What's the error in 1 repeated measurement?

[henry wang]

Homework Statement

I have only repeated a measurement once, I cannot assume it is distributed as a Gaussian because there is so few data. How can I estimate its combined uncertainty?

The Attempt at a Solution

Total data: x1, x2
I calculated the individual uncertainties in x1 and x2 using error propagation equation and found that they are essentially the same. Thus I used $$\Delta \bar{x}=\frac{\Delta x}{\sqrt{N}}$$ Where N is the number of repeats, which is 2.

 

[Simon Bridge]

The trick is to realize you are not calculating an exact uncertainty, you are estimating it.
The estimator for uncertainty depends on how the measurement was taken.

It is usually reasonable to assume gaussian errors unless you have reason to do otherwise simply because central limit theorem.
Most measurement errors are approximately to gaussian even if they are not strictly gaussian - and the uncertainty on the uncertainty is typically large.

The trouble, as you have realized, is that you cannot be sure of the uncertainty without lots of data points.
The best you can do is gamble.

I don't see how you found the individual errors though.
It is common to estimate errors on individual measurements by using the resolution of the instrument... this assumes that the instrument resolution is about the same or bigger than other contributions. It may be important to check that this is likely - ie. if you used a stopwatch, you can try timing other stuff to see how the errors are distributed.

 

[henry wang]

The trick is to realize you are not calculating an exact uncertainty, you are estimating it.
The estimator for uncertainty depends on how the measurement was taken.

It is usually reasonable to assume gaussian errors unless you have reason to do otherwise simply because central limit theorem.
Most measurement errors are approximately to gaussian even if they are not strictly gaussian - and the uncertainty on the uncertainty is typically large.

The trouble, as you have realized, is that you cannot be sure of the uncertainty without lots of data points.
The best you can do is gamble.

I don't see how you found the individual errors though.
It is common to estimate errors on individual measurements by using the resolution of the instrument... this assumes that the instrument resolution is about the same or bigger than other contributions. It may be important to check that this is likely - ie. if you used a stopwatch, you can try timing other stuff to see how the errors are distributed.

The measured varieble was used to calculate another quantity, the uncertainty calculated is really the uncertainty of the calculated quantity. Thank you for your help.

 

[Simon Bridge]

The tldr answer is "it depends" - the uncertainty estimator that is best used depends on the specifics of the situation.
It would help us advise you properly if we knew what the situation was and what you know about the likely physics being followed.

 

[henry wang]

The tldr answer is "it depends" - the uncertainty estimator that is best used depends on the specifics of the situation.
It would help us advise you properly if we knew what the situation was and what you know about the likely physics being followed.

I realized that the two measurements are not repeats since the variables were changed slightly due to human error, at the end I simply took the average of their uncertainties. Thank you very much for your help!