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Archived Weighted least-squares fit error propagation

  1. Nov 17, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose we measure N pairs of values (xi, yi) of two variables x and y that are supposed to statisfy a linear relation y = A + Bx suppose the xi have negligible uncertainty and the yi have different uncertainties [itex]\sigma_{i}[/itex]. We can define the weight of the ith measurement as [itex]w_{i} = 1/\sigma_{i}[/itex]. Then the best estimates of A and B are:

    A = \frac{\Sigma w x^{2}\Sigma w y - \Sigma w x \Sigma w x y}{\Delta}\\
    B = \frac{\Sigma w \Sigma w x y - \Sigma w x \Sigma w y}{\Delta}\\
    \Delta = \Sigma w \Sigma x^{2} - \left(\Sigma w x \right)^{2}

    Use error propagation to prove that the uncertainties in the constants A and B are given by

    \sigma_{A} = \sqrt{\frac{\Sigma w x^{2}}{\Delta}}\\
    \sigma_{B} = \sqrt{\frac{\Sigma w}{\Delta}}

    2. Relevant equations

    rules for sums and differences
    q = x \pm z\\
    \delta q = \sqrt{(\delta x)^{2} + (\delta z)^{2}}\\
    rules for products and quotients
    \delta q = \sqrt{(\delta x/x)^{2} + (\delta z/z)^{2}}\\

    3. The attempt at a solution
    What I'm thinking is because the uncertainty in x is negligible it will be treated like a constant. I'm not sure how to deal with all these sums. I feel like I don't know how to approach this question.
    Last edited: Nov 17, 2013
  2. jcsd
  3. May 1, 2016 #2


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    Science Advisor
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    Gold Member

    Try calculating E(A2) etc. The negligible uncertainty in the x allows you to treat the Δ denominators as constant.
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