Alternate expressions for the uncertainty propagation

[AndersF]

TL;DR

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?

 

[Vanadium 50]

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.