Question

If you have two uncertainties, and they are from two different sources and contribute to the uncertainty of a measurement, what formula should be used to combine these and find the total uncertainty for the measurement?

Answer 1

The propagation errors we can find the uncertainty of a given magnitude is the sum of the uncertainties of each magnitude.

                           Δm = ∑  $| \frac{dm}{dx_i} | \ \Delta x_i$

Physical quantities are precise values ​​of a variable, but all measurements have an uncertainty, in the case of direct measurements the uncertainty is equal to the precision of the given instrument.

When you have derived variables, that is, when measurements are made with different instruments, each with a different uncertainty, the way to find the uncertainty or error is used the propagation errors to use the variation of each parameter, keeping the others constant and taking the worst of the  cases, all the errors add up.

If m is the calculated quantity, x_i the measured values ​​and Δx_i the uncertainty of each value, the total uncertainty is

                      Δm = ∑  $| \frac{dm}{dx_i } | \ \Delta x_i$    | dm / dx_i | Dx_i

               

for instance:

If the magnitude is  a average of two magnitudes measured each with a different error

                     m = $\frac{m_1+m_2}{2}$

                     Δm = | $\frac{dm}{dx_1}$ |  Δx₁ + | $\frac{dm}{dx_2}$ | Δx₂

                     $\frac{dm}{dx_1}$ = ½

                     $\frac{dm}{dx_2}$ = ½

                     Δm = $\frac{1}{2}$ Δx₁ + ½ Δx₂

                     Δm = Δx₁ + Δx₂

In conclusion, using the propagation errors we can find the uncertainty of a given quantity is the sum of the uncertainties of each measured quantity.

Learn more about propagation errors here:

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