# Uncertainties with GUM for multiple measurements

I work with the GUM method to determine measurement uncertainties. As I understand the method, the model created for a particular measurement setup provides an uncertainty for a single measurement.

But normally I repeat such a measurement several times, e.g. I measure the weight of a sample five times and then average the values. With the GUM method, I have an uncertainty for each value:

$$m_1 \pm u_{m,1}$$

$$m_2 \pm u_{m,2}$$

$$m_3 \pm u_{m,3}$$

$$m_4 \pm u_{m,4}$$

$$m_5 \pm u_{m,5}$$

Calculating the mean value is trivial, but how can I take into account the five uncertainties associated with GUM? Here's what I'd like:

$$\bar{m} \pm u_{\bar{m}}$$

I know there are similar questions, but as far as I can see, none in the GUM context.

One of my thoughts is that I simply include random scattering as a normally distributed uncertainty in the budget. However, I could only create the budget after I have taken the measurements. That would also be a kind of cicle conclusion.

Thanks!