I Alternate expressions for the uncertainty propagation

AI Thread Summary
The discussion focuses on two formulas for calculating uncertainty propagation in measurements: one using absolute uncertainties and the other using standard deviations. The first formula provides a linear approximation, while the second incorporates the variances of the measurements, reflecting their correlation. An example using Ohm's law illustrates how both methods yield slightly different results for resistor values, emphasizing the importance of choosing the appropriate formula based on the context. The choice between the two methods should consider the level of accuracy required and the correlation between the variables involved. Understanding these differences is crucial for effective error propagation analysis.
AndersF
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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?

____________________________________________________________________________​

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?
 
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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.
 
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