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Standard Deviation

This topic has 6 expert replies and 8 member replies
nsharma215 Newbie | Next Rank: 10 Posts
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Standard Deviation

Post Sun Jul 20, 2014 6:54 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Hey, could you please help me in understanding the below question how to solve it

    Q-Sets A,B and C are shown below.If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their Standard Deviation, from largest to smallest?
    A {30,50,70,90,110}
    B {-20, -10,0,10,20}
    C {30,35,40,45,50}

    (A) A,C,B (B)A,B,C (c)C,A,B (d)B,A,C (e)B,C,A

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    Post Sun Jul 20, 2014 7:33 am
    A {30,50,70,90,110}
    B {-20, -10,0,10,20}
    C {30,35,40,45,50}

    Mean of A = 350/5 = 70
    mean of B = 0
    Mean of C = 200/5 = 40

    The farther the added value is from mean, the more is S.D

    So, B,C,A

    {E}?

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    nsharma215 Newbie | Next Rank: 10 Posts
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    Post Sun Jul 20, 2014 8:03 am
    Answer is correct thanks.

    So take away is that Mean is 0 and 100 is getting add in SD
    and 100(SD) is far from the 0(mean)

    Post Mon Jul 21, 2014 12:56 am
    nsharma215 wrote:
    Hey, could you please help me in understanding the below question how to solve it

    Q-Sets A,B and C are shown below.If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their Standard Deviation, from largest to smallest?
    A {30,50,70,90,110}
    B {-20, -10,0,10,20}
    C {30,35,40,45,50}

    (A) A,C,B (B)A,B,C (c)C,A,B (d)B,A,C (e)B,C,A

    Standard Deviation is the representation of Average Deviation of the terms in the given Set from the Mean of the set.


    Which means "the new Term is added to set" at a greater difference from the Mean of the set will increase the the standard deviation by a greater amount


    If we compare the Standard Deviation of the sets given then we realize that

    SD of C < SD of B = SD of A [Because in set A and B the consecutive terms are separated by a common difference 20 whereas the difference between consecutive terms of Set C are at a deviation of 5]

    100 is farthest from mean of B therefore Mean of B will be changed greatest after including 100 in set

    This eliminated Options A, B and C

    Between Set A and C, Set C has much lower standard deviation than Set A but after including 100 It will increase for both Sets but the substantial change will be noticed in SET C because 100 falls within the terms of Set A therefore
    after including 100 in each set
    SD of C > SD of A

    Therefore the correct Order of Change in SD will be B > C > A

    Answer: Option E

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    Last edited by GMATinsight on Mon Jul 21, 2014 6:27 am; edited 1 time in total

    Post Mon Jul 21, 2014 12:58 am
    One can also verify by inputting the values of the sets in on the page whose link is as follows

    http://easycalculation.com/statistics/standard-deviation.php

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    Post Mon Jul 21, 2014 6:01 am
    For the purposes of the GMAT, it's sufficient to think of Standard Deviation as the Average Distance from the Mean. Here's what I mean:

    Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
    The mean of set A = 10 and the mean of set B = 10
    How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

    Alternatively, let's examine the Average Distance from the Mean for each set.

    Set A {7,9,10,14}
    Mean = 10
    7 is a distance of 3 from the mean of 10
    9 is a distance of 1 from the mean of 10
    10 is a distance of 0 from the mean of 10
    14 is a distance of 4 from the mean of 10
    So, the average distance from the mean = (3+1+0+4)/4 = 2

    B {1,8,13,18}
    Mean = 10
    1 is a distance of 9 from the mean of 10
    8 is a distance of 2 from the mean of 10
    13 is a distance of 3 from the mean of 10
    18 is a distance of 8 from the mean of 10
    So, the average distance from the mean = (9+2+3+8)/4 = 5.5

    IMPORTANT: I'm not saying that the Standard Deviation of set A equals 2, and I'm not saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

    What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A.
    More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GMAT.

    ------------------------------
    Quote:
    Sets A,B and C are shown below.If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their Standard Deviation, from largest to smallest?
    A {30,50,70,90,110}
    B {-20, -10,0,10,20}
    C {30,35,40,45,50}

    (A) A,C,B (B)A,B,C (c)C,A,B (d)B,A,C (e)B,C,A
    So, for this question, we have:

    Mean of set A = 70
    Mean of set B = 0
    Mean of set C = 40

    100 is furthest away from the mean of 0 in set B, so this will cause the GREATEST change in standard deviation.
    100 is next furthest away from the mean of 40 in set C, so this will cause the 2nd greatest change in standard deviation.
    100 is closest to the mean of 70 in set A, so this will cause the LEAST change in standard deviation.

    Answer: E

    Cheers,
    Brent

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    Post Mon Jul 21, 2014 6:10 am
    With Standard Deviation questions on the GMAT, you will never be asked to calculate the actual standard deviation, because that involves some complex computation. To actually calculate, we'd find the difference between each term and the mean, then square that, then find the average of all of those, and square root that. For reference: http://en.wikipedia.org/wiki/Standard_deviation#Basic_examples

    The GMAT will never ask you to do that! The most that they can ask is that you understand that high SD means the data is spread out far from the mean, and a low SD means that most data is clustered closely around the mean. Whenever you see a SD question, just translate it as "how much is the data spread out?"

    When you add 100 to each set, the further 100 is from the mean, the more it will "spread" the data out from the mean.

