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RE: [ai-geostats] Why degree of freedom is n-1

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  • Harper, Bill
    Reza, If you are just asking why n-1 in the formula commonly found in stat books for computing the sample variance s^2, it is so that we have an unbiased
    Message 1 of 7 , Aug 25 1:06 PM
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      Reza,

       

      If you are just asking why n-1 in the formula commonly found in stat books for computing the sample variance s^2, it is so that we have an unbiased estimate of the population variance – look at a good calculus based probability and stat book.

       

      Other estimation methods (e.g., maximum likelihood) divide by n instead of n-1. 

       

      Oh, while the n-1 does make the sample variance s^2 an unbiased estimate of the population variance sigma^2, taking the square root and getting the sample standard deviation s does not result in an unbiased estimated of the population standard deviation sigma.  Another reason some prefer m.l.e.

       

      Best,

       

      Bill

       

      --

      William V Harper, Mathematical Sciences

      Otterbein College, Towers Hall 139, 1 Otterbein College

      Westerville OH 43081-2006   USA

      Office phone: 614-823-1417     Office Fax 614-823-3201

      Faculty page: http://www.otterbein.edu/home/fac/WLLVHRPR

      For the best in geostatistics: http://geoecosse.hypermart.net/

       


      From: Reza Nazarian [mailto:rnazarian@...]
      Sent: Thursday, August 25, 2005 3:23 PM
      To: ai-geostats@...
      Subject: [ai-geostats] Why degree of freedom is n-1

       

      Dear Experts
      Sorry may be the question is so basic .After searching my statistics books to find an answer with no great success, could you please explain me why we consider degree of freedom as n-1 in calculating variance. Thanks for your kind advises.


      Very Best Regards
      Reza Nazarian
      Schlumberger Information Solutions
      SONILS Oil Services Centre, Porto de Luanda , Angola

      (Via UK : +44 (0)207 576 6306
      * rnazarian@...
      http://www.sis.slb.com

    • Eric.Lewin@ujf-grenoble.fr
      This follow-up is slighlty aside the subject line of the mailing list, but as a geologist, this is the only statistically-flavoured one I am subscribed to.
      Message 2 of 7 , Aug 31 8:39 AM
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        This follow-up is slighlty aside the subject line of the mailing list, but
        as a geologist, this is the only statistically-flavoured one I am
        subscribed to. Therefore :

        Federico Pardo <federico.pardo@...> said:
        > Having N samples, and then n degrees of freedom.
        > One degree of freedom is used (or taken) by the mean calculation.
        > Then when you calculate the variance or the standard deviation, you only
        > have left n-1 degrees of freedom.

        Apart a rigorous calculation I am aware of that in this very case (cf.
        Peter Bossew's contribution on the same thread, that details it), gives a
        proof for this rule-of-thumb, what more or less rigourous statistical
        developments gives consistance to it ?

        I mean, for the empirical correlation coefficient,
        rhoXiYi = SUM_i=1..N( (x_i - mx).(y_i - my) / sx / sy ) / WHAT_NUMBER
        Must WHAT_NUMBER be, for a kind of unbiased estimate ("a kind of" meaning
        "with some eventual Fisher z-transform"...):
        * N for simplicity,
        * N-2 as I have most frequently seen in books that dare give this formula
        (N points, minus 1 for position and 1 for dispersion ?),
        * or 2N-4 -- 2N for the (x_i,y_i), minus 4 for {mx,my,sx,sy} -- as a
        strict application of the rule-of-thumb seems to suggest ?

        And what about, when fitting for instance a 3-parameter non-linear
        function, reducing the number of degrees of freedom, to N-3 (number of
        points, minus one for each function parameter ? I have never read any kind
        of explanation to support it, though it seems widely

        Thanks in advance for enlightments or simply tracks for other resources of
        explanations.
        -- √Čric L.
      • M.J. Abedini
        Dear Reza I was away from my office for quite a while. After surfing my folder, I came across your enquiry. I found it helpful to share the following thoughts
        Message 3 of 7 , Sep 17, 2005
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          Dear Reza

          I was away from my office for quite a while. After surfing my folder, I
          came across your enquiry. I found it helpful to share the following
          thoughts with you and other colleagues over the list.

          I prefer to approach your question from another angle.

          At first, one has to acknowledge that almost all measurements are
          corrupted by noise in one way or another. Furthermore, standard deviation is a
          measure uncertainty in measurement. Now, keeping These points in mind, look
          at the relation for calculating the standard deviation or for that matter
          variance when you have only ONE measurement. If you use
          the relation with n in the denominator, then you would get 0 for standard
          deviation implying your single measurement is exact and not corrupted by
          noise which is not true. On the other hand, relation with n-1 in the
          denominator would give you 0/0 which is indeterminate more compatible with
          preliminary propositions mentioned above.

          Another useful question might be the origin of that equation which has
          something to do with Normal probability distribution. The first chapter of
          "Nonlinear parameter estimation by Bard (1974)" might be useful to refer
          to as he was resorting to Entropy to derive Normal distribution and its
          associated parameters.

          Hope this helps.

          Thanks
          Abedini

          On Thu, 25 Aug 2005, Reza Nazarian wrote:

          > Dear Experts
          > Sorry may be the question is so basic .After searching my statistics books to
          > find an answer with no great success, could you please explain me why we
          > consider degree of freedom as n-1 in calculating variance. Thanks for your
          > kind advises.
          >
          >
          > Very Best Regards
          > Reza Nazarian
          > Schlumberger Information Solutions
          > SONILS Oil Services Centre, Porto de Luanda, Angola
          >
          > (Via UK: +44 (0)207 576 6306
          > * rnazarian@...
          > http://www.sis.slb.com
          >
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