Combined Uncertainty of an Indirect Measurement

S

shucksjay

Hello first time poster here. I have a measurement that is calculated by the formula z = a x b/(a + c). Each of the variables having a known uncertainty given in terms of standard deviations. Is the model equation the same as the formula to calculate the final value of z or is it something different?

I'm on the side that says the combined uncertainty is found by breaking the formula into three parts.
1. The a x b term
2. The a + c
3. The quotient of 1 and 2

The uncertainty of a x b would be Δx/x = sqrt[(Δa/a)^2 + (Δb/b)^2]

The uncertainty of a + c would just the RSS or Δy = sqrt(Δa^2 + Δb^2)

The combined uncertainty would then be Δz/z = sqrt[(Δx/x)^2 + (Δy/y)^2]

A colleague is on the side that the combined uncertainty is just the RSS of Δa, Δb, and Δc. They also don't agree with my approach to include the uncertainty of a in both equations 1 and 2.
 

BradM

Leader
Admin
Hello there!

Hello first time poster here. I have a measurement that is calculated by the formula z = a x b/(a + c). Each of the variables having a known uncertainty given in terms of standard deviations. Is the model equation the same as the formula to calculate the final value of z or is it something different?

I'm on the side that says the combined uncertainty is found by breaking the formula into three parts.
1. The a x b term
2. The a + c
3. The quotient of 1 and 2

The uncertainty of a x b would be Δx/x = sqrt[(Δa/a)^2 + (Δb/b)^2]

The uncertainty of a + c would just the RSS or Δy = sqrt(Δa^2 + Δb^2)

The combined uncertainty would then be Δz/z = sqrt[(Δx/x)^2 + (Δy/y)^2]

A colleague is on the side that the combined uncertainty is just the RSS of Δa, Δb, and Δc. They also don't agree with my approach to include the uncertainty of a in both equations 1 and 2.

One question I had was in this formula:

z = a x b/(a + c).

Is it:
z = (a x b)/(a + c) or z = a x [b/(a + c)]?

It would help (possibly at least from my perspective) to understand what kind of measurement this is, and how the uncertainty contributors affect the measurement. Is there any uncertainty literature provided with the measurement?
 
S

shucksjay

Hello there!



One question I had was in this formula:

z = a x b/(a + c).

Is it:
z = (a x b)/(a + c) or z = a x [b/(a + c)]?

It would help (possibly at least from my perspective) to understand what kind of measurement this is, and how the uncertainty contributors affect the measurement. Is there any uncertainty literature provided with the measurement?


Hello Brad,

The formula should read z = (a x b)/(a + c). The formula is used to calculate the diluted concentration of an analyte in a gas cylinder through mass flow controllers. One example I could give is that "a" would be 100ppm +/- 2ppm of an analyte balanced in an inert gas that's being flowed through flow controller "b" at a rate of 50 sccm +/- 0.5. There would then be a tee downstream from this that connects it to another mass flow controller("c") with a flow of 2950 sccm +/- 3.0. So the nominal concentration of the analyte shoud be z = (100 x 50)/(50 + 2950) = 1.666 ppm.
 

tacom

Involved In Discussions
Dear shucksjay
I advice that you should take the derivatives of model and find sensitivity cooefficient

tacom
 

tacom

Involved In Discussions
Dear shucksjay
I advice that you should take the derivatives of model and find sensitivity cooefficient


Z=a.b/(a+c)
Ca=b(a+c)-(a.b)/(a+c)^2
Cb=a/(a+c)
Cc=-a.b/(a+c)^2
Uz=(Ca^2.Ua^2+Cb^2.Ub^2+Cc^2.Ua^2)0,5

tacom
 
S

shucksjay

Tacom,

Is the last equation suppose to be Uz=(Ca^2.Ua^2+Cb^2.Ub^2+Cc^2.Uc^2)0,5?

shucksjay
 
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