Error Propagation Question

In summary, the goal is to find the uncertainty δd in an equation involving d, C_1, V, and D, where C_1 is a collection of constants, D and V have their own uncertainties, and d is inversely proportional to √(V). The formula for calculating δd is given by \frac{δd}{d}=|C_1| \sqrt{(\frac{δV}{V})^2+(\frac{δD}{D})^2 }, and this can be applied to equations 1 and 2, which have the same formula for δd.
  • #1
UncertaintyMan
2
0
My goal is to find the uncertainty [itex]δd[/itex] in the following equation.

[itex]d=C_1 \frac{1}{\sqrt{V}} \frac{1}{D}[/itex]

  • [itex]C_1[/itex] is the collection of constants [itex]\frac{2Lhc}{\sqrt{2m_e c^2 }}[/itex]
  • [itex]D[/itex] is a value measured in meters with an uncertainty [itex]δD = 0.001 m[/itex]
  • and [itex]V[/itex] is a value measured in volts with an uncertainty [itex]δV = 100 V[/itex]

My best guess on how to calculate [itex]δd[/itex] is

[itex]\frac{δd}{d}=|C_1| \sqrt{(\frac{δV}{V})^2+(\frac{δD}{D})^2 }[/itex]
... then plug in all the known values and solve for [itex]δd[/itex]

...Unfortunately I have no resources to tell me if I'm doing this right. I appreciate any helpful pointers any of you may have, I'm a big time noob when it comes to error analysis.

For those of you who are curious, this is from a Bragg Scattering lab and [itex]d[/itex] represents the distance between atoms in a polycrystalline graphite crystal.
 
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  • #2
UncertaintyMan said:
My goal is to find the uncertainty [itex]δd[/itex] in the following equation.

[itex]d=C_1 \frac{1}{\sqrt{V}} \frac{1}{D}[/itex]

  • [itex]C_1[/itex] is the collection of constants [itex]\frac{2Lhc}{\sqrt{2m_e c^2 }}[/itex]
  • [itex]D[/itex] is a value measured in meters with an uncertainty [itex]δD = 0.001 m[/itex]
  • and [itex]V[/itex] is a value measured in volts with an uncertainty [itex]δV = 100 V[/itex]

My best guess on how to calculate [itex]δd[/itex] is

[itex]\frac{δd}{d}=|C_1| \sqrt{(\frac{δV}{V})^2+(\frac{δD}{D})^2 }[/itex]
... then plug in all the known values and solve for [itex]δd[/itex]

...Unfortunately I have no resources to tell me if I'm doing this right. I appreciate any helpful pointers any of you may have, I'm a big time noob when it comes to error analysis.

For those of you who are curious, this is from a Bragg Scattering lab and [itex]d[/itex] represents the distance between atoms in a polycrystalline graphite crystal.
d is inversely prop. to √(V) , not V itself.

You should have something like:
[itex]\displaystyle \frac{δd}{d}=|C_1| \sqrt{\left(\frac{δ(\sqrt{V})}{\sqrt{V}}\right)^2+\left(\frac{δD}{D} \right)^2 }[/itex]​
 
  • #3
Awesome, thank you!

Quick side question: is it true that both of these equations have the same δd formula?

Equation 1. [itex]d = \sqrt{V}D[/itex]
Equation 2. [itex]d = \frac{1}{\sqrt{V}D}[/itex]

Error for either equation:
[itex]\frac{δd}{d} = \sqrt{(\frac{δ(\sqrt{V})}{\sqrt{V}})^2 + (\frac{δD}{D})^2}[/itex]
 
  • #4
UncertaintyMan said:
Awesome, thank you!

Quick side question: is it true that both of these equations have the same δd formula?

Equation 1. [itex]d = \sqrt{V}D[/itex]
Equation 2. [itex]d = \frac{1}{\sqrt{V}D}[/itex]

Error for either equation:
[itex]\frac{δd}{d} = \sqrt{(\frac{δ(\sqrt{V})}{\sqrt{V}})^2 + (\frac{δD}{D})^2}[/itex]

Yes, for reasonably small relative error.
 
  • #5


Your approach to calculating the uncertainty in d is correct. This is known as error propagation and it is a common method used in scientific measurements to determine the uncertainty in a calculated quantity.

To ensure that your calculation is correct, you can use the following steps:

1. Write out the equation for d in terms of the given variables:
d = C_1 * (1/√V) * (1/D)

2. Determine the partial derivatives of d with respect to each variable:
∂d/∂C_1 = (1/√V) * (1/D)
∂d/∂V = -C_1 * (1/2√V^3) * (1/D)
∂d/∂D = -C_1 * (1/√V) * (1/D^2)

3. Use these partial derivatives to calculate the uncertainty in d:
δd = √[(∂d/∂C_1)^2 * (δC_1)^2 + (∂d/∂V)^2 * (δV)^2 + (∂d/∂D)^2 * (δD)^2]

4. Substitute in the values for δC_1, δV, and δD, and solve for δd.

As a scientist, it is important to always consider the uncertainty in your measurements and calculations. This helps to ensure the accuracy and reliability of your results. Good luck with your Bragg Scattering lab!
 

1. What is error propagation?

Error propagation refers to the process of estimating the uncertainties in the results of a scientific measurement or calculation, based on the uncertainties in the input values and the mathematical relationships between them.

2. Why is error propagation important in scientific research?

Error propagation is important because it allows scientists to understand the potential errors and uncertainties in their data and results. This helps to ensure the accuracy and reliability of scientific findings, and allows for proper interpretation and communication of results.

3. How is error propagation calculated?

Error propagation is calculated using mathematical formulas that take into account the uncertainties in the input values and the relationships between them, such as addition, subtraction, multiplication, and division. These formulas can be complex, but there are also software programs and online calculators available to help with the calculations.

4. What are some common sources of error in scientific measurements?

Some common sources of error in scientific measurements include instrument error, human error, environmental conditions, and limitations of the experimental design. These can all contribute to the overall uncertainty in the measurement and should be considered in error propagation calculations.

5. How can error propagation be minimized?

Error propagation can be minimized by using precise and accurate measurement techniques, carefully designing experiments to minimize sources of error, and repeating measurements multiple times to assess the variability in the results. Additionally, using appropriate statistical methods and software can help to minimize error propagation and improve the accuracy of results.

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