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Standard Deviation – Why do we square the values

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This topic contains 32 replies, has 10 voices, and was last updated by  Mike Carnell 1 year, 2 months ago.

Viewing 33 posts - 1 through 33 (of 33 total)
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  • #53574

    wils
    Participant

    Ok…no beating me up on this question…just a curiosity that I can’t find the answer to.

    So, in the method for calculating the standard deviation at one point we subtract the mean from a data point and we do that with all the data points. Next we square the results of each subtraction, which effectively eliminates the negative signs.

    So, the question is…why do we square? Why not just take the absolute value.

    Thanks!

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    #190743

    MBBinWI
    Participant

    michsigma wrote:

    Ok…no beating me up on this question…just a curiosity that I can’t find the answer to.

    So, in the method for calculating the standard deviation at one point we subtract the mean from a data point and we do that with all the data points. Next we square the results of each subtraction, which effectively eliminates the negative signs.

    So, the question is…why do we square? Why not just take the absolute value.

    Thanks!

    OK, all together now – Variations are additive, std dev’s are not.

    It’s just how the math works. Kind of like asking why a right triangle is a^2 + b^2 = c^2, it just is.

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    #190744

    Rohatgi
    Participant

    Mich,

    Let’s add the difference as you said, what will you get Zero.
    Why Zero ?
    Because we are subtracting from Mean.

    Which will not give you any measure i.e. how much the data vary each side from mean.

    To overcome this, it makes sense to first square it and then take square-root.

    If statisticians had chosen to subtract Min. value from data points and then calculate deviation then there was no need to square it.

    So it is because , how we define it. e.g. Left is -ve, Right is +ve etc.

    Other option could have been to take modulus and then sum it then it would be a different measure.

    Anupam

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    #190745

    Robert Butler
    Participant

    You could run an analysis using the absolute deviations instead of the measure that is used. I was trying to find the original referernce so I could give you all of the gory details but, at the moment, I can’t seem to locate the book I want. By way of an anemic substitute I’ll offer the following from pp.29 Statistical Methods 7th Edition Snedecor and Cochran:

    “The average of the absolute deviations could be used [to measure the amount of variation] but for several reasons a different measure was adopted around 1800 – the population standard deviation.”

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    #190754

    Paulonis
    Participant

    There was some influential work that concluded that the sample standard deviation was a more efficient estimator of the population statistic than the sample mean deviation was of it’s population statistic. There is a great deal of debate regarding under which assumptions this is true. Nevertheless, standard deviation caught on and became the generally accepted statistic.

    More importantly in my mind, the square operation is much nicer mathematically than the absolute value operation. The absolute value makes analytical calculations much more difficult when using the mean deviation.

    A nice reference is:

    http://www.leeds.ac.uk/educol/documents/00003759.htm

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    #190768

    wils
    Participant

    Thanks everyone….especially Mr. Butler and Paulonis for the refernces….I knew the truth was out there!

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    #198222

    vishal

    The other reasoning may be:-

    Mean is the central point of the data points and the difference on -ve and +ve side can be expressed as mean, if we end up taking absolute value the we will be gaving sd “2” as a result every time

    2*mean/mean

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    #198223

    Amit Kumar Ojha
    Participant

    The main difference between Mean Absolute Deviation (calculated by taking the absolute value of difference around mean) and standard deviation (calculated by squaring the differences and then adding them up and finally taking the Square Root) is that Standard Deviation gives more weightage to the extreme value and hence is considered a better estimation than Mean Absolute Deviation.

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    #198227

    Chris Seider
    Participant

    You can add squares and do proper statistics such as ANOVA, etc. using sums of squares.

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    #200688

    Kevin

    Hi Wils,

    I think the answer is in the DNS of this website. We love saying “Six Sigma”! lol

    Best,
    Kevin

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    #200709

    MBBinWI
    Participant

    Wow, six and a half years later, Kevin comes through with a witty answer. Wils is long gone, and you’re answer wasn’t particularly witty.

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    #200712

    Mike Carnell
    Participant

    That is a lot of answers for a really simple question.

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    #200719

    Andrew Parr
    Participant

    @MBBinWI maybe it just proves that even though Kevin is older now it still isn’t wise to leave him Home Alone.

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    #200745

    MBBinWI
    Participant

    @Robert-Butler – are you trying to assume the persona of a well respected forum member?

    @Katie-barry – I think we have an imposter here.

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    #200746

    MBBinWI
    Participant

    @katiebarry – sorry, messed up your handle. Looks like we have an imposter trying to assume the identity of our friend @rbutler

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    #200748

    Robert Butler
    Participant

    @MBBinWI – no, we don’t. When the forum shifted over to ID’s with the @ sign attached I assumed the new ID had to be all one word so I chose rbutler. There is a problem with this because if you try to search my older posts you have to use the full name – the 2010 post above was made before the @ switch and it looks like the @ was just appended to my name in the posts before the change.

