Error Propagation: Solutions to Complex Equations

In summary, the conversation discusses how to propagate errors in equations when only one term has an uncertainty. The suggested method is to multiply and divide the uncertainty value by the constants in the equation. It is also mentioned that when there are multiple uncertain terms, the exact error can be calculated by finding the maximum and minimum values. A rule of thumb is also mentioned, stating that when adding or subtracting measurements, errors add, and when multiplying or dividing measurements, percentage errors add.
  • #1
coregis
17
0
Always the easy things we forget...
I know how errors propogate through multiplication or division when every term has an error, but how do I propagate errors in equations when only one term has an uncertainty? I want to say just multiply and divide the uncertainty value by the constants, i.e plug my value in the equation, then plug the uncertainty. This is the same as if I just found the % uncertainty, and multiplied the final product by that, correct? Is this the right way to go about this? And what if two (or more) terms have uncertainties? Would I find the uncertainty between those terms and then apply that % to the final number? Thanks.
 
Last edited:
Mathematics news on Phys.org
  • #2
If you mean something like y= ax+ b where a and b are exactly defined constants and x is measurement: x= m+/- e, then the largest possible value is a(m+e)+b= am+ ae+ b= (am+b)+ ae and the smallest possible is a(m-e)+ b= am-ae+ b= (am+b)- ae.

That is: (am+ b)+/- ae. Any added constants you can ignore. Constants multiplied by x multiply the error. Same for percentage error.

With more than one "uncertain" number you can get the exact error by calculating the maximum and minimum. A "rule of thumb" (good approximation but not exact) is that when you add or subtract measurements, the errors add, when you multiply or divide measurements, the percentage errors add.
 
  • #3


You are correct in your approach to propagating errors in equations when only one term has an uncertainty. In this case, you can simply multiply or divide the uncertainty value by the constants in the equation. This is equivalent to finding the percentage uncertainty and multiplying it by the final product.

However, when multiple terms in the equation have uncertainties, the process becomes a bit more complex. In this case, you would need to find the uncertainty between those terms and apply that percentage to the final result. This can be done by finding the partial derivatives of the equation with respect to each variable and then using the formula for error propagation, which involves taking the square root of the sum of the squares of each partial derivative multiplied by its corresponding uncertainty.

It is important to remember that error propagation can become quite complicated for complex equations, and it is always a good idea to check your calculations and assumptions to ensure accuracy. Additionally, it is important to keep track of significant figures throughout the calculations, as this can also affect the final result.

In summary, your approach is generally correct for propagating errors in equations, but it may become more involved when dealing with multiple uncertainties. It is always a good idea to double check your calculations and assumptions to ensure accuracy.
 

1. What is error propagation?

Error propagation is the process of determining the uncertainty or error in a final result that is calculated from a set of measured values or variables. It takes into account the uncertainties associated with each of the input values and uses mathematical methods to calculate the overall uncertainty in the final result.

2. Why is error propagation important?

Understanding and accounting for error propagation is crucial in scientific research and experimentation. It allows for a more accurate and realistic representation of the data and results obtained. Ignoring error propagation can lead to incorrect conclusions and inaccurate predictions.

3. How is error propagation calculated?

Error propagation is typically calculated using the rules of error propagation, also known as the "propagation of uncertainty" or "uncertainty propagation" theory. This involves finding the partial derivatives of the equation and using them to calculate the overall uncertainty in the result.

4. What are some common sources of error in error propagation?

There are many factors that can contribute to error propagation, including measurement errors, instrument uncertainties, human errors, and systematic errors. It is important to identify and minimize these sources of error to improve the accuracy of the final result.

5. How can I minimize error propagation in my experiments?

To minimize error propagation, it is important to use precise and accurate measuring instruments, carefully record and analyze data, and repeat experiments multiple times to account for random errors. It is also important to understand and account for any systematic errors that may be present in the experimental setup.

Similar threads

  • General Math
Replies
5
Views
980
Replies
8
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
15
Views
943
  • General Math
Replies
2
Views
2K
  • Classical Physics
Replies
6
Views
2K
  • Other Physics Topics
Replies
1
Views
2K
Replies
3
Views
674
  • Classical Physics
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
2K
Back
Top