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Can I use root-sum-square for this sort of problem?

  1. Sep 2, 2015 #1
    1. The problem statement, all variables and given/known data
    This isn't actually a problem I came across in a textbook, but close enough. Let's say I have a rod made of n sections. Each section has a 3-simga manufacturing uncertainty of +/- some value normally distributed about the mean. What is the uncertainty of the final length of the assembled rod?

    2. Relevant equations
    Root-sum-square equation

    3. The attempt at a solution
    My thought is to simply plug in each of the uncertainties into the root-sum-square equation to get the final uncertainty on the length of the assembles rod. I'm not entirely confident in this as I only ever recall that equation being applied to measurements, although I see this problem as being analogous. Assuming, the equation does apply, does the final answer maintain the 3-sigma uncertainty on the length of the rod?
     
  2. jcsd
  3. Sep 2, 2015 #2

    andrewkirk

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    If by '3-sigma uncertainty = x' you just mean that the std dev is x/3, and the lengths of the sections are assumed to be independent (which is a questionable assumption in a manufacturing process), then the answer is Yes.

    However 'terms like '3-sigma uncertainty' are often understood to imply a certain percentile of the distribution. For a normal dist, 99.73% of points lie within 3-sigma of the mean. The 3-sigma figure obtained by summing the squares and then square rotting will not necessarily still be a 99.73 percentile if the distribution of section lengths is not normal, because most non-normal distributions are not additive - ie the distribution changes shape upon addition. So if a percentile/confidence level is implied and the distribution of section lengths is not normal, the answer is No.
     
  4. Sep 2, 2015 #3

    Ray Vickson

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    If section ##i## has 3-sigma uncertainty ##u_i##, then (presumably) it has 1-sigma uncertainty ##u_i/3##, so the standard deviation is ##\sigma_i = u_i/3##. If the uncertainties in the individual sections are "independent"---which you are claiming is the case---then the variance in the final length is
    [tex] \sigma^2 = \sum_{i=1}^n \sigma_i^2 = \frac{1}{9} \sum_{i=1}^n u_i^2 [/tex]
    Therefore, the standard deviation of the total length is
    [tex] \sigma = \frac{1}{3} \sqrt{ \sum_{i=1}^n u_i^2}. [/tex]
    So, yes, indeed, the 3-sigma uncertainty in the total length is ##u = \sqrt{\sum_i u_i^2},## as you want.

    For more on this, Google "variance of sum".
     
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