Calculating uncertainty of a measurement

Question

When measuring some quantity we express it with a value followed by an uncertainty:

For example: The answers of a previous task I completed. $$0.250±0.001$$ $$0.75±0.1$$ $$1.20±0.02$$ $$68.73±0.03$$ $$0.770±0.006$$ $$153±1$$

I'm trying to understand how the value and uncertainty are rounded. From my observation, I've noticed that the uncertainty is always rounded to 1sf. Furthermore, I've also noticed that the value is always rounded to the same decimal place as the uncertainty. Are these two observations true in general?

I'm also trying to understand why my second observation is the case. I think if the value contained more decimal places than the uncertainty, it wouldn't really mean anything. In a sense the uncertainty would kind of render the additional decimal places useless. Could this be correct? Is there a nicer intuition for why this is the case or a refined explanation?

Answer 1

You are correct in all accounts and this is usually taught in lab intros. The intuition you've presented is spot on. A lot of the time you'll also see uncertainties with two sigfigs if the uncertainty's most significant figure is 1 or 2, as this leaves less wiggle room for the uncertain digits at the end of the value (e.g. $1.56\pm0.13$ instead of $1.6\pm0.1$).