# Error Propagation Calculator

Created by Juhi Raj, PhD
Reviewed by Steven Wooding
Last updated: Feb 02, 2023

This error propagation calculator can be used to evaluate the uncertainty (or error) of any mathematical operation containing uncertain quantities. This error propagation calculator makes calculations easier by automating the most basic operations. You can use this error propagation calculator to figure out how the inaccuracy in the primary parameter(s) affects the final parameter.

If you would rather like to estimate the standard error of the mean of any dataset with up to 30 numbers, you can use our standard error calculator. Furthermore, you can explore our percent error calculator and relative error calculator for further calculations.

In this article below, you can also find the formula to propagate the error for various basic arithmetic operations.

## What is error propagation?

Error propagation occurs when you measure some quantities X and Y with uncertainties ΔX and ΔY, respectively. Then you want to calculate some other quantity Z using the measurements of X and Y. It turns out that the uncertainties ΔX and ΔY will propagate to the uncertainty of Z.

Or statistically, if f is a function of the independent variables X and Y, represented as f(X, Y), the uncertainty in f can be calculated by taking the partial derivatives of f with respect to each variable, multiplying by the uncertainty in that variable, and then adding these separate terms in quadrature.

## What is the error propagation formula for addition and subtraction?

The error propagation formulae depend on the mathematical operation used for calculation.

Let’s say you measured the length of two rods (X ± ΔX) as 2.00 ± 0.03 m and (Y ± ΔY) is 0.88 ± 0.04 m. This means that the sum of lengths of the two rods would be Z = X + Y = 2.00 m + 0.88 m = 1.12 m.

Using the error propagation rules:

\quad \begin{align*} \Delta Z &= \sqrt{(\Delta X)^2 + (\Delta Y)^2}\\ &= \sqrt{(0.03)^2 + (0.04)^2}\\ &= 0.05 \end{align*}

## How to use this error propagation calculator?

You can simply compute this using the error propagation calculator by following the steps below:

• Select the mathematical operation as Addition.
• Enter the length of the first rod as X = 2.0.
• Enter the length of the second rod as Y = 0.88.
• Input the error of the length of the first rod as ΔX = 0.03.
• Input the error of the length of the second rod as ΔY = 0.04.
• Using the error propagation equation, this calculator will compute the sum as Z and propagate the total error as ΔZ.

Similarly, in the case of subtraction, the error propagation equation is precisely the same as above.

## Error propagation for multiplication and division operations.

In the case of error propagation for multiplication and division operation, the uncertainty is propagated slightly differently from what we have seen so far.

For example, a bird flies a distance (X ± ΔX) = 120 ± 3 m in a time (Y ± ΔY) = 20 ± 1.2 s. What is the velocity of the bird?
Since the velocity Z = X/Y, select Division as the mathematical operation.

• Enter the distance the bird flies as X = 120.
• Enter the time of flight by the bird as Y = 20.
• Input the error of the distance as ΔX = 3.
• Input the error of the time as ΔY = 1.2.

Using the error propagation rules, the uncertainty for multiplication and division operation is computed as shown below:

$\quad \Delta Z = Z \cdot \sqrt{\bigg(\frac{\Delta X}{X}\bigg)^2 + \bigg(\frac{\Delta Y}{Y}\bigg)^2}$

## FAQ

### How do you propagate errors?

Error propagation determines how the inaccuracy in the primary parameter(s) of any mathematical operation can affect the final parameter. Then, given the data of X and Y, you can derive any other quantity Z. It turns out that the X and Y uncertainties will propagate to the Z uncertainty.

### How do you propagate division errors?

To propagate division errors, you just need to add all the relative errors to get the relative error in the result.

For example, when you measure some quantities X and Y with uncertainties ΔX and ΔY, respectively. Then if you want to calculate some other quantity Z = X/Y, the error is propagated as below:

• ΔZ = Z⋅√((ΔX/X)2 + (ΔY/Y)2)

### What is the error of adding two numbers that have a 1% error?

To determine the error while adding two numbers, you just have to compute the quadrature of the individual uncertainties. Hence, the total error of adding two numbers with an uncertainty of 1% is √2%.

Juhi Raj, PhD
Operator
X
ΔX
Y
ΔY
Z
Δ Z
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