During an experiment I had to measure the speed of light $c$ (it's an indirect measurement, because i had $c(x,y,z)$ as a function of some quantities $x,y,z$ I could measure). I took several measures $c_1,c_2, \dots , c_n$ and I calculated the mean value $c_{mean}$.

Now i would like to find the error associated with $c_{mean}$.

If each quantity $x,y,z$ is affected by an error $\sigma_x,\sigma_y,\sigma_z$ , then each measurement $c_i$ is affected by and error $$\sigma_{c_i}=\sqrt{\biggl( \frac{\partial c}{\partial x} \biggr) ^2 \cdot \sigma_x ^2 \ + \ \biggl( \frac{\partial c}{\partial y} \biggr) ^2 \cdot \sigma_y ^2 \ + \ \biggl( \frac{\partial c}{\partial z} \biggr) ^2 \cdot \sigma_{z} ^2 }$$ that i can calculate.

So now I have $n$ different errors $\sigma_{c_i}$, how can I use them to calculate the error associated with the $c_{mean}$ ?


So you have:

$$ c_1, c_2, ..., c_n $$

From that, compute an observed standard deviation, $\sigma_c$. The standard error of the mean is then:

$$ \sigma_{\bar c} = \sigma_c / \sqrt n $$

Note that if the observed standard deviation differs significantly from the expectation, as computed from the observed $\sigma_x$, $\sigma_y$, $\sigma_z$, then the measurements of $x$, $y$, and $z$ are correlated.


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