Combination of Errors

  • Thread starter charlesltl
  • Start date
  • #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
 

Answers and Replies

  • #2
ehild
Homework Helper
15,540
1,907
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
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.
 
  • #4
ehild
Homework Helper
15,540
1,907
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
 
  • #5
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
34,852
6,121
= k2√[ΔA2 + 2k1ΔA2 +k12(ΔA2+ΔB2 + ΔC2)]
I agree with ehild. That is the correct answer.
 
  • #6
Thank you for your help ehild and haruspex.
 

Related Threads on Combination of Errors

Replies
3
Views
800
Replies
1
Views
5K
Replies
9
Views
3K
Replies
1
Views
2K
  • Last Post
Replies
0
Views
5K
  • Last Post
Replies
3
Views
13K
  • Last Post
Replies
8
Views
7K
  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
1
Views
4K
Top