# How do uncertainties in digital measurements work?

Suppose I stand on an electronic weighing scale, and it reports my weight as 74.5 kilos.

Usually a lot of books would say that the last digit is uncertain, and that the uncertainty is 0.1kg, and that my 'true' weight is between 74.4 and 74.6 kilos. But that kinda bothers me, because how do we know? Isn't the uncertainty depend on the type of equipment used, or various other factors?

Is the 0.1 of a unit some kind of standard when designing digital equipment? And, by the way, how do these equipment actually report the data? And how would the equipment 'guess' the last digit? How would the equipment 'decide' what to report as the final digit?

If let's say my true weight is precisely 74.5200000...., and obviously the equipment can only detect the smallest increment of 0.1kg, then how would the weighing scale decide the last digit? It would make more sense to me if the scale reports 74.5, because the extra bit (0.020000...) wouldn't be detectable by the scale? But thinking this way, the scale would've reported 74.5 even if my true weight is something like 74.5600000.... (even though it is now closer to 75.6, but the machine can't tell because it still cannot detect the extra 0.06000.... bit)

Thus taking these considerations into consideration, I don't really get the rationale behind interpreting the data as 74.5 +- 0.1, because if the machine works the way I've just described, then wouldn't it be better to just say that the true value lies between 74.5 and 74.6? Unless, of course, the machine doesn't work that way at all, which once again begs the same question: how does the equipment work?

Thanks.

The basics on error analysis on measure is about rounding.

If you design an equipment to show a certain number of digit (say to the $10^{-1}$ of the unit), then in order to be certain of your analysis, you have to know how the plot is made with regard to rounding. In your example, a scale showing $74.5\ \mathrm{kg}$ for a weight of $74.52$ and $74.56$ is simply dropping the last digit. I could design a scale rounding to the closest value and in this case I would see a value of $74.5$ in the first case and $74.6$ in the second.

But what if I don't know anything about the scale or how the values I've got have been rounded by the people doing some measurement? Taking my two scales, a value of $74.5$ can be obtain by the first scale measuring a object with a mass equal to $74.52$ or with the second scale with an object of mass equal to $74.48$. How do I know?

Error analysis is just a way to know where the true result might be with regards to the values we've got and their precision. There is several ways to round the results into a readable output in a given set of units but without any information on the precision of the measurements, $74.5\pm 0.1\ \mathrm{kg}$ is the best guess you can make on the error bar.
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