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    Post Mon Jul 21, 2014 6:13 am

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    Post Mon Jul 21, 2014 6:21 am
    Here are a few more practice questions where we can apply the concept of "average distance from the mean" as an approximation for Standard Deviation:

    http://www.beatthegmat.com/standard-deviation-more-than-s-t274384.html
    http://www.beatthegmat.com/standard-deviation-t269584.html
    http://www.beatthegmat.com/range-and-sd-t89159.html

    Cheers,
    Brent

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    nsharma215 Newbie | Next Rank: 10 Posts
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    Post Tue Jul 22, 2014 12:47 am
    Thanks all.
    It is very useful

    Anaira Mitch Master | Next Rank: 500 Posts
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    Post Mon Dec 12, 2016 9:18 pm
    Brent@GMATPrepNow wrote:
    For the purposes of the GMAT, it's sufficient to think of Standard Deviation as the Average Distance from the Mean. Here's what I mean:

    Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
    The mean of set A = 10 and the mean of set B = 10
    How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

    Alternatively, let's examine the Average Distance from the Mean for each set.



    Set A {7,9,10,14}
    Mean = 10
    7 is a distance of 3 from the mean of 10
    9 is a distance of 1 from the mean of 10
    10 is a distance of 0 from the mean of 10
    14 is a distance of 4 from the mean of 10
    So, the average distance from the mean = (3+1+0+4)/4 = 2

    B {1,8,13,18}
    Mean = 10
    1 is a distance of 9 from the mean of 10
    8 is a distance of 2 from the mean of 10
    13 is a distance of 3 from the mean of 10
    18 is a distance of 8 from the mean of 10
    So, the average distance from the mean = (9+2+3+8)/4 = 5.5

    IMPORTANT: I'm not saying that the Standard Deviation of set A equals 2, and I'm not saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

    What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A.
    More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GMAT.

    ------------------------------
    Quote:
    Sets A,B and C are shown below.If number 100 is included in each of these sets, which of the following represents the correct ordering of the sets in terms of the absolute increase in their Standard Deviation, from largest to smallest?
    A {30,50,70,90,110}
    B {-20, -10,0,10,20}
    C {30,35,40,45,50}

    (A) A,C,B (B)A,B,C (c)C,A,B (d)B,A,C (e)B,C,A
    So, for this question, we have:

    Mean of set A = 70
    Mean of set B = 0
    Mean of set C = 40

    100 is furthest away from the mean of 0 in set B, so this will cause the GREATEST change in standard deviation.
    100 is next furthest away from the mean of 40 in set C, so this will cause the 2nd greatest change in standard deviation.
    100 is closest to the mean of 70 in set A, so this will cause the LEAST change in standard deviation.

    Answer: E

    Cheers,
    Brent
    Amazing explanation Brent.

    neha shekhawat Junior | Next Rank: 30 Posts Default Avatar
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    20 Jan 2017
    Posted:
    14 messages
    Post Tue Aug 01, 2017 3:38 pm
    Hey Brent,
    I am confused with the language of this question ( the absolute increase in their Standard Deviation, from largest to smallest? )
    Is this question asking to arrange the S.D of all set in descending order?
    I totally understand the concept of S.D, and looking at the given set and their spread I can easily see that first set has largest S.D and last set has the least S.D.
    So if this question is asking about arranging the S.D in descending order then it would have been a,b,c.
    But this is not the answer.
    what is absolute increase in this case?
    As when we calculate S.D of each set (which is not required ,But I calculated out of curiosity ) are:
    a) 31.6
    b) 15.8
    c) 7.9

    why are you subtracting each value of mean from 100?

    Post Tue Aug 01, 2017 9:02 pm
    neha shekhawat wrote:
    Hey Brent,
    I am confused with the language of this question ( the absolute increase in their Standard Deviation, from largest to smallest? )
    Is this question asking to arrange the S.D of all set in descending order?
    I totally understand the concept of S.D, and looking at the given set and their spread I can easily see that first set has largest S.D and last set has the least S.D.
    So if this question is asking about arranging the S.D in descending order then it would have been a,b,c.
    But this is not the answer.
    what is absolute increase in this case?
    As when we calculate S.D of each set (which is not required ,But I calculated out of curiosity ) are:
    a) 31.6
    b) 15.8
    c) 7.9

    why are you subtracting each value of mean from 100?
    Hi Neha,

    "Is this question asking to arrange the S.D of all set in descending order?" is incorrect. The question does not ask this.

    This question asks to arrange the ABSOLUTE INCREASE IN THE VALUES OF S.D of all set in descending order?

    We need not be bothered about the values of the SDs of the three sets.

    Say, currently the SDs of Sets A, B and C are a, b, and c, respectively. We need not know which value among a, b, and c is the largest and which is rhe smallest.

    We are concerned with what happens when a new entrant '100,' is included in each set.

    Say after the inclusion of '100,' the new SDs of three sets A, B, and C are (a + a'), (b + b'), and (c + c'), respectively; where a', b' and c' are the absolute increase in respective SD values.

    The question asks us to arrange the values of a', b' and c' in descending order.

    Hope this helps!

    Relevant book: Manhattan Review GMAT Sets & Statistics Guide

    -Jay
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    Thanked by: neha shekhawat
    neha shekhawat Junior | Next Rank: 30 Posts Default Avatar
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    14 messages
    Post Wed Aug 02, 2017 9:35 am
    Thanks jay
    I just realized that I missed the sentence (If number 100 is included in each of these sets) while reading the question.I straight away jumped on the sets.I have to pay more attention while reading such questions now.

    GMAT/MBA Expert

    Post Sun Aug 06, 2017 11:06 pm
    neha shekhawat wrote:
    Thanks jay
    I just realized that I missed the sentence (If number 100 is included in each of these sets) while reading the question.I straight away jumped on the sets.I have to pay more attention while reading such questions now.
    This is such a great lesson to learn from this problem. On the GMAT, of all places, is the devil IS in the details: subtle misreadings are deadly, and many of the questions are written to ENCOURAGE such misreadings!

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