# Why is the uncertainty of a digital instrument always the smallest scale division?

For a mass balance that can give readings to 2 decimal places, the uncertainty is assumed to be $\pm 0.01$. Can someone explain the rationale for this, as I would think it more logical for the uncertainty to be given as $\pm 0.005$.

• Well, the value is recorded as a binary value, so back computing the accuracy is not trivial. Presumably $\pm 0.01$ describes the actual faith you should have in the value. No one said it will be correct, but they claim it won't be off by more than 0.01 units.
– Zhe
Dec 20 '16 at 13:23
• If I'm following correctly, you're assuming it's like a safety net of sorts? As in the uncertainty is unlikely to ever be as large as +/- 0.01, but it's written that way anyway just in case it's larger than +/- 0.005? Dec 20 '16 at 13:29
• The uncertainty is in some binary digit. Suppose it's $\pm 0.0001_{2}$. That translates to $\pm 0.0625$. But you might round that up to $\pm 0.1$ if your scale only has 1 place after the decimal point. Maybe I should just write this up as an answer.
– Zhe
Dec 20 '16 at 13:30
• Whatever the details of the measurement as a binary number translated to decimal, the only sure way to know what the manufacturer of your scales intends by an error of $\pm 001$ is to look at the specifications. If you work in a certified lab the scales should adhere to some standard such as ISO9000. If not, all you can assume is that the error is literally what you mention. You could always check by repeated measurements of the same object. Also, you could try measuring in groups as this will reduce the noise introduced by the scales and see if this has any effect. Dec 21 '16 at 16:03

Then you're in a scenario where the uncertainty is something like $\pm 0.0001_{2}$, which is $\pm 0.0625$. In this case, I would say that the error is in the first decimal place after the decimal point.
You can't give an error of $\pm 0.005$, partly for the reason I gave, but also because the error in measurement is not necessary an error in rounding. It's possible that the transducer is temperature sensitive, so within a reasonable range, it might over or under measure by $0.01$ units.