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    #200749

    Robert Butler
    Participant

    P.S. it would be nice if the site could change all of my post ID’s to one or the other but I suspect that would be more trouble than it is worth.

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    #200752

    Mike Carnell
    Participant

    @rbutler I still keep up with your posts. I love your answers. Very straight forward responses to seemingly complex questions.

    What would be great would be to have someone pull together all your responses into a book. That was done with some totally inane blogs years ago for another person. On a scale of 1-10 in terms of helpful it was about a -100. At least yours would provide some useful information.

    Take care.

    @KatieBerry you are tough. You have to have over 10 years refereeing these food fights.

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    #200755

    Katie Barry

    @mike-carnell I’m definitely tough!! Hard as nails. Unless you see me watching a sappy commercial when I *might* cry. :)

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    #200757

    Andy-Parr

    @katiebarry you make whether to cry or not seem like a choice. If so you are definitely as hard as nails or maybe just a smiling assassin.

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    #200758

    Mike Carnell
    Participant

    @katiebarry Consuelo watches that stuff too. Hallmark and WE channel. I think there are others with that same romance novel programing. If I have to sit through Pride and Prejudice one more time……

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    #200761

    Robert Butler
    Participant

    @mike-carnell you said “If I have to sit through Pride and Prejudice one more time……”…then, of course, you, like me, will do it! :-) Question: does Consuelo own just one version or does she have all the different performances of PandP?

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    #200762

    Mike Carnell
    Participant

    @rbutler I will do it just like you said but if you put on John Wick as soon as it is over they recognize cause and effect pretty quickly. That whole puppy thing is tougher on them than the entire rest of the movie.

    Near as I can tell there is just one version in the house but I don’t watch close enough to tell. I did draw the line at P&P with Zombies. I am not doing that one ever again.

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    #200763

    Katie Barry
    Keymaster

    @andy-parr I am TOTALLY going with smiling assassin from now on! Love it.

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    #200764

    Katie Barry
    Keymaster

    @mike-carnell and @rbutler Consuelo is right. You can’t watch P&P enough times. My preference is the BBC version with Colin Firth. The newer movie with Keira Knightley is good, too, but I’m loyal to the earlier version.

    I heard there’s a new dog in John Wick 2. (Spoiler alert) I don’t think I’ve ever recovered from the puppy dying in the first one.

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    #200765

    Mike Carnell
    Participant

    @katiebarry I have seen the Keira Knightly version. That woman is way to skinny. You do stories from back then you need curves.

    There is a new dog in JW2. That whole thing “You make me watch P&P again then we are doing JW.” Life is a negotiation.

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    #200766

    MBBinWI
    Participant

    @katiebarry – nevermind

    @rbutler – the thing that caused my suspicions was that Robert-butler didn’t have any points. Glad to know you’re not being spoofed.

    @mike-carnell – you’ve given me a new movie to search out – and not PandP ;-}

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    #200777

    Katie Barry

    @mike-carnell Fully agree re Keira casting. She’s a great actress, but was not a good physical representative of Elizabeth.

    Adding JW2 to my list of movies to see. It’s quite long, but I’ll get there eventually.

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    #200781

    Katie Barry
    Keymaster

    @rbutler said:

    P.S. it would be nice if the site could change all of my post ID’s to one or the other but I suspect that would be more trouble than it is worth.

    — Done! :)

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    #200788

    MBBinWI
    Participant

    @katiebarry – of course, ’cause you’re the forum goddess. ;-}

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    #200789

    Robert Butler
    Participant

    Thanks Katie – much appreciated. Interestingly enough (I just got home from a very very long work day) when I checked my home e-mail a minute or so ago MBBinWI’s query to the @robert Butler name showed up in my home e-mail.

    Oh yes, PandP for Katie and Consuelo – My wife’s collection

    1. BBC Video – PandP starring Elizabeth Garvie and David Rintoul
    2. A&E PandP The Special Edition (2 discs) Starring Colin Firth and Jennifer Ehle
    3. Focus Features PandP Starring Keira Knightley and Matthew Macfadyen

    She likes all of them equally – zero variation.

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    #202681

    Carlos tarnowski

    If you take the mean of the absolute deviation you will have exactly what you did, which is the mean of the absolute deviations for your sample. But the inferences you could make about your population gets limited, because You can’t use the properties from the distribution curve it comes from, as it uses sd as an input parameter. So I think, first they tried to work out what parameters were related to the distribution population curve and sd was always there, and the absolute mean value no, then, sd is used for that purpose, not eliminating the information the absolute deviation gives, too.

    But it is just an intuitive opinion, not based on any book.

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    #202683

    Mike Carnell
    Participant

    Carlos You are responding to a question from over 7 years ago. The last post from Robert Butler was 16 months ago and we had all deviated from the original questions.

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