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Weighted lest square fit

  1. Feb 1, 2007 #1
    1. The problem statement, all variables and given/known data

    I have a project for one of my class and I have been given a sheet to do the statistical analyst of my data. I am not convince this sheet is proper and I need someone to look over it it. I don't understand where my Delta R goes...

    2. Relevant equations
    \chi^2 =-\frac{1}{2} \sum_{i=1}^{N}{\left(\frac{y_i-ax_i}{\sigma_i^2}\right)^2} \\
    \frac{\partial \chi^2}{\partial a}[/tex]
    [tex] = \sum_{i=1}^{N}{\frac{x_i}{\sigma_i^2}(y_i-ax_i)} = 0 \\
    \sum_{i=1}^{N}{\frac{x_i y_i}{\sigma_i^2}}[/tex]
    [tex] = a\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} \\
    a= \sum_{i=1}^{N}{\frac{x_iy_i}{x_i^2}} \\
    \sigma_a^2 = \sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} S^2 \\
    S^2 [/tex]
    [tex]= \frac{1}{N-1}\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}} \\
    \sigma_a^2 =[/tex]
    [tex] \frac{1}{N-1}\left(\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}}\right)\left(\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}}\right) \\
    r_n^2 = \left(n-\frac{1}{2}\right)\lambda R \\
    a= \sum_{n=1}^{15}{\frac{r_n^2\left(n-\frac{1}{2}\right)}{(r_n^2)^2}} \\
    [tex] = \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2}{\sigma_r^2}}\right)\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{\sigma_r^2}}\right) \\
    \Psi(x) = x^2 \\
    \sigma_\Psi^2 = (\frac{\partial}{\partial x}(x^2)\Delta x)^2 [/tex][tex]= 4x^2(\Delta x)^2 \\
    \sigma_{r_n^2} = 4r_n^2(\Delta r)^2 \\
    \sigma_a^2 = \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2}{4r_n^2}}\right)/\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{4r_n^2}}\right)

    3. The attempt at a solution

    The error on r comes from the fact it's a measurement and we square it. I think Delta R should be there somewhere even though it is constant...
    Last edited by a moderator: Feb 2, 2007
  2. jcsd
  3. Feb 1, 2007 #2

    PDF of latex file.

    Attached Files:

  4. Feb 2, 2007 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    Logik, do not put the spaces in [ tex ] and [ /tex ].

    And do NOT write a LONG formula as a single Tex statement.
    It will wind up all on one page.
  5. Feb 2, 2007 #4
    Logik - regarding your enclosure, I don't understand the step from equation (3) to equation (4). I also don't know if this is relevent to your question.
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