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    bounds on the value of the SD




    I learned stats in a very applied, algebra-based way...so for those of you who've taken a series of more rigorous stats courses, this will likely be an easy question with an easy answer.

    The standard deviation can be thought of as "roughly" the average deviation (although the average absolute deviation would be a more exact measure of the same thing).

    Therefore it makes intuitive sense that the value of the S.D. should always fall in the range (smallest absolute deviation, largest absolute deviation). Or perhaps the range [smallest absolute deviation, largest absolute deviation], since a uniform distribution would have deviations of 0 and an SD of 0, thus making that a closed interval.

    But, can this range restriction on the SD be said without any qualifications, i.e., mathematically proved? Can it be proved for the population SD? For the sample SD?

    Thanks.
    Last edited by bruin; 08-29-2016 at 05:03 PM.

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    Re: bounds on the value of the SD

    Quote Originally Posted by bruin View Post
    Therefore it makes intuitive sense that the value of the S.D. should always fall in the range (smallest absolute deviation, largest absolute deviation). Or perhaps the range [smallest absolute deviation, largest absolute deviation], since a uniform distribution would have deviations of 0 and an SD of 0, thus making that a closed interval.
    would you mind posting an actual example of what you're saying, just to make sure i'm understanding your question?

    because if you're saying what i think you're trying to say then it's not quite right. but i wanna make sure i'm understanding you first before i make a doofus out of myself.
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    Re: bounds on the value of the SD

    Sure thing. I mean this...if you have a dataset like this:

    X (X-M)
    1 -3
    2 -2
    3 -1
    4 0
    5 1
    6 2
    7 3

    M=4


    The greatest absolute deviation is 3, the least absolute deviation is 0

    Being that the SD can be thought of as a measure of "average" distance from the mean, it just makes intuitive sense that the SD would be between the smallest and greatest individual distances from the mean. And in this dataset, it is 2.16, which is between 0 and 3 as we might expect.

    Can we say that (without exception) the SD will be between the smallest and greatest absolute deviations for a data set?

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    Re: bounds on the value of the SD

    Quote Originally Posted by bruin View Post
    Can we say that (without exception) the SD will be between the smallest and greatest absolute deviations for a data set?
    consider the following dataset:

    Code: 
    a<-c(1,0,1,0,1,0)
    
    1
    0
    1
    0
    1
    0
    
    > mean(a)
    [1] 0.5
    > abs(1-mean(a))
    [1] 0.5
    > abs(0-mean(a))
    [1] 0.5
    the mean is 0.5, the smallest and greatest absolute deviations are also 0.5. BUT

    Code: 
    sd(a)
    0.5477226
    so the standard deviation is larger than the mean and the absolute deviations.

    i'm pretty sure that with some cleverly placed 0s and repeated values here and there you can come up with a number of counter-examples to your claim.
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    Re: bounds on the value of the SD

    Quote Originally Posted by bruin View Post
    The standard deviation can be thought of as "roughly" the average squared deviation (although the average absolute deviation would be a more exact measure of the same thing).
    The part in bold has been added.
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    Re: bounds on the value of the SD


    Quote Originally Posted by gianmarco View Post
    The part in bold has been added.
    well, if we're being formal we'd have to add the words "the square root of" as well (i.e. the square root of the the average squared deviation) because the average squared deviation (from the mean) is, in itself, not the standard deviation...


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