# Alternate expressions for the uncertainty propagation

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• AndersF
In summary, there are two different formulas for calculating the propagation of uncertainty in a measurement. These are equation (1) and equation (2), which involve different methods for estimating the uncertainty in a quantity that depends on other quantities. The main difference between the two expressions is the degree of correlation between the quantities involved. When choosing between these two equations, it is important to understand the level of error propagation and to carefully consider which formula is most appropriate for the specific situation.
AndersF
TL;DR Summary
What criteria to use to decide which of the two formulas to use.
I have seen that there are two different formulas that we can use when calculating the propagation of uncertainty in a measurement. If ##X=f(A, B, C, \ldots)## is the quantity whose uncertainty we want to estimate, which depends on the quantities ##A,B,C,...##, then we could calculate the propagation of uncertainty either by (1) or by (2):

$$\Delta X=\left|\frac{\partial f}{\partial A}\right| \cdot \Delta A+\left|\frac{\partial f}{\partial B}\right| \cdot \Delta B+\left|\frac{\partial f}{\partial C}\right| \cdot \Delta C+\cdots \tag{1}$$

$$\sigma_{X}=\sqrt{ \left(\frac{\partial f}{\partial A} \right)^{2}\sigma_{A}^2+\left(\frac{\partial f}{\partial B} \right)^{2}\sigma_{B}^2+\left(\frac{\partial f}{\partial C} \right)^{2}\sigma_{C}^2+\cdots } \tag{2}$$

With ##\Delta X, \Delta A,\Delta B,\Delta C,...## the uncertainties in the values of ##X, A,B,C,...## and ##\sigma_X,\sigma_A,\sigma_B,\sigma_C,...## the standard deviation in these measurements.

What is the difference between these two expressions? When do we choose equation (1) or equation (2) for estimating the propagation of uncertainty in a value?

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For example, if we wanted to estimate the uncertainty in the calculation of the value of some resistors from the measured values in current and voltage, ##V = 2.04,\space 2.10,\space 2.19 \space V##, ##I =16.8,\space 28.7 ,\space 63.7\space mA##, whit ##\Delta V = 0.01\space V## and ##\Delta I =0.1\space mA## the uncertainties in these measurements, applying (1) and (2) to Ohm's law would give (3) and (4), next to next:

$$\Delta R=\left|\frac{1}{I}\Delta V\right|+\left|\frac{V}{I^{2}} \Delta I\right| \tag{4}$$
$$\sigma_{R}=\sqrt{ \frac{1}{I^2}\sigma_{I}^2 + \frac{V^2}{I^{4}}\sigma_{B}^2} \tag{2}$$

Then, for the ##R=121.42857..., \space 73.17073..., \space 34.37990... \space \Omega## calculated values, if we take ##\sigma_V= \Delta V## and ##\sigma_I= \Delta I##, we would get these values, which in this case are slightly different:

$$\Delta R=1,\space 0.6,\space 0.2\space \Omega$$
$$\sigma_{R} = 0.9,\space 0.4,\space 0.2\space \Omega$$

Therefore, what would be the criteria for deciding whether to use (1) and (2), for this example and for a general case?

Picking out formulas willy-nilly is going to cause you trouble. If you need more than an estimate- and either will do that - you need to understand error propegation at the level of, say, Taylor's book. The difference between those two expressions is the degree of correlkation.

## 1. What is uncertainty propagation?

Uncertainty propagation is the process of quantifying and propagating uncertainties from input parameters to output variables in a mathematical model or experiment.

## 2. Why is it important to consider alternate expressions for uncertainty propagation?

Alternate expressions for uncertainty propagation allow for a more comprehensive and accurate analysis of uncertainties, as different approaches may be better suited for certain types of models or experiments.

## 3. What are some common alternate expressions for uncertainty propagation?

Some common alternate expressions for uncertainty propagation include Monte Carlo simulation, sensitivity analysis, and interval analysis.

## 4. How do these alternate expressions differ from traditional uncertainty propagation methods?

Traditional uncertainty propagation methods, such as the Taylor series expansion, assume that the input parameters are independent and normally distributed. Alternate expressions take into account non-normal distributions and correlations between input parameters.

## 5. When should I use alternate expressions for uncertainty propagation?

Alternate expressions should be used when the assumptions of traditional methods are not met, or when a more detailed and accurate analysis of uncertainties is needed. They may also be useful for complex models or experiments with many input parameters.

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