I Instrumental error and statistical error

[gnegnegne]

In a lab experiment we had to measure some angles. Every angle measure is the difference between two angular positions and the instrument we used had a resolution of 1', so the uncertainty due to the instrument is $$\sigma_{instr}=\sqrt2'=0.02357... deg$$.
We measured the same angle a few times and we obtained values such as $$\theta_1=60.03 deg, \theta_2=60.00 deg, \theta_3=59.97 deg$$
The mean is 60 deg and the standard deviation of the mean is $$\sigma_{stat}=0.0173... deg$$.

What should I consider as the final uncertainty of the measure? The highest uncertainty? Or the sum of squares? I would consider the sum of squares because the uncertainties are independent, but our professor told us that the highest includes the other, and I haven't really understood why.

 

[BvU]

Hello gne³,

Suppose you had three observations of something, namely: -1, 0 and 1, and the extra bit of information is that the resolution of the measuring device was 0.5 . What would you report as your result ?

There are a lot of comments to be made on this:

• basically the resolution of an instrument is something quite different from its accuracy
• the statistics of error estimation tell you the estimated error in the error estimate is enormous for just a few observations
• The values you obtained were NOT like 60.03 deg but like $60^\circ 2'$ etcetera. The 60.03 is a calculated value.
• ...

 

[gnegnegne]

Hi, thank you very much!
Well, the mean is 0 and the standard deviation of the mean is 0.5773... (devstd/sqrt(3)) . I would consider as the total uncertainty $$\sqrt{0.5773^2+0.5^2}=0.76376...$$ Therefore as my result I would report $$0.0 \pm 0.8$$. However I'm not sure whether the standard deviation and the resolution should be summed.

In some situations it's pretty obvious: when I use a chronometer the instrument uncertainty is irrelevant. Vice versa when I use a ruler in a standard situation I will always measure the same quantity, the standard deviation would be 0 (but that doesn't mean that my measure has no uncertainty, I'd use the instrument uncertainty).
My situation however is in the middle: the order of magnitude of the SDOM and the instrument resolution are comparable.

 

[BvU]

A (cheap) $3\;\scriptsize {1/2}$ digital multimeter may have a resolution of 1 mV but the accuracy can be only 1%, so that 1.302 V becomes 1.302 $\pm$ 0.013 V, or (imho better): 1.30 $\pm$ 0.01 V.
(see what http://www.d.umn.edu/~snorr/ece2006f7/Lab1.pdf write about the error reporting)

Your angle measurement device may be analog with a vernier scale -- I wondered why your measurements were even minutes --
and then the 1' is rather optimistic

Your estimate (0.017 deg) of the standard deviation carries a relative uncertainty of $1/\sqrt 3$ (or worse), so you should report $60.00 \pm 0.02$ degrees.

And we are back to your original question: is the resolution error included in the estimate of the standard deviation as calculated from the sample of observations. I'd be inclined to say yes (with some reservations -- depending on the type of measurement, the equipment, the noise, the circumstances, ...): the resolution is a random contribution to the scatter in the observations.

 

[Dale]

I would consider as the total uncertainty $$\sqrt{0.5773^2+0.5^2}=0.76376...$$ Therefore as my result I would report $$0.0 \pm 0.8$$. However I'm not sure whether the standard deviation and the resolution should be summed.

This is the correct procedure. The definitive reference for error handling is the NIST paper:

https://www.nist.gov/sites/default/files/documents/2017/05/09/tn1297s.pdf

 

[gnegnegne]

Thanks for both the answers! Yes, we used a spectrometer with a nonius to measure the angles. I understand your explanation, however my problem isn't the estimation of the instrument error, which of course it's pretty rough. The problem is what do I have to do once I have a statistical standard deviation and an uncertainty due to the instrument, and the NIST paper seems clear to me: you have to consider both the "statistical uncertainty" and the "instrument uncertainty", summing their squares. Thanks for your help!

 

[Dale]

and the NIST paper seems clear to me: you have to consider both the "statistical uncertainty" and the "instrument uncertainty", summing their squares.

Yes, and they give some good hints about how to figure out the non statistical uncertainty also.