Let's take a concrete example. One is dealing with a wire through which a DC is running. One measures its voltage and obtain its current by measuring the magnetic field around it in some way such that the standard deviation of the current is $\sigma_\text{I, raw}=0.179222... A$. According to several sources, this means that one should report the error/uncertainty of the current as $\sigma_\text{I, rounded} = 0.18 A$. For the voltage, it is easier, one considers $\sigma_V = 0.001V$ which is the precision of the apparatus (if the reading doesn't change with time while doing the experiment, that is). However let's say one is interested to calculate the resistance of the wire via Ohm's law $R=\frac{V}{I}$. One has that the standard deviation of the resistance is worth $\sigma_R$ is worth $R\sqrt{\left ( \frac{\sigma_I}{I} \right )^2 + \left ( \frac{\sigma_V}{V} \right )^2}$. My question is, which value of $\sigma_I$ should one use? I have been taught to use $\sigma_\text{I, rounded}$ but I feel this is not correct, because the rounding is arbitrary and therefore biased. An apparently very popular (more than 800 citations according to Google Scholar) source that also indicates to use rounded values for calculations is https://eurachem.org/images/stories/Guides/pdf/QUAM2012_P1.pdf (page 39 for instance), I feel this is not quite correct. In a real case example, this can cause $\sigma_R$ to deviate by a very large percentage (more than 50%*) than if one had used $\sigma_\text{I, raw}$. I am tempted to think that it is only when reporting the value of the standard deviation that one should round off, not when performing calculations involving said standard deviation. Am I correct? If I am correct, this mean that, as a rule of thumb, one should not perform computations of propagation of uncertainties/errors with rounded values of standard deviations and rounded mean values. It is only at the very end, when presenting/reporting the result, that one should round the mean and standard deviation of the mean. And one should not take these rounded values as a starting point for further calculations.
- Take for example $\sigma_\text{I, raw}=0.1550$, in that case $\sigma_\text{I, rounded}=0.2$, the values differ by more than 20% and this will impact greatly $\sigma_R$.