I am hopelessly confused about the very basic question of how to propagate error if you want to take into consideration the error that each data point already has. For example, say I have time measurements in seconds of 30+/-1 s, 32+/-5 s, and 30+/-2 s, and I find a final answer for time by averaging the three values of 30, 32 and 30. Now, I want to calculate the uncertainty of time, how would I factor in the +/-1 s, +/-5 s, and +/-2 seconds?
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$\begingroup$ You want the standard error on the mean? Can you explain what you did for $\bar t$, and any guesses at, say, the standard deviation (which is not the standard error)? $\endgroup$– JEBCommented Apr 24, 2020 at 15:43
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1 Answer
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The standard error of measurements, $t_i$, is estimated by:
$$ SEM = \frac{\sigma}{\sqrt N}$$
where $\sigma$ is the standard deviation, and $N$ is the number of measurements.
The only nuance here is that each $t_i$ come with a weight:
$ w_i = \frac{1}{\sigma^2_i}$
So you need to compute the weighted standard deviation, via:
$$ \sigma^2 = \langle t_i^2\rangle - \langle t_i\rangle^2$$
where $\langle f(t_i) \rangle$ is the weighted average of $f(t_i)$.
Note that you do not have 3 degrees-of-freedom, because they are not all of the same significance. You need to compute the effective degrees of freedom, which is given by:
$$ N = \langle 1 \rangle \approx 1.56 $$
