Calculating Uncertainty in z with Method 1 & Method 2

In summary, when calculating the propagated uncertainty of a multivariable function, there are two methods to consider: Method 1, which involves finding the partial derivatives and using them in a formula, and Method 2, which involves finding the square root of the sum of the squares of the partial derivatives. Method 2 is the more commonly used statistical approach and is considered to represent the uncertainty better than Method 1, which does not take into account the possibility of negative or positive errors. However, to reconcile the two methods, it is suggested to modify Method 1 by using absolute values for both terms before adding.
  • #1
opticaltempest
135
0
Assume we have the function [tex]z = x\sin y[/tex]
Our best guest for our measurement is x=1.0 and y=2.0. The uncertainty in x is 0.05. The uncertainty in y is 0.10.

We want to calculate the final uncertainty as the initial uncertainties propagate through the function.

***** Method 1 *****
In Calculus III we find the propagation of uncertainties in multivariable functions using the following method:

[tex]
dz = \frac{{\partial z}}{{\partial x}}dx + \frac{{\partial z}}{{\partial y}}dy
[/tex]

So the uncertainty would be

[tex]
\begin{array}{l}
dz = \sin \left( y \right)dx + x\cos \left( y \right)dy \\
dz = \sin \left( {2.0} \right)\left( {0.05} \right) + \left( {1.0} \right)\cos \left( {2.0} \right)\left( {0.10} \right) \\
dz = 0.0039 \\
\end{array}
[/tex]***** Method 2 *****

According to
https://www.amazon.com/dp/093570275X/?tag=pfamazon01-20

It says we should use this formula to calculate the propagated uncertainty:

[tex]
\delta z = \sqrt {\left( {\frac{{\partial z}}{{\partial x}}dx} \right)^2 + \left( {\frac{{\partial z}}{{\partial y}}dy} \right)^2 }
[/tex]

Using this method the uncertainty is

[tex]
\begin{array}{l}
\delta z = \sqrt {\left[ {\sin \left( {2.0} \right)\left( {0.05} \right)} \right]^2 + \left[ {\left( {1.0} \right)\cos \left( {2.0} \right)\left( {0.10} \right)} \right]^2 } \\
\delta z = 0.062 \\
\end{array}
[/tex]

The uncertainty in method 2 is nearly 16 times larger than the uncertainty in method 1.
I am assuming method 2 represents the uncertainty better than method 1.

My question is: What is method 2 taking into account that method 1 isnt? Why does method 2 represent the uncertainty better than method 1?
 
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  • #2
To reconcile the 2 approaches, I suggest you modify method 1 to use absolute value for both terms and then add. This would bring them closer.

Method 2 is the usual statistical approach, since errors can be negative or positive.
 
  • #3
I believe the law of averages would rapidly assert itself in this scenario. The Chi squared probability is the most reliable method, IMO.
 

1. What is the purpose of calculating uncertainty in z with Method 1 and Method 2?

The purpose of calculating uncertainty in z with Method 1 and Method 2 is to determine the range of values within which the true value of z lies. This helps to provide a measure of the reliability and accuracy of the calculated value of z.

2. How does Method 1 differ from Method 2 in calculating uncertainty in z?

Method 1 involves using the standard deviation of the sample to calculate the uncertainty in z, while Method 2 involves using the standard error of the mean to calculate the uncertainty. Method 2 is typically used when the sample size is small, while Method 1 is used when the sample size is large.

3. What factors can affect the uncertainty in z calculated with Method 1 and Method 2?

The sample size, the variability of the data, and the accuracy of the measurements can all affect the uncertainty in z calculated with Method 1 and Method 2. Additionally, any errors or biases in the data collection process can also impact the uncertainty.

4. Is one method more accurate than the other in calculating uncertainty in z?

It depends on the specific circumstances and data being analyzed. Method 1 is generally considered more accurate for large sample sizes, while Method 2 is more accurate for small sample sizes. It is important to consider the characteristics of the data and choose the appropriate method accordingly.

5. Can uncertainty in z be reduced or eliminated?

No, uncertainty is an inherent part of any measurement or calculation. However, it can be minimized by using more precise and accurate measurements and by increasing the sample size. It is important to report the uncertainty in z along with the calculated value to provide a complete understanding of the results.

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