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Combination of Errors

  1. Mar 19, 2013 #1
    Hello Everyone,

    I am trying to compute the standard error of a stress related calculation.

    Let's consider the following:

    A ± ΔA
    B ± ΔB

    where A and B are the mean values while ΔA and ΔB are the respective standard errors.

    The common combination of errors formulas are as follows:

    y = A + B then, Δy = √(ΔA2+ΔB2)

    y = k. A then, Δy = k.ΔA

    Now my equation is,

    y = k2 . [ A + k1 . ( A + B + C ) ]

    May I know if my following working is correct?

    ( A + B + C )
    = √(ΔA2+ΔB2 + ΔC2)

    k1 . ( A + B + C )
    = k1 . √(ΔA2+ΔB2 + ΔC2)

    A + k1 . ( A + B + C )
    = √[ΔA2 + (k1 . √(ΔA2+ΔB2 + ΔC2))2]
    = √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

    k2 . [ A + k1 . ( A + B + C ) ]
    = k2 . √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]


    I am confused because I was suggested that it should be,
    k2 . [ A + k1 . ( A + B + C ) ]
    = k2 . √ [k1. ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

    Your input in highly appreciated.

    Charles
     
  2. jcsd
  3. Mar 19, 2013 #2

    ehild

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    The function f=y = k2 . [ A + k1 . ( A + B + C ) ] is the same as
    y = k2 [ A(k1+1) + k1 B + k1C ) ]. You can take y as linear combination of A, B, C with the constants p and q (p=k2(1+k1), q=k1k2):

    y= pA+qB+qC

    ehild
     
  4. Mar 19, 2013 #3
    I am sorry but I noticed that there are some mistakes in my first post. The subscripts and superscripts are not clearly shown. So the correct equations are as follows:

    The common combination of errors formulas are as follows:

    y = A + B then, Δy = √(ΔA2+ΔB2)

    y = k. A then, Δy = k.ΔA

    Now my equation is,

    y = k2 . [ A + k1 . ( A + B + C ) ]

    May I know if my following working is correct?

    ( A + B + C )
    = √(ΔA2+ΔB2 + ΔC2)

    k1 . ( A + B + C )
    = k1 . √(ΔA2+ΔB2 + ΔC2)

    A + k1 . ( A + B + C )
    = √[ΔA2 + (k1 . √(ΔA2+ΔB2 + ΔC2))2]
    = √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

    k2 . [ A + k1 . ( A + B + C ) ]
    = k2 . √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]


    I am confused because I was suggested that it should be,
    k2 . [ A + k1 . ( A + B + C ) ]
    = k2 . √ [k1. ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

    ------------------------------------------------------------------------------------
    Thanks ehild,
    I tried you suggestion and ended up as follows:
    Taking,
    p=k2(1+k1),
    q=k1k2

    pA+qB+qC
    = √[p2ΔA2+q2ΔB2 + q2ΔC2]
    = √[(k2(1+k1))2ΔA2+(k1k2)2ΔB2 + (k1k2)2ΔC2]
    = k2√[(1+k1)2ΔA2+(k1)2ΔB2 + (k1)2ΔC2]
    = k2√[(1+2k1+k12)ΔA2+(k1)2ΔB2 + (k1)2ΔC2]
    = k2√[ΔA2 + 2k1ΔA2 +k12(ΔA2+ΔB2 + ΔC2)]

    It ends up to be different than the earlier solutions that I found and was suggested. Please could you (or anyone) enlighten me.

    Thank you in advance.
     
  5. Mar 19, 2013 #4

    ehild

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    As far as I know, that is the correct expression of the error of your function.

    If you have a function f(x,y,z) of variables x, y, z and you know the mean values and standard deviations X±Δx, Y±Δy, Z±Δz, than the error of the function is [tex]Δf=\sqrt{(\frac{\partial f}{\partial x}\Delta x)^2+(\frac{\partial f}{\partial y}\Delta y)^2+(\frac{\partial f}{\partial z}\Delta z)^2}[/tex]

    ehild
     
  6. Mar 19, 2013 #5

    haruspex

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    I agree with ehild. That is the correct answer.
     
  7. Mar 20, 2013 #6
    Thank you for your help ehild and haruspex.
     
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