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Error Propagation Calculator

Enter the values X and Y and their relative changes in the calculator & it will calculate the error of propagation.

\( Z = X + Y\)

\( ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

\( Z = X − Y\)

\( ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

\( Z = X \cdot Y\)

\( ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 + (\dfrac{ΔY}{Y})^2}\)

\( Z = \dfrac {X} {Y}\)

\( ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 + (\dfrac{ΔY}{Y})^2}\)

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The error propagation calculator is designed to calculate the uncertainty due to the change in the final outcome of physical quantities. This provides the estimation more validity due to the removal of errors.

Standard Error Propagation:

The standard Propagation of Error (or Propagation of Uncertainty) is defined as the effect on the function of the variable uncertainty.  The propagation of error calculator deals with the calculation of the error in the final result from the errors in measured quantities x, y, and z dimensions. The change or the uncertain future values are combined together to provide an accurate result of measurements.

The Error Propagation Formulas:

The propagated standard deviation formula for various operations is given as: The formula for error propagation for addition:
  • Z = X + Y
  • \(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)
The formula for error propagation for subtraction:
  • Z = X − Y
  • \(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)
The formula for error propagation for multiplication:
  • Z = X * Y
  • \(ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 +  (\dfrac{ΔY}{Y})^2}\)
The formula for error propagation for division:                                    
  • \(Z = \dfrac{X}{Y}\)
  • \(ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 +  (\dfrac{ΔY}{Y})^2}\)
The error propagation calculator computes the uncertainty for addition, subtraction, multiplication, and division.                          

Practical Example:

Let’s suppose the length of two rods is 850 and ΔX 50 cm the second-rod length is 850 cm and ΔY is equal to 30cm. The length of the first rod is 900 cm and the second rod is 880 cm Then propagated standard deviation of the two rods for addition is given by: Given: X = 850 cm ΔX = 50 cm  X = 850 cm ΔX = 30 cm  Solution: Formula to calculate Addition Z = X + Y Z = 850 +850 Z = 1700 \(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\) \(ΔZ = \sqrt{(900)^2 +(880)^2}\)  ΔZ = 1258.73 Measuring error propagation standard deviation assists us in removing the uncertainty about the outcome. You can reduce the uncertainty of the outcome by the uncertainty propagation calculator. The natural log error propagation is useful to calibrate the minor or large changes in physical quantities.

Working of Error Propagation Calculator:

Find the combined errors for various mathematical operations by the propagation of the uncertainty calculator. Let's learn how! Input:
  • Choose the math operator 
  • Enter X and Change Δ X
  • Enter Y and Change Δ Y
  • Tap calculate
Output:
  • Value of Z and ΔZ 
  • Step-by-step calculations 

FAQs:

What is Meant by Measurement Error?

It is the difference between a measured quantity and its true propagated standard deviation or propagated errors. The error propagation standard deviation is a true reflection of change when gauging it by the propagation of error calculator physics. 

What are the Types of Errors?

The following are the types of errors:
  • Gross Errors
  • Random Errors
  • Systematic Errors

What is Absolute Error?

Absolute error is the variation between the actual values and measured values. It is given by the propagated standard deviation formula for absolute error and can be measured by the greatest possible error calculator. Absolute error = |VA-VE|
  • VA =  actual values 
  • VE= measured values 

References:

From the source of chem.libretexts.org: Propagation error From the source of sciencedirect.com: Error standard propagation
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