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Tue May 22 10:24:19 PDT 2012

Contents


    Statistics

      Also See

      [ math_81_Probabillity.html ]

      Online Statistics and Probability Calculators

      [ statistics.php ] [ http://statpages.org/ ] [ ../tools/stats.scm ] (Local Scheme functions)

      Notes and definitions on statistics.

    1. STATISTICS::=following,
      Net

        Definition of a (finite) sample

      1. I:: Finite_Sets=given. I standards for index. Typically it is a range 1..n where n is the sample size. However in some languages/cultures I could be 0..n-1. In fact there is no reason to limit I to a range. Any finite set of indices will work.

      2. Sample::=I>->Real. Each index item has a measured value.

        Example. For example the list

      3. (1,2,3) has I = 1..3 and size=3.

      4. size::Real= |I|. n:=size. Local shorthand.

      5. For p:Real, p::Sample = I +> p. A coercion that converts a single value into a sample with the same value for each index. This turns out to be useful when we subtract the mean of a sample (a number) from every item in the sample.

      6. For x, y::Sample I will use x and y as the names of samples of data.

        Statistics on one sample

      7. mean(x)::= +x/n. In STANDARD (+) is a serial operator that adds up all the items in its arguments.
      8. +(1,2,3) = (1+2+3) = 6.
      9. mean((1, 2, 3)) = +(1, 2, 3)/3 = 6/3 = 2.

        min, max, range, mode, histogram are to be done. [click here [socket symbol] stats1 if you can fill this hole]

      10. ss(x)::=+(x*x). Sum of squares.
      11. ss((1, 2, 3)) = +(1*1, 2*2 , 3*3) = +(1,4,9) = 14.

      12. ms(x)::= ss(x - mean(x) )/n. Mean squares about mean.
      13. (-1)|-ms(x) =( ss(x) - (+x)*mean(x))/n. Better for small hand calculations.
      14. ms((1,2,3)) = (14 - 6*2)/3 = 2/3.

      15. var(x)::= n * ms(x)/(n-1). Sample variance -- rescale to allow for estimating the mean.

      16. root_mean_square(x)::=sqrt(ms(x)).
      17. rms::= root_mean_square.

      18. standard_deviation(x)::=sqrt( var (x) ).
      19. sd(x)::=standard_deviation(x).

        Statistics on Two samples

      20. SP(x,y)::=+((x-mean(x))*(y-mean(y))).
      21. (-1)|-ss(x) = SP(x,x).

      22. MS(x,y)::=SP(x,y)/n.

      23. r(x,y)::= MS(x,y)/( sd(x)*sd(y)). Correlation coefficient -- Pearson.

        More... [click here [socket symbol] stats2 if you can fill this hole]


      (End of Net STATISTICS)

    . . . . . . . . . ( end of section Statistics in MATHS) <<Contents | End>>

    Notes on MATHS Notation

    Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

    Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

    The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.

    For a complete listing of pages in this part of my site by topic see [ home.html ]

    Notes on the Underlying Logic of MATHS

    The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

    For a more rigorous description of the standard notations see

  1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

    Glossary

  2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
  3. given::reason="I've been told that...", used to describe a problem.
  4. given::variable="I'll be given a value or object like this...", used to describe a problem.
  5. goal::theorem="The result I'm trying to prove right now".
  6. goal::variable="The value or object I'm trying to find or construct".
  7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
  8. hyp::reason="I assumed this in my last Let/Case/Po/...".
  9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
  10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
  11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.

End