What is Error Propagation?

Error propagation, also known as propagation of uncertainty, is a method used in scientific and engineering calculations to determine how the uncertainty in individual measurements affects the overall uncertainty of a calculated result.

When performing calculations with measured quantities, each measurement has an associated uncertainty or error. These individual uncertainties combine and propagate through the calculation, resulting in an uncertainty in the final result.

The error propagation calculator is a tool designed to help users determine the uncertainty in a calculated result based on the uncertainties of the input variables. It applies the principles of error propagation to provide a quantitative measure of the reliability of the final calculated value.

How to Use the Error Propagation Calculator?

1. Enter the Function
Start by entering the mathematical function for which you want to calculate the error propagation. Use standard mathematical notation, such as “ax + by” for ax + by. The calculator supports basic arithmetic operations (+, -, *, /), as well as more complex functions like sin(), cos(), exp(), log(), etc.

2. Specify the Variables
List all the variables used in your function, separated by commas. For example, if your function uses x and y, enter “x,y” in the Variables field. Make sure to list only the variables for which you have measured values and associated uncertainties.

3. Input the Values
Enter the measured values for each variable, in the same order as you listed the variables. Separate the values with commas. For instance, if x = 2 and y = 3, enter “2,3” in the Values field.

4. Provide the Errors
Enter the uncertainties (or errors) associated with each measured value, again in the same order as the variables. These should be the absolute errors, not relative or percentage errors. For example, if the error in x is ±0.1 and in y is ±0.2, enter “0.1,0.2” in the Errors field.

5. Calculate
Click the “Calculate” button to perform the error propagation calculation.

6. Interpret the Results
The calculator will display three key pieces of information:

• Function value: This is the result of evaluating your function with the given input values.
• Propagated error: This is the calculated uncertainty in the final result, taking into account all input uncertainties.
• Final result: This is expressed in the form “value ± error”, representing the best estimate of the true value and its associated uncertainty.

7. Complex Functions
For more complex calculations, you can use advanced mathematical functions supported by the calculator. This includes trigonometric functions (sin, cos, tan), exponentials (exp), logarithms (log, ln), and more.

8. Error Checking
The calculator includes error checking to ensure that the number of variables matches the number of values and errors provided. If there’s a mismatch, it will alert you to correct your input.

By following these steps, users can effectively utilize the error propagation calculator to determine the uncertainty in their calculated results.

This tool is invaluable for anyone working with experimental data, as it provides a quantitative measure of the reliability of derived quantities based on measured values with known uncertainties.