# Propagation of uncertainty for a mean value

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 $$c_{mean}$$ ?

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