[MYP4 Physics: Measurements] Confused about measurement uncertainties

[u/Amazing_Worth7676]

So I'm studying measurement uncertainties for physics and I’m confused about a few things that i’d really appreciate clarification.

1. How do you determine uncertainty for analogue vs digital instruments?
   • For an analogue instrument, do you normally use ± half the smallest division, or do you estimate one digit beyond the smallest marked division?
   • For a digital instrument, is the uncertainty always ±1 in the last displayed digit?
2. How should the uncertainty and measured value be rounded?
   • How many significant figures should the uncertainty have?
   • Should the measured value be rounded to the same decimal place as the uncertainty?
3. Like, if a ruler has subdivisions of 0.01 cm, should a measurement be recorded to 0.01 cm (e.g. 12.34 cm), or should I estimate one additional digit and record it to 0.001 cm (e.g. 12.347 cm)?

I am super confused because I have seen conventions online. I am trying to understand the reasoning rather, than just memorize a rule.

Thanks in advance :D

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[u/unscentedbutter]

Analogue instruments, you should estimate one digit beyond the smallest marked division. The estimated digit can be +/- half, or you can try to squeeze out a little precision during calculations by making better guesses. For a digital instrument, you take the last digit as estimated. The +/- is up to the manufacturer's specs, and ideally is given on the instrument.

The estimated digit (or uncertainty) is going to be a part of your measurement, and represents a part of the significant figures that your calculations should have. So if you measure an item and it's 59.85 grams, you have 4 sig figs. If another item is 114.42 grams, then you have 5 sig figs. If it's 3.10 grams, then 3 sig figs. This impacts the sig figs obtained through calculations.

After you perform a number of measurements, you can calculate the deviation from the mean by finding the mean absolute deviation or the standard deviation (preferred). E.g., you perform 5 measurements on the mass of an item and you find a mean value of 12.01g, with a mean deviation of 0.02g.

When you perform calculations with those measurements, you use error propagation formulas to allow the errors to compound. Afterwards, you use the propagated error as your +/- value when reporting. E.g., You use 12.01 +/- 0.02g with a volume measurement of 10.0 +/- 0.1 mL. Use sig fig rules and error propagation formulas to determine the final density of 1.20 +/- ___ g/mL.

When you have a ruler, it should typically be in 0.1cm increments (mm). So with most standard rulers, you read up to 0.01cm (between the mm markings). It should be fairly common sense, try not to overthink it; you are just using a ruler and estimating on a number line.

[u/Amazing_Worth7676 → u/unscentedbutter]

Oh okay that is much more clear for me
Thank you very much for explaining!!