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I would like to determine the combined standard measurement uncertainty for a mean mass ##\bar{m}## computed from a mean volume ##\bar{V}## and a constant density ##\rho## with $$\bar{m} = \bar{V} \, \rho.$$

I know the mean volume ##\bar{V}##, volume standard deviation ##\sigma_V##, number of volume samples ##n_V## and constant density ##\rho##. The density ##\rho## is assumed to have no uncertainty.

I also have a small set of relative errors ##\epsilon_i## with ##n_\epsilon## samples that describe the deviation between masses ##\tilde{m}_i## obtained with the applied measurement method (same system to determine the volume and the same constant density) and the masses ##\hat{m}_i## obtained with an assumingly more reliable reference measurement method. The relative error ##\epsilon_i## is computed by $$\epsilon_i = \frac{\tilde{m}_i - \hat{m}_i}{\hat{m}_i}.$$

All uncertainties are assumed to be normally distributed.

Approach

My approach to compute the combined standard measurement uncertainty ##u_c(\bar{m})## is $$u_c(\bar{m}) = \sqrt{\frac{\rho^2 \, u(V)^2 + u(m)^2}{n_V}}$$

with

$$u(V) = \sigma_V,$$

$$u(m) = \frac{1}{n_\epsilon - 1} \sum_{i = 0}^{n_\epsilon} \; (\epsilon_i \, \bar{m} - 0)^2.$$

Here, I assume that the error of the measurement method is zero-mean and I use the computed mean mass ##\bar{m}## to convert the given relative errors ##\epsilon_i## with unknown reference masses ##\hat{m}_i## into an absolute error.

Is this approach reasonable?

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# I Combined measurement uncertainty for mass computation

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