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?

  • $\begingroup$ 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)? $\endgroup$
    – JEB
    Apr 24, 2020 at 15:43

1 Answer 1


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:

$$ N = \langle 1 \rangle \approx 1.56 $$


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