Uncertainties in a set of measurements

[KDPhysics]

Homework Statement

Error analysis for standard introductory physics experiment.

Relevant Equations

Arithmetic mean, standard deviation, half range.

Suppose I measure the distance between two objects for three trials. The two objects then get farther away, and I measure the distance between them again for three trials. I repeat this for 3 more different distances, getting a total of 15 measurements (3 trials for 5 distances).

I then compute the average and the half-range (approximates the uncertainty) for each distance. For example, the first distance will have average 3.3m and uncertainty 0.15m . In this, case, since the uncertainty is fairly small, I have been told to keep the two significant figures and simply make the number of digits after the comma in the average to coincide with the number of digits after the comma of the uncertainty. So, i would write 3.30±0.15. That is clear to me.

The problem is, what if the second distance is 5.3±0.7. In this case, I only keep one significant figure for the uncertainty. But then, the measurements will have a different number of significant figures. Is this ok? Or should I report the first measurement as 3.3±0.2?

Also, to compute the uncertainty, should I use the half range or the standard deviation (i'm in high school)?

 

[Gaussian97]

Well, I'm not an expert on how to treat uncertainties, but in the lab courses I've been, there seem to be different conventions on how to express them. First of all, I think that uncertainty is always computed using the sample standard deviation with 1 degree of freedom:
$$\sigma_{\bar{x}}=\frac{1}{\sqrt{N^2-N}}\sqrt{\sum_{i=1}^N(x_i-\bar{x})^2}$$ with $$\bar{x}=\frac{1}{N}\sum_{i=1}^Nx_i$$.

To then express the results I've seen different conventions, I usually give my results with the uncertainty having 2 significant digits, but in some labs, they prefer to give 2 significant digits if the first digit is 1 or 2, but only one significant digit if it's 3 or more.
I think that the reason for that is Benford's law that says that almost 50% of the time you uncertainty will start by 1 or 2, but I don't know very much.

I don't know if this helps you very much.

 

[KDPhysics]

So even in high school I should use the standard deviation?
Also, is it ok if my measurements have different significant figures e.g. 5.3±0.7 and 3.30±0.15 in the previous post?

 

[Gaussian97]

So even in high school I should use the standard deviation?
Also, is it ok if my measurements have different significant figures e.g. 5.3±0.7 and 3.30±0.15 in the previous post?

No, I don't think there's any problem, and nothing of this depends on whether you are in High School or not. As long as you are clear and consistent with your method (and you explain it properly) it should be okay. Another issue would be that your professor wants it in another way, then of course you should always adapt your conventions to those imposed to you.

 

[KDPhysics]

alright, thank you!

 

[haruspex]

if the second distance is 5.3±0.7

Your problem arises from the two digit precision (0.15) quoted for the error in the first distance. That is an artefact of the division by 2.
Now, that halving does represent an increase in precision, just not by an entire factor of 10. Using the standard deviation methods you would reduce the uncertainty by a factor $\sqrt{N-1}$, so in this case $\sqrt 2$. That's not enough to be adding a precision digit, so round the .15 up to .2.