# Calculating uncertainty using a set of measured values

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?

• 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)?
– JEB
Apr 24, 2020 at 15:43

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:

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