5-4)^2)/7) = âˆš42/7 ~âˆš(6)
Standard deviation of the set 0,2,4,6,8,2,6 = âˆš((40+(2-4)^2 + (6-4)^2)/7) = âˆš48/7 ~âˆš(7)
Standard deviation of the set 0,2,4,6,8,0,8 = âˆš((40+(0-4)^2 + (8-4)^2)/7) = âˆš72/7 ~âˆš(10.x)

The SD of the set with 2,6 is the closest to âˆš8. So, the correct answer is option D

Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Aug 25, 2012 7:17 am

mehaksal 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

Alternative approach.

Having said that . . .

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 D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 amFollowed by: 52 members; Location: Utrecht, The Netherlands; Thanked: 318 times

by neelgandham » Sat Aug 25, 2012 7:32 am

Since you mentioned that you are bad at statistics, I have collected a few urls from the library which might be helpful in your prep.

Statistics overview by Vivian Kerr: https://www.beatthegmat.com/mba/2011/03/ ... s-overview
An Important Feature of the Median and the Mean by Brent Hanneson:https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean
Standard Deviation by Ron Purewal: https://www.beatthegmat.com/mba/2010/10/ ... -deviation
Standard Deviation by Kaplan GMAT: https://www.beatthegmat.com/mba/2009/08/ ... -deviation
BEATtheGMAT Library: https://www.beatthegmat.com/mba/category ... statistics

Let me know if you need any further help

Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

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kheba: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Sun Aug 10, 2008 5:54 am

by kheba » Wed Aug 29, 2012 3:44 am

Just a quick question .. why not option B? When we add two 4s , the SD remains unchanged. What am I missing here

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dhirajdas53: Senior | Next Rank: 100 Posts; Posts: 32; Joined: Sat Apr 07, 2012 3:18 am

by dhirajdas53 » Wed Aug 29, 2012 4:59 am

Kheba

When you include 4,4 in the list. Total deviation from the mean will remain same but standard deviation, which is the mean of deviation will change as the number of item now will be 7.

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