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Re: AI-GEOSTATS: Standard deviation, Variance

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  • Digby Millikan
    Apologise, The original email, ... was not written by Isobel, it came from W.D. Allen on sci.stat.math which I posted in the summary of my replies. Thankyou
    Message 1 of 9 , Dec 5, 2002
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      Apologise,
      The original email,

      > > >>The reason is simple and comprehensive....
      > > >>
      > > >>Assume a population with ANY distribution of
      > > >>elements. Then randomly select
      > > >>a number of sample elements from the population to
      > > >>characterize the
      > > >>underlying population. That distribution of sample
      > > >>elements ALWAYS tends
      > > >>toward a normal [Gaussian] distribution. And the
      > > >>mean and standard deviation
      > > >>of the sample distribution are unbiased
      > > >>representations of the mean and
      > > >>standard deviation of the underlying population.

      was not written by Isobel, it came from W.D. Allen on sci.stat.math
      which I posted in the summary of my replies.
      Thankyou both for your help in this matter, I am currently reading
      Practical Geostatistics 2000 and have ordered the statistics books
      as recommended by Donald.


      Regards Digby Millikan B.Eng

      Geolite Mining Systems
      U4/16 First Ave.,
      Payneham South SA 5070
      Australia.
      Ph: +61 8 84312974

      digbym@...
      http://www.users.on.net/digbym


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    • Ruben Roa
      Isobel did not write the careless paragraph about the central limit theorem (CLT) Don replied to, as pointed out by Digby. I wish to add something to what Don
      Message 2 of 9 , Dec 5, 2002
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        Isobel did not write the careless paragraph about the central limit theorem
        (CLT) Don replied to, as pointed out by Digby. I wish to add something to
        what Don said about the conditions under which the CLT applies, and that
        people usually miss in considering the universality of the CLT. See below.

        >> Let X_1,...., X_n be a sequence of independent,
        >> identically distributed
        >> random variables with common mean m and common
        >> standard deviation
        >> sigma. Let Z_n be defined as a normalized sum
        >>
        >> Z_n = [S_n - m]/ (sigma/sqt root of n),
        >> S_n = [Z_1
        >> +.....+ X_n]/n
        >>
        >> S_n is the sample mean
        >>
        >> Let F_n(z) be the cumulative probability
        >> distribution function for Z_n
        >> and let G(z) be the cumulative probability
        >> distribution function for the
        >> standard Normal,. Then F_n(z) --> G(z) as n
        >> increases.
        >>
        >> Note two things about this statement, (1) the
        >> theorem does not say how
        >> "fast" the cdf for Z_n approaches the standard
        >> Normal, (2) the speed of
        >> convergence depends on z. Also the speed of
        >> convergence depends on the
        >> distribution type of the X_i's

        Note also the sum operation. The CLT, more precisely called the Additive
        CLT, applies to sums of pairwise independent random variables as n tends to
        infinity. But if the operation is multiplication with equal-signed r.v.,
        then convergence in distribution is towards the lognormal, not the normal.
        It might well be that when considering natural phenomena, multiplicative
        processes be more or equally common than additive ones, as we oftenly
        observed skewed continuous data.
        Rubén
        http://webmail.udec.cl

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      • Isobel Clark
        Thanks to Rubén and Digby for pointing out what I had misunderstood about Don Myers email. It had not occurred to me (duh) that the lines starting would
        Message 3 of 9 , Dec 6, 2002
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          Thanks to Rubén and Digby for pointing out what I had
          misunderstood about Don Myers' email.

          It had not occurred to me (duh) that the lines
          starting '>' would be read as being from me rather
          than part of a forwarded email.

          Another score on the dumb side. Apologies for the
          strong reaction to Don's email if (on this occasion)
          he was not criticising my contribution.

          Isobel

          http://uk.geocities.com/drisobelclark


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