Combination of Errors

1. Mar 19, 2013

charlesltl

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)]

Charles

2. Mar 19, 2013

ehild

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

3. Mar 19, 2013

charlesltl

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.

4. Mar 19, 2013

ehild

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 $$Δf=\sqrt{(\frac{\partial f}{\partial x}\Delta x)^2+(\frac{\partial f}{\partial y}\Delta y)^2+(\frac{\partial f}{\partial z}\Delta z)^2}$$

ehild

5. Mar 19, 2013

haruspex

I agree with ehild. That is the correct answer.

6. Mar 20, 2013

charlesltl

Thank you for your help ehild and haruspex.