# When to round off standard deviations in uncertainty propagation?

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 $\sigma_R$.

When the first digit is small, especially "$1$", it's good practice to report the uncertainty with two digits, exactly for the reason you point out (greater impact of rounding).

It's always important to keep in mind that, although the statistics are hard math, their application to uncertainty estimation in science is in part an art, as it's highly dependent on the specificities of the observable and how it's being measured. That's why there are so many conventions (like the arbitrary 5$\sigma$ for discoveries in high energy physics), suggestions, good practices, and so on.

Thus, as long as you're sure on the math, you're free (although it's still generally a good idea to follow the practices of your field) to apply some common sense.

So, finally: you should round off the uncertainties in the end of the calculation. Otherwise you can only loose precision and, worse, accuracy, due to possible systematic bias.

This much has already been said here:

"Just don't believe the additional digits - but there is no reason to drop them."

And it's also the recommendation at the Science Learning Center at the University of Michigan-Dearborn web page:

"Do not round off until the end of the problem. Carry at least one extra significant figure."

And the recommendation too of Duane Deardorff (director of the physics undergrad labs at the University of North Carolina)

If a calculated number is to be used in further calculations, it is good practice to keep one extra digit to reduce rounding errors that may accumulate. Then the final answer should be rounded according to the above guidelines.

• Fine. But when should I do the round off? In the very end or for instance mid-way through the calculations (as suggested by some sources)? – thermomagnetic condensed boson Nov 12 '17 at 17:44
• I'd say at the very end. A common first approximation for getting the uncertainty of the final value is to consider its variation under that of the observables it depends on. By the way, you did check the Wikipedia page on Propagation of uncertainty, right? – stafusa Nov 12 '17 at 17:54
• Yes I did check out the Wikipedia's example from which I based on my own example. Ok, so you have the same opinion than I have, but this goes against what seems to be common practice. I gave a very cited link that clearly use rounded values mid-way through the calculations to reach the uncertainty, introducing a bias in the process. I'd appreciate if you could edit your answer and state clearly that one should do the round off only at the very end (explaining why) and that there are possibly a lot of bad practice going around. This would answer my question. – thermomagnetic condensed boson Nov 12 '17 at 17:59
• I am on the phone right now. I'll try to take a look into that and hopefully find some source later on. – stafusa Nov 12 '17 at 18:01
• Ok great! Take your time, I will not accept any answer before you have the time to fully improve yours. – thermomagnetic condensed boson Nov 12 '17 at 18:05