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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Tue May 21, 2013 9:23 pm

Which of the following data sets has the third largest standard deviation?
{1, 2, 3, 4, 5}
{2, 3, 3, 3, 4}
{2, 2, 2, 4, 5}
{0, 2, 3, 4, 6}
{-1, 1, 3, 5, 7}

My take E, experts please suggest without calculating SD

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Source: Beat The GMAT — Problem Solving | Tue May 21, 2013 9:23 pm

Atekihcan: Master | Next Rank: 500 Posts; Posts: 149; Joined: Wed May 01, 2013 10:37 pmFollowed by: 9 members; Thanked: 54 times

by Atekihcan » Tue May 21, 2013 10:56 pm

Note that mean of all the sets are 3.
Now, standard deviation is nothing but the average measure of how far away are the elements from the mean.

Now, elements of option A are uniformly distributed.
Elements of options E and D are more away from 3 than the elements of A, so their standard deviation must be greater than A.
Similarly elements of options B and C are closer to 3 than the elements of A, so their standard deviation must be less than A.

So, option A must have the third highest standard deviation.

Answer : A

Here is solution with standard deviation calculation.

A. {1, 2, 3, 4, 5} # mean = 3 # stdev = âˆš[(2Â² + 1Â² + 0Â² + 1Â² + 2Â²)/5] = âˆš2
B. {2, 3, 3, 3, 4} # mean = 3 # stdev = âˆš[(1Â² + 0Â² + 0Â² + 0Â² + 1Â²)/5] = âˆš(2/5) = âˆš(0.4)
C. {2, 2, 2, 4, 5} # mean = 3 # stdev = âˆš[(1Â² + 1Â² + 1Â² + 1Â² + 2Â²)/5] = âˆš(8/5) = âˆš(1.6)
D. {0, 2, 3, 4, 6} # mean = 3 # stdev = âˆš[(3Â² + 1Â² + 0Â² + 1Â² + 3Â²)/5] = âˆš4
E. {-1, 1, 3, 5, 7} # mean = 3 # stdev = âˆš[(4Â² + 2Â² + 0Â² + 2Â² + 4Â²)/5] = âˆš10

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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 » Wed May 22, 2013 6:07 am

vipulgoyal wrote:Which of the following data sets has the third largest standard deviation?
{1, 2, 3, 4, 5}
{2, 3, 3, 3, 4}
{2, 2, 2, 4, 5}
{0, 2, 3, 4, 6}
{-1, 1, 3, 5, 7}

My take E, experts please suggest without calculating SD

So far, all of the official GMAT questions involving SD (comparing the SD's of several lists of numbers) that I've seen can be answered by "eyeballing" the values and comparing how spread apart they are.

So, for example, the values in E are spread apart the most, so E has the greatest SD.
The values in D are spread apart the second most, so D has the second greatest SD.

Now let's look at the values are packed together the most.
The values in B are the most tightly packed, so B has the smallest SD
The values in C are next most tightly packed, so C has the second smallest SD

So, the SDs from least to greatest are: B, C, A, D, E

So A has the third largest standard deviation..

Cheers,
Brent

Brent Hanneson - Creator of 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 » Wed May 22, 2013 6:14 am

If you'd like more practice comparing the SDs of sets of values, you can try this one:

https://www.beatthegmat.com/standard-dev ... 22516.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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nav88: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Mon Nov 26, 2012 5:08 amFollowed by: 1 member

by nav88 » Thu May 30, 2013 10:39 am

Brent@GMATPrepNow wrote:
vipulgoyal wrote:Which of the following data sets has the third largest standard deviation?
{1, 2, 3, 4, 5}
{2, 3, 3, 3, 4}
{2, 2, 2, 4, 5}
{0, 2, 3, 4, 6}
{-1, 1, 3, 5, 7}

My take E, experts please suggest without calculating SD
So far, all of the official GMAT questions involving SD (comparing the SD's of several lists of numbers) that I've seen can be answered by "eyeballing" the values and comparing how spread apart they are.

So, for example, the values in E are spread apart the most, so E has the greatest SD.
The values in D are spread apart the second most, so D has the second greatest SD.

Now let's look at the values are packed together the most.
The values in B are the most tightly packed, so B has the smallest SD
The values in C are next most tightly packed, so C has the second smallest SD

So, the SDs from least to greatest are: B, C, A, D, E

So A has the third largest standard deviation..

Cheers,
Brent

Thanks Brent for the solution

One query though

Can we say the smaller the range the smaller the SD for the set( provided the comparison is made with sets containing equal number of elements)? I know on actual GMAT we wont be calculating the SD for Each set. But i have tried so many problems using the above (so called) rule.And my guess was correct each time.

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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 » Thu May 30, 2013 11:44 am

nav88 wrote: Can we say the smaller the range, the smaller the SD for the set( provided the comparison is made with sets containing equal number of elements)? I know on actual GMAT we wont be calculating the SD for Each set. But i have tried so many problems using the above (so called) rule.And my guess was correct each time.

I'd say your suggestion that "the smaller the range the smaller the SD for the set( provided the comparison is made with sets containing equal number of elements)" will be true for many of the easier SD questions.

However, it's not always true.

For example, consider set A: {1, 2, 3, 4, 5} and B: {1, 3, 3, 3, 5}
Here, the ranges are equal, but the SDs are not equal.
Can you tell which one has a greater SD?

Answer: [spoiler]The SD of a set of numbers can be very loosely defined as the average distance the elements are away from the mean. Since the numbers in set B are more closely packed near the mean, this set must have a smaller SD.

So, set A has the greater SD.
[/spoiler]
Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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